Download Continuity and Separation Axioms Based on βc

The Islamic University of Gaza
Deanery of Higher Studies
Faculty of Science
Department of Mathematics
Continuity and Separation Axioms Based
on βc-Open Sets
by
Ayman Y. Mizyed
Supervisor
Dr. Ahmad A. El-Mabhouh
A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENT
FOR THE DEGREE OF MASTER OF MATHEMATICS OF COLLEGE OF SCIENCE
AT THE ISLAMIC UNIVERSITY OF GAZA, PALESTINE
.
2015 - 1436
i
Contents
Contents
i
Acknowledgements
iii
Abstract
v
Introduction
vi
1 Preliminaries
1
1.1
Topological Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1
1.2
Near Open Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6
2 βc-Open Sets
2.1
Definition and Characterizations
10
. . . . . . . . . . . . . . . . . . . . . . . 10
2.2
The Family βCO(X) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
2.3
βc-Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
2.4
The Topology Generated by βc-Open Sets . . . . . . . . . . . . . . . . . . 35
3 βc-Continuous Functions
41
3.1
Definitions and Characterizations . . . . . . . . . . . . . . . . . . . . . . . 41
3.2
Properties of βc-Continuous Functions . . . . . . . . . . . . . . . . . . . . 49
ii
4 βc-Separation Axioms
55
4.1
βc-Generalized Closed Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
4.2
New Types of Separation Axioms . . . . . . . . . . . . . . . . . . . . . . . 59
Conclusion
63
Bibliography
64
iii
Acknowledgements
On the very outset of this work, I would like to thank Allah for giving me power to complete this work.
I am ineffably indebted to my supervisor Dr. Ahmad A. El-mabhouh for his constructive suggestions and to the point guidance. I also extend my gratitude to every member
in the department of Mathematics for their endless support through out my study.
I also express my gratitude to my reverend parents and members of my family who have
always supported me morally. I really extend my sincere and heartfelt obligation towards
all the people and especially all my friends who have helped me in this endeavor. Without
their active guidance, help, cooperation and encouragement, I would not have made this
work come true.
iv
Abstract
In this thesis, we study a new class of sets called βc-open sets which is contained in the
class of β-open sets and contains the class of bc-open sets. We prove that the topology
generated by the family of βc-open sets in a natural way is the same as the topology of
θ-open sets in extremally disconnected Alexandroff spaces. Moreover, we introduce βccontinuous functions as a new class of continuous functions and give a characterization of
these functions. Finally, we define βc-g.closed sets and investigate new separation axioms
βc-Tk , k ∈ {0, 12 , 1, 2}.
v
Introduction
The most important tools in general topology are the separation axioms and the continuity between topological spaces. Several authors have studied so many types of continuous
functions in topological spaces by depending on different kinds of sets which they defined
and introduced. The second more important tool in general topology is the notion of separation axioms and the most familiar axioms T0 , T1 and T2 . By rewriting these concepts,
using different types of sets, a large number of these spaces were defined and studied.
The notions of semi-open sets, α-open sets, pre-open sets, semi-preopen sets or β-open
sets have been introduced and studied respectively by Levine(1963), Njastad(1965), Mashhour(1982) and Andrijevic(1986). Andrijevic(1996) gave a new class of generalized open
sets which contained semi-open sets and pre-open sets but smaller than semi-pre-open
sets call it b-open sets. In last five years, new generalized open sets appeared such as
sc-open ( semi-open and union of closed sets), pc-open ( pre-open and union of closed
sets) and bc-open sets ( b-open and union of closed sets) which due to Khalaf [1], Ameen
[25] and Hairwan [7] respectively.
The aim of this thesis is to introduce a new type of β-open sets named βc-open sets and
establish the related most important tools in general topology, continuity between topological spaces and separation axioms.
This thesis comprises four chapters. In chapter one, we give preliminaries that will be
vi
used in the remainder of this thesis and it consists of two sections. The first section deals
with basic definitions and theorems of topological spaces. The second section is devoted
to the main definitions and results of generalized open sets.
Chapter two studies βc-open sets that satisfies β-open sets which is a union of closed
sets. It includes four sections. The first section we define βc-open sets and compare it
with other generalized open sets. In the second section, we concern when the family of
βc-open sets forms the topology and particularly when it forms the discrete and indiscrete topology. In additions, we investigate the relation among the family of βc-open
sets and other families of sets. In the third section, we study the concepts of βc-closure,
βc-neighborhood, βc-interior, βc-limit points, βc-derived set and βc-frontier of a set and
some of their properties that are analogous to those of open sets. In the fourth section, we
show that the family of βc-open sets generates in the natural way the same topology as
the topology τθ which consists of all θ-open sets as members in extremally disconnected
Alexandroff spaces.
Chapter three talks about a new class of continuous functions based on βc-open sets
called βc-continuous functions. This chapter falls into two sections. The first section,
we introduce βc-continuous functions and give some characterizations of βc-continuity.
Besides, we compare this function with some types of continuity. In the second section,
we give some properties of βc-continuity and study some conditions for composition of
two functions to be βc-continuous.
Chapter four focuses on βc.g-closed and βc-separation axioms. It contains two sections.
In the first section, some properties of βc.g-closed sets are defined and studied. The
second section contains the definitions and investigations of βc-T0 , βc-T 1 , βc-T1 and βc2
T2 separation axioms in topological spaces and study their fundamental properties and
relationships to other types of spaces.
vii
Chapter 1
Preliminaries
1.1
Topological Spaces
In this section, we give the basic definitions and theorems for topological spaces that will
be used in the remainder of the thesis. For more details see [18] and [22].
Definition 1.1.1. A topology on a set X is a collection τ of subsets of X, called the open
sets satisfying:
(1) Any union of elements of τ belongs to τ ,
(2) Any finite intersection of elements of τ belongs to τ ,
(3) ∅ and X belong to τ .
Remark 1.1.2. We say (X, τ ) is a topological space, sometimes abbreviated ” X is a
topological space ” when no confusion can be result on τ .
Definition 1.1.3. Let (X, τ ) be a topological space. A set A ⊆ X is closed iff X\A is
an open set of X.
1
Definition 1.1.4. If X is a topological space and E ⊆ X, the closure of E in X is the
set
cl(E) =
T
{K ⊆ X|K is closed and E ⊆ K}.
Theorem 1.1.5. Let A and B be subsets of the space X. Then,
(a) cl(∅) = ∅.
(b) cl(A ∪ B) = cl(A) ∪ cl(B).
(c) cl(cl(A)) = cl(A).
(d) If A ⊆ B, then cl(A) ⊆ cl(B).
Definition 1.1.6. If X is a topological space and E ⊆ X, the interior of E in X is the
set
int(E) =
S
{G ⊆ X|G is open and G ⊆ E}.
Theorem 1.1.7. Let A and B be subsets of the space X. Then,
(a) int(A) ⊆ A.
(b) int(int(A)) = int(A).
(c) int(A ∩ B) = int(A) ∩ int(B).
(d) int(X) = X.
(e) A is open if and only if int(A) = A.
(f ) If A ⊆ B, then int(A) ⊆ int(B).
Definition 1.1.8. If X is a topological space and E ⊆ X, the frontier of E in X is the
set
F r(E) = cl(E) ∩ cl(X\E).
Theorem 1.1.9. For any subset E of a topological space X:
(a) cl(E) = E ∪ F r(E).
2
(b) int(E) = E\F r(E).
(c) X = int(E) ∪ F r(E) ∪ int(X\E).
Definition 1.1.10. If X is a topological space and x ∈ X, a neighborhood of x is a set
U which contains an open set V containing x.
Definition 1.1.11. An accumulation point ( cluster point ) of a set A in a topological
space X is a point x ∈ X such that each neighborhood of x contains some points of A,
other than x. The set D(A) of all cluster points of A is called the derive set of A.
Definition 1.1.12. Let (X, τ ) be a space. A base for τ is a collection B of subsets of X
such that:
(a) Each member of B is also a member of τ and
(b) If U ∈ τ and U 6= ∅, then U is a union of sets belonging to B.
0
Definition 1.1.13. If (X, τ ) is a topological space and A ⊆ X, the collection τ =
{G ∩ A|G ∈ τ } is a topology for A, called the relative topology for A. The fact that a
subset of X is being given this topology is signified by referring as a subspace of X.
Theorem 1.1.14. If A is a subspace of a topological space X, then:
(a) H ⊆ A is open in A iff H = G ∩ A, where G is open in X,
(b) F ⊆ A is closed in A iff F = K ∩ A, where K is closed in X,
(c) If E ⊆ A, then clA (E) = A ∩ clX (E).
Corollary 1.1.15. Let (Y, τY ) be a subspace of (X, τ ). If A is closed in X and A ⊆ Y ,
then A is closed in Y .
Proposition 1.1.16. A closed subset of a closed subspace is closed.
Theorem 1.1.17. Let (X1 , τ1 ), (X2 , τ2 ),..., (Xn , τn ) be a finite collection of topological
spaces, and let X = X1 × X2 × .... × Xn . If B is the collection of all sets in X of the form
3
U1 × U2 × ... × Un , where Uk is an open set in Xk for k = 1, 2, ..., n, then B is the base for
a topology on X.
Definition 1.1.18. If the collection B described in Theorem 1.1.17, then the topology
τ (B) generated by B and having B as a base is called the product topology on X =
X1 × X2 × .... × Xn .
Theorem 1.1.19. Let (X1 , τ1 ) and (X2 , τ2 ) be spaces and consider the product topology
on X1 × X2 . Let A1 ⊆ X1 and A2 ⊆ X2 . Then, cl(A1 × A2 ) = cl(A1 ) × cl(A2 ).
Corollary 1.1.20. Let (X1 , τ1 ) and (X2 , τ2 ) be spaces and consider the product topology
on X1 × X2 . Let A1 ⊆ X1 and A2 ⊆ X2 . Then, if A1 is closed in X1 and A2 is closed in
X2 , then A1 × A2 is closed in the product space X1 × X2 .
Definition 1.1.21. Let X and Y be topological spaces and let f : X → Y . Then, f is
continuous at x0 ∈ X if and only if for each neighborhood V of f (x0 ) in Y , there is a
neighborhood U of x0 in X such that f (U ) ⊆ V . We say f is continuous on X if and only
if f is continuous at each x0 ∈ X.
Theorem 1.1.22. If X and Y are topological spaces and f : X → Y , then the following
are equivalent:
(a) f is continuous,
(b) For each open set H in Y , f −1 (H) is open in X,
(c) For each closed set K in Y , f −1 (K) is closed in X,
(d) For each E ⊆ X, f (cl(E)) ⊆ cl(f (E)).
Definition 1.1.23. If f : X → Y and A ⊆ X, we will use f |A ( the restriction of f to A
) to denote the map of A into Y defined by (f |A)(a) = f (a) for each a ∈ A.
4
Definition 1.1.24. Let f : X → Y be a given function from the space X to the space
Y . Then, f is open iff for each open set U ⊆ X, f (U ) is open in Y . The function f is
closed iff for each closed set A ⊆ X, f (A) is closed in Y .
Definition 1.1.25. Let f : X → Y be a bijective function from the space X to the space
Y . If f is open and continuous, then f is called homeomorphism. If f is a homeomorphism
from X to Y , then the spaces X and Y are said to be homeomorphic and denoted by
X∼
= Y.
Theorem 1.1.26. Let (X, τX ) and (Y, τY ) be topological spaces and let (a, b) be any point
in the product topological space X × Y . Then, the subspace X × {b} is homeomorphic to
X and the subspace {a} × Y is homeomorphic to Y .
Definition 1.1.27.
(1) A topological space X is a T0 space iff whenever x and y are
distinct in X, there is an open set containing one and not the other.
(2) A topological space X is a T1 space iff whenever x and y are distinct in X, there is
a neighborhood of each not containing the other.
(3) A topological space X is a T2 space (Hausdorff space) iff whenever x and y are distinct
in X, there are disjoint open sets U and V in X with x ∈ U and y ∈ V .
Definition 1.1.28. A space X is a Urysohn space if and only if whenever x 6= y in X,
there are neighborhoods U of x and V of y such that cl(U ) ∩ cl(V ) = ∅.
Remark 1.1.29. Every regular, T1 space is Urysohn and every Urysohn space is Hausdorff.
Theorem 1.1.30. The following are equivalent for a topological space X:
(a) X is T1 ,
(b) Each one point set in X is closed,
(c) Each subset of X is the intersection of the open sets containing it.
5
Theorem 1.1.31. The following are equivalent for a topological space X:
(a) X is Hausdorff,
(b) Limits in X are unique,
(c) The diagonal ∆ = {(x, x) : x ∈ X} is closed in X × X.
1.2
Near Open Sets
It is well-known that a large number of papers and theses is devoted to the study of
classes of subsets of topological spaces containing the class of open sets and possessing
properties more or less similar to those of open sets. Such sets are denoted as near open
sets or generalized open sets. Through this section, we give the required definitions and
results in generalized open sets which will used during this thesis.
Definition 1.2.1. Let X be a topological space. A subset A of X is called:
1. semi-open[14] if A ⊆ cl(int(A)).
2. pre-open[9] if A ⊆ int(cl(A)).
3. α-open[16] if A ⊆ int(cl(int(A))).
4. Semi-preopen[4] or β-open[13] if A ⊆ cl(int(cl(A))).
5. b-open[2] if A ⊆ cl(int(A)) ∪ int(cl(A)).
6. Regular-open[22] if A = int(cl(A)).
Definition 1.2.2. The complement of semi-open sets, α-open sets, pre-open sets, β-open
sets, b-open sets and regular open sets are called semi-closed sets[14], α-closed sets [16],
pre-closed sets [9], β-closed sets [13], b-closed[2] and regular closed sets [22] respectively.
Remark 1.2.3. The family of semi-open sets, α-open sets, pre-open sets, β-open sets,
b-open, regular open and regular closed sets are denoted by SO(X), αO(X), P O(X),
βO(X), BO(X), RO(X) and RC(X) respectively.
6
Definition 1.2.4. [1] A subset A of a space X is called sc-open if for each x ∈ A ∈ SO(X),
there exists a closed set F such that x ∈ F ⊆ A. The family of all sc-open subsets of a
topological space (X, τ ) is denoted by SCO(X).
Definition 1.2.5. [25] A preopen subset of a space X is called pc-open if for each x ∈ A,
there exists a closed set F such that x ∈ F ⊆ A. The family of all pc-open subsets of a
topological space (X, τ ) is denoted by P CO(X).
Definition 1.2.6. [6] A b-open subset of a space X is called bc-open if for each x ∈ A,
there exists a closed set F such that x ∈ F ⊆ A. The family of all bc-open subsets of a
topological space (X, τ ) is denoted by BCO(X).
Definition 1.2.7. [15] A set A of a topological space is called θ-open if for each x ∈ A
there exists an open set G such that, x ∈ G ⊆ cl(G) ⊆ A.
Definition 1.2.8. [15] A set A of a topological space is called δ-open if for each x ∈ A
there exists an open set G such that, x ∈ G ⊆ int(cl(G)) ⊆ A.
Remark 1.2.9. The collection of θ-open sets in a topological space (X, τ ) forms a topology
τθ which is coarser than τ . Also, the family of δ-open sets in a topological space (X, τ )
forms a topology τδ such that τδ ⊆ τ .
Theorem 1.2.10. [20] τ = τθ if and only if (X, τ ) is regular.
Definition 1.2.11. [12] A subset A of a space X is θ-semi-open if for each x ∈ A, there
exists a semi-open set G such that x ∈ G ⊆ cl(G) ⊆ A.
Definition 1.2.12. A topological space X is called:
(a). A locally indiscrete[9] if and only if every open set is closed.
(b). A regular space[22] if for each x ∈ X and for each open set G containing x, there
7
exist an open set H such that x ∈ H ⊆ cl(H) ⊆ G.
(c). An extremally disconnected[9] if the closure of any open set is open.
Definition 1.2.13. [17] A topological space X is called an Alexandroff space, if any
arbitrary intersection of open sets is open.
Remark 1.2.14. Note that a space is Alexandroff space if f arbitrary union of closed sets
is closed.
Lemma 1.2.15. [4] Arbitrary union of β-open sets is β-open set.
Lemma 1.2.16. [4] For every open set U in a topological space X and every A ⊆ X we
have, cl(A) ∩ U ⊆ cl(A ∩ U ).
Theorem 1.2.17. [4] If V is open and A is β-open, then V ∩ A is β-open.
Proposition 1.2.18. [5] If A ⊆ X0 ⊆ X and X0 ∈ βO(X). Then A ∈ βO(X) if and
only if A ∈ βO(X0 ).
Proposition 1.2.19. [5] Let X1 , X2 be topological spaces. If Ai is β-open set in Xi for
each i = 1, 2, then A1 × A2 is β-open set in the product space X1 × X2 .
Definition 1.2.20. A function f : X → Y is called:
(1) Semi-continuous[14] if the inverse image of each open subset of Y is semi-open in X.
(2) β-continuous[5] if the inverse image of each open subset of Y is β-open in X.
(3) Strongly θ-continuous[19] if the inverse image of each open subset of Y is θ-open in X.
(4) Contra continuous[11] if the inverse image of each open subset of Y is closed in X.
(5) Perfectly continuous[8] if the inverse image of each open subset of Y is clopen in X.
(6) RC-continuous[23] if the inverse image of each open subset of Y is regular closed in
X.
8
Theorem 1.2.21. [5] If f : X → Y is β-continuous function and X0 is an open set in
X, then the restriction f |X0 : X0 → Y is β-continuous function.
Theorem 1.2.22. [5] Let Xi and Yi be topological spaces and fi : Xi → Yi be β-continuous
functions for i = 1, 2. Then, the function f : X1 × X2 → Y1 × Y2 defined by putting
f ((x1 , x2 )) = (f (x1 ), f (x2 )) is β-continuous.
9
Chapter 2
βc-Open Sets
In this chapter, we study a new class of β-open sets called βc-open sets in topological spaces. In addition, we consider when the family of βc-open sets forms a topology
and particularly the discrete and indiscrete topology. Moreover, βc-closure, βc-interior,
βc-frontier, βc-limit points and βc-derived sets are studied. Finally, we introduce the
topology τβc , which is the topology generated in a natural way by the family of βc-open
sets.
2.1
Definition and Characterizations
The first section contains the introduction to the class βc-open sets which is a new class
of β-open sets. In addition, some properties of βc-open sets are given in this section.
Moreover, the relations among βc-open sets and other types of sets are investigated.
Definition 2.1.1. Let X be a topological space. A subset A of X is called βc-open set
if :
(a). A is β-open and
10
(b). for each x ∈ A there is a closed set F such that x ∈ F ⊆ A.
A subset B of X is βc-closed if and only if X\B is βc-open set. We denote to the
families of βc-open sets and βc-closed sets in a topological spaces (X, τ ) by βCO(X) and
βCC(X) respectively.
Proposition 2.1.2. A subset A in a topological space X is βc-open if and only if A is
β-open and it is a union of closed sets.
Proof. Let A be a βc-open. Then, A is β-open and for each x ∈ A there exists a closed
S
S
set Fx in X such that x ∈ Fx ⊆ A. This implies that x∈A Fx ⊆ A ⊆ x∈A Fx . Thus,
S
A = x∈A Fx where Fx is closed set for each x ∈ A. The converse is direct from the
definition of βc-open sets.
Remark 2.1.3. It is clear from the definitions that every bc-open is βc-open. The converse
is not true in general as shown in the following example.
Example 2.1.4. Consider the space R with usual topology. If A = (0, 1] ∩ Q, then
cl(int(cl(A))) = cl(int([0, 1])) = cl((0, 1)) = [0, 1] which contains A so that, A is β-open
and for each x ∈ A, {x} is closed set since the singletons are closed in R. So we have,
x ∈ {x} ⊆ A and A is βc-open. On the other hand, cl(int(A)) ∪ int(cl(A)) = cl(∅) ∪
int([0, 1]) = (0, 1) does not contain A. So A is not b-open and hence, it is not bc-open.
Remark 2.1.5. Every βc-open is β-open but the converse need not be true in general,
consider the following example.
Example 2.1.6. Let X = {a, b, c} with a topology τ = {∅, X, {a}, {b}, {a, b}}, then:
i. The family of closed sets is: {∅, X, {c}, {a, c}, {b, c}}.
ii. The family of β-open sets is: {∅, X, {a}, {b}, {a, b}, {a, c}, {b, c}}.
11
iii. The family of βc-open sets is: {∅, X, {a, c}, {b, c}}.
Note that {a} is β-open set but not βc-open set.
Remark 2.1.7. A βc-open set need not be a closed set as shown in the following example.
Example 2.1.8. Consider the set R of real numbers with usual topology and let A =
S
(0, 1). Then cl(int(cl(A))) = [0, 1] ⊇ (0, 1) = A and A = n∈N [ n1 , 1 − n1 ]. So A is
βc-open set but not a closed set.
Proposition 2.1.9. Arbitrary union of βc-open sets is βc-open set.
Proof. Let {Aα : α ∈ ∆} be a family of βc-open sets in a topological space X. Then Aα is
S
S
β-open set for each α ∈ ∆ and by Lemma 1.2.15, α∈∆ Aα is β-open set. If x ∈ α∈∆ Aα ,
then their exists γ ∈ ∆ such that x ∈ Aγ . Since Aγ βc-open set, there exists a closed set
S
S
F such that x ∈ F ⊆ Aγ ⊆ α∈∆ Aα . Hence, α∈∆ Aα is βc-open set.
Remark 2.1.10. The intersection of even two βc-open sets need not be open. Consider
the following example.
Example 2.1.11. In Example 2.1.6, A = {a, c} and B = {b, c} are βc-open sets but
A ∩ B = {c} is not βc-open set.
Corollary 2.1.12. Arbitrary intersection of βc-closed sets is βc-closed set.
Proof. Using Proposition 2.1.9 and De Morgan’s Law.
Proposition 2.1.13. A subset A in a topological space X is βc-open if and only if for
each x ∈ A, there exists βc-open set B such that x ∈ B ⊆ A.
Proof. Direct from Definition 2.1.1 and Proposition 2.1.9.
Proposition 2.1.14. Every regular closed set is βc-open set.
12
Proof. Let A be regular closed set, then A = cl(int(A)) but cl(int(A)) ⊆ cl(int(cl(A)))
and so, A is β-open. Since A is closed, by Definition 2.1.1, A is βc-open set.
Proposition 2.1.15. Let X be a T1 -space, then βCO(X) = βO(X).
Proof. Let X be a T1 -space and A ∈ βO(X). If A = ∅, then A ∈ βCO(X). If not,
then for each x ∈ A, {x} is closed. Hence, x ∈ {x} ⊆ A. Therefore, by Definition 2.1.1,
A ∈ βCO(X). By Remark 2.1.5, βCO(X) ⊆ βO(X). Thus, βCO(X) = βO(X).
Proposition 2.1.16. If the topological space X is locally indiscrete, then every semi-open
set is βc-open.
Proof. Let A be a semi-open. Then A ⊆ cl(int(A)) ⊆ cl(int(cl(A))). So A is β-open.
Since X is locally indiscrete, int(A) is closed and A ⊆ cl(int(A)) = int(A) which implies,
A is open set and for any x ∈ A, x ∈ int(A) ⊆ A. Hence, by Definition 2.1.1, A is
βc-open.
Remark 2.1.17. Since every open set is semi-open, then by Proposition 2.1.16, in a locally
indiscrete space, every open set is βc-open set.
Proposition 2.1.18. Let X be a topological space, if X is regular space, then every open
set is a βc-open set.
Proof. Let A be an open, then A is β-open. Since X is regular, then by Definition 1.2.12,
for each x ∈ A, there exists an open set G such that x ∈ G ⊆ cl(G) ⊆ A. So that,
x ∈ cl(G) ⊆ A. Therefore, by Definition 2.1.1, A is βc-open set.
Proposition 2.1.19. Let X be an extremally disconnected topological space. If A is a
βc-open set and B a regular open. Then A ∩ B is a βc-open set.
13
Proof. Let X be an extremely disconnected topological space and let A and B be subsets of
X. If A is βc-open and B is regular open then, B is open and A∩B ⊆ cl(int(cl(A)))∩B ⊆
cl(int(cl(A)) ∩ B) = cl(int(cl(A) ∩ B)) ⊆ cl(int(cl(A ∩ B))). So that, A ∩ B is β-open. If
x ∈ A ∩ B then, since A is βc-open, there exists a closed set F such that x ∈ F ⊆ A and
so, x ∈ F ∩ B ⊆ A ∩ B. Since B is regular open in an extremely disconnected space so,
B is closed and hence F ∩ B is closed set. Therefore, A ∩ B is a βc-open set.
Corollary 2.1.20. Let X be an extremely disconnected topological space and let B be a
regular open subset of X then, B is a βc-open set.
Proof. Using Proposition 2.1.19. Take A = X and B is a regular open set. Then A ∩ B
= B is a βc-open set.
Proposition 2.1.21. Let X be an extremally disconnected topological space and let A be
δ-open subset of X, then A is a βc-open set.
Proof. Let X be an extremally disconnected topological space and let A be δ-open subset
of X. Then for any x ∈ A there exists an open set Gx such that x ∈ Gx ⊆ int(cl(Gx )) ⊆
S
A which implies x∈A Gx = A is an open set and so a β-open, but X is extremally
disconnected so that, int(cl(Gx )) = cl(Gx ) and x ∈ cl(Gx ) ⊆ A. Therefore, A is βc-open
set.
Proposition 2.1.22. Let X be a topological space with a subset A. If A is θ-semi open
set. Then A is βc-open set.
Proof. Let A be a θ-semi open set. Then for each x ∈ A there exists a semi open set
S
Gx such that x ∈ Gx ⊆ cl(Gx ) ⊆ A which implies that A = x∈A Gx which means A
is a union of semi open sets and therefore, A is semi open and so β-open set. Also,
S
A = x∈A cl(Gx ) which is a union of closed sets. Therefore, A is βc-open set.
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Corollary 2.1.23. Let X be a topological space with a subset A. If A is θ-open set, then
A is βc-open set.
Proof. Since every θ-open set is θ-semi open set, then by Proposition 2.1.22, we get the
result.
Remark 2.1.24. Since semi-open sets ⇒ b-open sets ⇒ β-open sets, then sc-open sets ⇒
bc-open sets ⇒ βc-open sets.
Remark 2.1.25. Since pre-open sets ⇒ b-open sets ⇒ β-open sets, then pc-open sets ⇒
bc-open sets ⇒ βc-open sets. This implies the following.
Proposition 2.1.26. Let (X, τ ) be a topological space, then SCO(X) ∪ P CO(X) ⊆
BCO(X) ⊆ βCO(X).
Proposition 2.1.27. Let (X, τ ) be an Alexandroff space, then SCO(X) = BCO(X)
= βCO(X).
Proof. By Proposition 2.1.26, SCO(X) ⊆ BCO(X) ⊆ βCO(X). If A ∈ βCO(X), then
A is a union of closed sets in an Alexandroff space, which implies that A is closed.
Hence, A ⊆ cl(int(cl(A))) = cl(int(A)) and so that, A is semi-open set. Therefore, A ∈
SCO(X). Thus, SCO(X) ⊆ BCO(X) ⊆ βCO(X) ⊆ SCO(X) and we get that SCO(X)
= BCO(X) = βCO(X).
The following diagram shows the relations among SCO(X), P CO(X), BCO(X),
βCO(X), SO(X), P O(X), BO(X), βO(X), RO(X), RC(X), θSO(X), SθO(X), τδ ,
τα , τθ and τ .
15
set.png
Remark 2.1.28. From the above diagram and in any topological space (X, τ ), we notice
the following facts:
(1.) τ is incomparable with βCO(X).
(2.) τα is incomparable with βCO(X).
(3.) SθO(X) is incomparable with βCO(X).
(4.) τδ is incomparable with βCO(X).
(5.) RO(X) is incomparable with βCO(X).
Proposition 2.1.29. Let (Y, τY ) be a subspace of a space (X, τ ). If A is βc-open set in
space X and A ⊆ Y such that Y is β-open, then A is βc-open in subspace Y .
Proof. Suppose that A is a βc-open in X, then A is β-open in X but A ⊆ Y and Y is
β-open. So that, by Proposition 1.2.18, A is β-open in subspace Y . Also, for each x ∈ A
16
there exists a closed set F in X such that x ∈ F ⊆ A but A ⊆ Y so by Corollary 1.1.15,
F is closed in subspace Y . Hence, A is βc-open in (Y, τY ).
Corollary 2.1.30. Let A and Y be any subsets of a space X. If A is βc-open and Y is
clopen set in X, then A ∩ Y is βc-open in a subspace (Y, τY ).
Proof. Let A be a βc-open, then A is β-open. Since Y is clopen. By Theorem 1.2.17,
A ∩ Y is β-open. Since A is βc-open, then for each x ∈ A, there exists closed set F in X
such that x ∈ F ⊆ A. Hence, x ∈ F ∩ Y ⊆ A ∩ Y and therefore, A ∩ Y is βc-open set
such that A ∩ Y ⊆ Y . Thus, by Proposition 2.1.29, A ∩ Y is βc-open in Y .
Proposition 2.1.31. Let (Y, τY ) be a subspace of a space (X, τ ) and A ⊆ Y . If A is
βc-open in a subspace (Y, τY ) and Y is clopen, then A is βc-open set in X.
Proof. Let A be a βc-open in a subspace (Y, τY ), then A is β-open in a subspace (Y, τY )
and for x ∈ A, there exists a closed set F in Y such that x ∈ F ⊆ A. Since Y is clopen,
Y is β-open in X and A is β-open in Y , then by Proposition 1.2.18, A is β-open in X.
Also, Y is closed in X and F is closed in Y , then by Proposition 1.1.16, F is closed in X.
Hence, A is βc-open in (X, τX ).
Proposition 2.1.32. Let X, Y be topological spaces and X × Y be the product topology.
If A ∈ βCO(X) and B ∈ βCO(Y ), then A × B ∈ βCO(X × Y ).
Proof. Let (x, y) ∈ A × B, then x ∈ A and y ∈ B. Since A ∈ βCO(X), then A is
β-open and there exists a closed set F in X such that x ∈ F ⊆ A. Since B ∈ βCO(Y ),
then B is β-open and there exists a closed set E in Y such that y ∈ E ⊆ B. Therefore,
(x, y) ∈ F × E ⊆ A × B and by Proposition 1.2.19, A × B ∈ βO(X × Y ). Since F and E
are closed in X and Y respectively, then by Corollary 1.1.20, F × E is closed in X × Y .
Therefore, A × B ∈ βCO(X × Y ).
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2.2
The Family βCO(X)
In this section, we view the cases when the family βCO(X) forms the discrete and indiscrete topology. In addition, we investigate some relations among βCO(X) and other
families of sets.
Proposition 2.2.1. Let (X, τ ) be a topological space. If the family βO(X) forms a
topology on X, then the family βCO(X) also forms a topology on X.
Proof. Let X be a topological space and suppose that the family βO(X) forms a topology
X. Then:
(1) ∅ , X ∈ βCO(X).
(2) By Proposition 2.1.9, we have arbitrary union of βc-open sets is βc-open set.
(3) Let A, B ∈ βCO(X), then A and B ∈ βO(X). Since βO(X) forms a topology in
X, A ∩ B ∈ βO(X). If x ∈ A ∩ B where both A and B are βc-open sets, then there
exist closed sets F and E in X such that x ∈ F ⊆ A and x ∈ E ⊆ B which implies,
x ∈ F ∩ E ⊆ A ∩ B. Since finite intersection of closed sets is closed, F ∩ E is closed and
hence, A ∩ B ∈ βcO(X). Therefore, βCO(X) is a topology on X.
Example 2.2.2. Let X = {a, b, c} with the topology τ = {∅, X, {a}}, then:
(1) C(X)= {∅, X, {b, c}}.
(2) βO(X) = {∅, X, {a}}.
(3) βCO(X) = {∅, X}.
Note that the family βCO(X) forms the indiscrete topology. We can generalize this example in the following theorem.
Theorem 2.2.3. Let (X, τ ) be a topological space such that τ = {∅, X, {a}} where a is
fixed in X, then βCO(X) = {∅, X}.
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Proof. Let a be a fixed in X and let A be a nonempty subset of X such that A ∈ βCO(X).
Then a ∈ A. To see this, suppose a ∈
/ A, then A ⊆ {a}c . Since {a} is open, {a}c is a closed
set and so, cl(A) ⊆ {a}c . Since the only nonempty proper open set is {a}, int({a}c ) = ∅.
Hence, cl(int(cl(A))) ⊆ cl(int({a}c )) = cl(∅) = ∅. But A is βc-open and so A is β-open;
that is, A ⊆ cl(int(cl(A))) = ∅ which implies, A = ∅ which is a contradiction. Since A
is βc-open set, there is a closed set F in X such that a ∈ F ⊆ A but the only closed set
contains a is X which implies, X ⊆ A. Therefore, A = X and βCO(X) = {∅, X}.
Example 2.2.4. Let X = {a, b, c, d}, with a topology τ = {∅, X, {a}, {a, b}, {a, b, c}},
then:
(1) C(X) = {∅, X, {d}, {c, d}, {b, c, d}}.
(2) βO(X) = {∅, X, {a}, {a, b}, {a, c}, {a, d}, {a, b, c}, {a, b, d}}.
(3) βCO(X) = {∅, X}.
Note that {a} ⊆ {a, b} ⊆ {a, b, c} ⊆ X and βCO(X) forms the indiscrete topology which
lead to the following two generalizations.
Lemma 2.2.5. Let X be a topological space, if there exists a ∈ X such that {a} is the
smallest nonempty element of τ with direct inclusion, then every nonempty β-open set
contains {a}.
Proof. Let A be a nonempty β-open set such that {a} * A then by the same argument
in Theorem 2.2.3, we have, cl(int(cl(A))) = ∅ + A, which contradicts with A is β-open
set. Hence, any nonempty β-open set must contains {a}.
Theorem 2.2.6. Let X be a topological space, if there is a ∈ X such that {a} is the
smallest nonempty element of a space X with direct inclusion, then βCO(X) = {∅, X}.
Proof. Let A ∈ βCO(X), if A = ∅, then A is βc-open set. If A 6= ∅, then A is β-open set
and by Lemma 2.2.5, a ∈ A. But A is βc-open set so that, there is a closed set F in X
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such that a ∈ F ⊆ A but the only non empty closed set contains a is X which implies, X
⊆ A. Therefore, A = X and βCO(X) = {∅, X}.
Proposition 2.2.7. Let (X, τ ) be the indiscrete topological space, then βCO(X) forms
the indiscrete topology.
Proof. Let (X, τ ) be the indiscrete topological space and let A ∈ βCO(X), then A must
be union of closed sets. But the only closed sets in X are ∅ and X. Moreover, ∅ and X
are β-open sets and hence, ∅ and X are βc-open sets. Therefore, βCO(X) = {∅, X}.
Remark 2.2.8. The converse of Proposition 2.2.7 need not be true in general. Consider
Example 2.2.2 in which βCO(X) is the indiscrete topology but (X, τ ) is not.
Proposition 2.2.9. Let X be a topological space and A ⊆ X. If A is a clopen set, then
A ∈ βCO(X).
Proof. Let X be a topological space and A ⊆ X such that A is a clopen set. Then A is
open so it is β-open and A is closed so it is a union of closed sets and hence, it is βc-open.
Therefore, A ∈ βCO(X).
Corollary 2.2.10. Let (X, τ ) be a topological space. Then, βCO(X) forms the discrete
topology on X if and only if (X, τ ) is the discrete topology.
Proof. (⇒) Let (X, τ ) be a topological space and let βCO(X) forms the discrete topology
on X. Then, for any a ∈ X we have, {a} is βc-open set which implies, {a} is the
union of closed sets and hence, {a} is a closed and β-open set. If {a} is not open, then
{a} ⊆ cl(int(cl({a}))) = cl(int({a})) = ∅ which is a contradiction. So, {a} is a clopen
set and therefore, X is the discrete topology.
(⇐) Since each set is clopen. Then by Proposition 2.2.9, we have, βCO(X) is the discrete
topology.
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Remark 2.2.11. If τ1 and τ2 be topologies on X such that τ1 ⊆ τ2 , then there is no relation
between the families βCO(X, τ1 ) and βCO(X, τ2 ). Consider the following example.
Example 2.2.12. Let X = {a, b, c} with topologies τ1 = {∅, X, {a}, {b}, {a, b}} and τ2 =
{∅, X, {a}, {b}, {a, b}, {a, c}}. Then,
(1) βCO(X, τ1 ) = {∅, X, {a, c}, {b, c}}.
(2) βCO(X, τ2 ) = {∅, X, {b}, {a, c}}.
Note that τ1 ⊆ τ2 but βCO(X, τ1 ) * βCO(X, τ2 ) because {b, c} ∈ βCO(X, τ1 ) and {b, c} ∈
/
βCO(X, τ2 ). Moreover, βCO(X, τ2 ) * βCO(X, τ1 ) because {b} ∈ βCO(X, τ2 ) and {b} ∈
/
βCO(X, τ1 ).
Remark 2.2.13. Let τ1 and τ2 be two topologies on the space X such that βCO(X, τ1 ) ⊆
βCO(X, τ2 ), then it not necessary that, τ1 ⊆ τ2 as shown in the following examples.
Example 2.2.14. Let X = {a, b, c} with topologies τ1 = {∅, X, {a}, {a, b}}, τ2 = {∅, X, {a}},
then βCO(X, τ1 ) = {∅, X} and βCO(X, τ2 ) = {∅, X}.
Note that βCO(X, τ1 ) ⊆ βCO(X, τ2 ). But τ1 * τ2 because {a, b} ∈ τ1 but {a, b} ∈
/ τ2 .
Example 2.2.15. Let X = {a, b, c} with topologies τ1 = {∅, X, {a}, {a, b}}, τ2 = {∅, X, {a}, {b}, {a, b}},
then βCO(X, τ1 ) = {∅, X} and βCO(X, τ2 ) = {∅, X, {a, c}, {b, c}}.
Note that βCO(X, τ1 ) ⊆ βCO(X, τ2 ). But τ2 * τ1 because {b} ∈ τ2 but not in τ1 .
Proposition 2.2.16. Let A and B be two subsets of a topological space X. Then, if A
∈ SCO(X) and B ∈ P CO(X), then A ∩ B ∈ βCO(X).
Proof. Let A ∈ SCO(X) and B ∈ P CO(X). Then, A ∩ B ⊆ cl(int(A)) ∩ int(cl(B)) ⊆
cl(int(A) ∩ int(cl(B))) = cl(int(A ∩ cl(B) ⊆ cl(int(cl(A ∩ B))) which implies that, A ∩ B
is a β-open set. If x ∈ A ∩ B, then since A is sc-open set, there exists a closed set F in X
such that x ∈ F ⊆ A and since B is pc-open, there exists a closed set G in X such that
21
x ∈ G ⊆ B. Hence, x ∈ F ∩ G ⊆ A ∩ B where F ∩ G is closed in X. Therefore, A ∩ B is
βc-open set and A ∩ B ∈ βCO(B).
Proposition 2.2.17. Let A and B be two subsets of a topological space X. If A is open
set which union of closed sets and B ∈ βCO(X), then A ∩ B ∈ βCO(X).
Proof. Let A be an open set and B be a βc-open set then A ∩ B ⊆ A ∩ cl(int(cl(B)))
⊆ cl(A ∩ in(cl(B))) = cl(int(A ∩ cl(B))) ⊆ cl(int(cl(A ∩ B))) which implies, A ∩ B is a
β-open set. If x ∈ A ∩ B then, since A is a union of closed sets, there exists a closed set
F in X such that x ∈ F ⊆ A and since B is βc-open set, there exists a closed set G in
X such that x ∈ G ⊆ B. Hence, x ∈ F ∩ G ⊆ A ∩ B where F ∩ G is a closed set in X.
Therefore, A ∩ B is a βc-open set and A ∩ B ∈ βCO(X).
Proposition 2.2.18. Let (X, τ ) be a topological space with subsets A and B. Then, if A
is clopen and B ∈ βCO(X), then A ∩ B ∈ βCO(X).
Proof. Let A be a clopen set and B be a βc-open set then, A ∩ B ⊆ A ∩ cl(int(cl(B))) ⊆
cl(A ∩ int(cl(B)) ⊆ cl(int(A ∩ cl(B)) ⊆ cl(int(cl(A ∩ B))) which implies, A ∩ B is β-open
set. If x ∈ A ∩ B, then since B is βc-open set, there exists a closed set G in X such that
x ∈ G ⊆ B but A is clopen such that x ∈ A ⊆ A. So that, x ∈ A ∩ G ⊆ A ∩ B where
A ∩ G is a closed set in X. Therefore, A ∩ B is a βc-open set and A ∩ B ∈ βCO(X).
2.3
βc-Operators
In this section, we define and study βc-operators named, βc-closure, βc-interior, βcfrontier. Also, we extend our study to βc-neighborhood, βc-limit points and βc-derived
sets which are based on the notation of βc-open sets. At the end of this section, we
22
give new equivalent definitions for βc-open sets and βc-closed sets using the concept of
βc-frontier.
Definition 2.3.1. Let (X, τ ) be a topological space and A ⊆ X. The βc-closure of A in
X is the set,
βc-cl(A) =
T
{ K ⊆ X : K is βc-closed set and A ⊆ K }.
Remark 2.3.2. The following two examples show that there is no relations between cl(A)
and βc-cl(A).
Example 2.3.3. Consider X = {a, b, c} with a topology τ = {∅, X, {a}}. Then, βccl({b, c}) = X and cl({b, c}) = {b, c}. But X * {b, c}, which implies, βc-cl(A) * cl(A).
Example 2.3.4. Let X = {a, b, c} with a topology τ = {∅, X, {a}, {b}, {a, b}}. Then,
cl({b}) = {b, c} and βc-cl({b}) = {b}. But {b, c} * {b}, which implies, cl(A) * βc-cl(A).
Hence, in general we have, cl(A) 6= βc-cl(A).
Theorem 2.3.5. Let A be a subset of a space X. Then, βc-cl(A) is the smallest βc-closed
set containing A.
Proof. Let { Fα : α ∈ ∆ } be the collection of all βc-closed sets in X containing A.
T
T
Then, by Corollary 2.1.12, βc-cl(A) = α∈∆ Fα is βc-closed set and A ⊆ α∈∆ Fα . Since
T
T
α∈∆ Fα ⊆ Fα for each α ∈ ∆, then βc-cl(A) =
α∈∆ Fα is the smallest βc-closed set
containing A.
Theorem 2.3.6. A subset A of a space X is βc-closed if and only if βc-cl(A) = A.
Proof. Let A be βc-closed set. By Theorem 2.3.5, βc-cl(A) ⊆ A. But from Definition
2.3.1, A ⊆ βc-cl(A). Hence, A = βc-cl(A). Conversely, let βc-cl(A) = A. Since βc-cl(A)
is βc-closed set, A is βc-closed set.
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Example 2.3.7. In the usual topology R, βc-cl((0, 1]) = (0, 1] because (0, 1] is βc-closed
and if Q is the set of rationales, then βc-cl(Q) = Q.
Theorem 2.3.8. Let A be a subset of X and x ∈ X. Then, the following are equivalent:
(i) For any βc-open set G in X such that x ∈ G we get, A ∩ G 6= ∅.
(ii) x ∈ βc-cl(A).
Proof. (i ⇒ ii) If x ∈
/ βc-cl(A), then there exists βc-closed set F such that A ⊆ F and
x∈
/ F . But X\F is βc-open set containing x and therefore, A ∩ (X\F ) ⊆ A ∩ (X\A) = ∅
which is a contradiction and hence, x ∈ βc-cl(A).
(ii ⇒ i) Straightforward.
Theorem 2.3.9. Let A and B be two subsets of the space X. Then,
(a) A ⊆ βc-cl(A).
(b) βc-cl(∅) = ∅ and βc-cl(X) = X.
(c) If A ⊆ B, then βc-cl(A) ⊆ βc-cl(B).
(d) βc-cl(βc-cl(A)) = βc-cl(A).
(e) If βc-cl(A) ∩ βc-cl(B) = ∅, then A ∩ B = ∅.
(f ) βc-cl(A) ∪ βc-cl(B) ⊆ βc-cl(A ∪ B).
(g) βc-cl(A ∩ B) ⊆ βc-cl(A) ∩ βc-cl(B).
Proof. (a) From Theorem 2.3.5.
(b) From Definition 2.3.1.
(c) Let A ⊆ B and assume x ∈ βc-cl(A). Then by Theorem 2.3.8, for any βc-open set G
containing x, we have A ∩ G 6= ∅. But A ⊆ B which implies, B ∩ G 6= ∅ where B is any
βc-open set containing x. Hence, x ∈ βc-cl(B) and therefore, βc-cl(A) ⊆ βc-cl(B).
(d) Since βc-cl(A) is βc-closed set, βc-cl(βc-cl(A)) = βc-cl(A).
(e) Let A ∩ B 6= ∅, then there is x ∈ A ∩ B which implies x ∈ A and x ∈ B. By Part (1),
24
x ∈ βc-cl(A) and x ∈ βc-cl(B) and so βc-cl(A) ∩ βc-cl(B) 6= ∅.
(f ) Since A ⊆ A ∪ B and B ⊆ A ∪ B, by Part (c), βc-cl(A) ⊆ βc-cl(A ∪ B) and βc-cl(B)
⊆ βc-cl(A ∪ B). Hence, βc-cl(A) ∪ βc-cl(B) ⊆ βc-cl(A ∪ B).
(g) Since A∩B ⊆ A and A∩B ⊆ B, by Part (c), βc-cl(A∩B) ⊆ βc-cl(A) and βc-cl(A∩B)
⊆ βc-cl(B). Hence, βc-cl(A ∩ B) ⊆ βc-cl(A) ∩ βc-cl(B).
Remark 2.3.10. The converse of parts (c), (e) and the reverse inclusions of (f ), (g) need
not be true in general. Consider the following examples.
Example 2.3.11. [ βc-cl(A) ⊆ βc-cl(B) ; A ⊆ B and A ∩ B = ∅ ; βc-cl(A)∩ βccl(B) = ∅ ]
Let X = {a, b, c} with the topology τ = {∅, X, {a}}. If A = {a}, B = {b}, then X = βccl(A) ⊆ βc-cl(B) = X but A * B. Moreover, A ∩ B = ∅ but βc-cl(A)∩ βc-cl(B) =
X 6= ∅.
Example 2.3.12. [βc-cl(A ∪ B) * βc-cl(A)∪ βc-cl(B)]
Let X = {a, b, c} with the topology τ = {∅, X, {a}, {b}, {a, b}}. If A = {a} and B = {b},
then A∪B = {a, b} which implies, βc-cl(A) = {a}, βc-cl(B) = {b} and so, βc-cl(A∪B) =
X * {a, b} = βc-cl(A) ∪ βc-cl(B).
Example 2.3.13. [βc-cl(A) ∩ βc-cl(B) * βc-cl(A ∩ B)]
In Example 2.3.11, A ∩ B = ∅ and so, βc-cl(A ∩ B) = ∅. But βc-cl(A) = X and βccl(B) = X which implies, βc-cl(A) ∩ βc-cl(B) = X * βc-cl(A ∩ B) = ∅.
Definition 2.3.14. Let (X, τ ) be a topological space. A set Ux ⊆ X is said to be βcneighborhood of a point x ∈ X if and only if there exists βc-open set V in X such that x
∈ V ⊆ Ux .
25
Definition 2.3.15. Let (X, τ ) be a topological space. A point x ∈ X is said to be βcinterior point of a set A ⊆ X if and only if there exists βc-open set V in X such that x
∈ V ⊆ A.
Definition 2.3.16. The union of all βc-open sets which are contained in A is called the
βc-interior of A and is denoted by βc-int(A).
Remark 2.3.17. Since their is no relations between βc-open set and open sets, their is
no relations between interior points and βc-interior points of A. Consider the following
example.
Example 2.3.18. Consider (X, τ ) defined in Example 2.1.6. Then, if A = {a, c}, then
int(A) = {a} and βc-int(A) = {a, c} which implies, βc-int(A) * int(A). On the other
hand, if B = {a}, then int(B) = {a} and βc-int(B) = ∅ that is, int(B) * βc-int(B). So
that, in general, int(B) 6= βc-int(B).
Theorem 2.3.19. Let X be a topological space and A ⊆ X. Then, βc-int(A) is the
largest βc-open subset of X contained in A.
Proof. Let U = {Uα : α ∈ ∆} be a collection of all βc-open sets contained in A. Then, by
S
Definition 2.3.16, α∈∆ Uα = βc-int(A) and so, by Proposition 2.1.9, βc-int(A) is βc-open
set. Let V be βc-open set such that V ⊆ A, then for any y ∈ V we have, y ∈ V ⊆ A and
so, y ∈ βc-int(A). Therefore, V ⊆ βc-int(A) and hence, βc-int(A) is the largest βc-open
set contained in A.
Proposition 2.3.20. Let X be a topological space and A ⊆ X. Then, A is βc-open set
if and only if βc-int(A) = A.
Proof. Let A be βc-open set, then for any x ∈ A, x ∈ βc-int(A) and A ⊆ βc-int(A). But
by Proposition 2.3.19, we have βc-int(A) ⊆ A and hence, βc-int(A) = A. Conversely, if
26
βc-int(A) = A, then by Proposition 2.3.19, βc-int(A) is βc-open and so, A is βc-open
set.
Theorem 2.3.21. Let A and B be two subsets of the space X, then:
(a) βc-int(A) ⊆ A.
(b) βc-int(∅) = ∅ and βc-int(X) = X.
(c) If A ⊆ B, then βc-int(A) ⊆ βc-int(B).
(d) βc-int( βc-int(A)) = βc-int(A).
(e) If A ∩ B = ∅, then βc-int(A) ∩ βc-int(B) = ∅.
(f ) βc-int(A) ∪ βc-int(B) ⊆ βc-int(A ∪ B).
(g) βc-int(A ∩ B) ⊆ βc-int(A) ∩ βc-int(B).
Proof. (a) From Theorem 2.3.19.
(b) From Definition 2.3.16.
(c) Let A ⊆ B and x ∈ βc-int(A), then there exists βc-open set V such that x ∈ V ⊆ A.
But A ⊆ B which implies, x ∈ V ⊆ B. Hence, x ∈ βc-int(B) and therefore, βcint(A) ⊆ βc-int(B).
(d) Since βc-int(A) is βc-open set, βc-int(βc-int(A) = βc-int(A).
(e) If βc-int(A) ∩ βc-int(B) 6= ∅, then there is x ∈ βc-int(A) ∩ βc-int(B). So there
exist βc-open sets U and V such that x ∈ U ⊆ A and x ∈ V ⊆ B which implies,
x ∈ U ∩ V ⊆ U ⊆ A and x ∈ U ∩ V ⊆ V ⊆ B. Hence, x ∈ A ∩ B and therefore,
A ∩ B 6= ∅.
(f ) Since A ⊆ A ∪ B and B ⊆ A ∪ B, by Part (c), βc-int(A) ⊆ βc-int(A ∪ B) and
βc-int(B) ⊆ βc-int(A ∪ B). Hence, βc-int(A) ∪ βc-int(B) ⊆ βc-int(A ∪ B).
(g) Since A ∩ B ⊆ A and A ∩ B ⊆ B, by Part (c), βc-int(A ∩ B) ⊆ βc-int(A) and
βc-int(A ∩ B) ⊆ βc-int(B). Hence, βc-int(A ∩ B) ⊆ βc-int(A) ∩ βc-int(B).
27
Remark 2.3.22. The converse of parts (c), (e) and the reverse inclusions of (f ), (g) need
not be true in general. Consider the following examples.
Example 2.3.23. [βc-int(A) ⊆ βc-int(B) ; A ⊆ B]
Let X = {a, b, c} with the topology τ = {∅, X, {a}}, then if A = {a} and B = {b}, we
have βc-int(A) = βc-int(B) = ∅ but A * B.
Example 2.3.24. [βc-int(A) ∩ βc-int(B) = ∅ ; A ∩ B = ∅]
In Example 2.3.23, if A = {a} and B = {a, b}, then βc-int(A) = βc-int(B) = ∅ which
implies, βc-int(A) ∩ βc-int(B) = ∅ but A ∩ B = {a} =
6 ∅.
Example 2.3.25. [βc-int(A ∪ B) * βc-int(A) ∪ βc-int(B)]
In Example 2.3.23, if A = {a} and B = {b, c}, then βc-int(A) = βc-int(B) = ∅ which
implies, βc-int(A) ∪ βc-int(B) = ∅. But A ∪ B = X and βc-int(A ∪ B) = X.
Example 2.3.26. [βc-int(A) ∩ βc-int(B) * βc-int(A ∩ B)]
Let X = {a, b, c} with the topology τ = {∅, X, {a}, {b}, {a, b}}, then if A = {a, c} and
B = {b, c}, then βc-int(A) = {a, c} and βc-int(B) = {b, c} which implies, βc-int(A) ∩ βcint(B) = {c}. But A ∩ B = {c} and βc-int(A ∩ B) = ∅.
The relations between the βc-closure and βc-interior can be considered in the following
theorem.
Theorem 2.3.27. Let X be a topological space, and A ⊆ X. Then the following statements hold.
(1) [βc-cl(A)]c = βc-int(Ac ).
(2) [βc-int(A)]c = βc-cl(Ac ).
(3) βc-cl(A) = [βc-int(Ac )]c .
(4) βc-int(A) = [βc-cl(Ac )]c .
28
Proof. Only we prove the first one and the other parts can be proved similarly.
T
[βc-cl(A)]c = [ {K : K is βc-closed set and A ⊆ K}]c .
S
= [ {K c : K c is βc-open set and K c ⊆ Ac }].
S
= [ {B : B is βc-open set and B ⊆ Ac }].
= βc-int(Ac ) .
Proposition 2.3.28. Let X be a topological space, then for any subset A of X:
(1) If G is clopen set, then βc-cl(A) ∩ G ⊆ βc-cl(A ∩ G).
(2) If F is clopen-set, then βc-int(A ∪ F ) ⊆ βc-int(A) ∪ F .
Proof. (1) Let x ∈ βc-cl(A) ∩ G, then x ∈ βc-cl(A) and x ∈ G. Let V be any βc-open
set containing x, then by Proposition 2.2.18, V ∩ G is βc-open set containing x. Since
x ∈ βc-cl(A), then by Theorem 2.3.8, (V ∩ G) ∩ A 6= ∅ which implies V ∩ (G ∩ A) 6= ∅
and this true for any βc-open set which contains x. Hence, by Theorem 2.3.8, x ∈ βccl(A ∩ G). Therefore, βc-cl(A) ∩ G ⊆ βc-cl(A ∩ G)
(2) Using Theorem 2.3.27 and part (1) of Proposition 2.3.28.
Definition 2.3.29. Let A be a subset of a topological space (X, τ ). A point x ∈ X is said
to be βc-limit point of A if for each βc-open subset G of X containing x, G ∩ A\{x} =
6 ∅.
The set of all βc-limit points of A is called the βc-derived set of A and its denoted by
βc-D(A). Note that for a subset A of X, a point x ∈ X is not βc-limit point of A if and
only if there exists βc-open set G in X such that,
x ∈ G and G ∩ (A\{x}) = ∅
or, equivalently,
x ∈ G and G ∩ A = ∅ or G ∩ A = {x}
29
or, equivalently,
x ∈ G and G ∩ A ⊆ {x}.
Proposition 2.3.30. Let A be a subset of a topological space X. If for each closed set F
of X containing x satisfies F ∩ (A\{x}) 6= ∅, then the point x ∈ X is βc-limit point of
A.
Proof. Let U be any βc-open set containing x, then there exists closed set F such that
x ∈ F ⊆ U . But from hypothesis, F ∩ (A\{x}) 6= ∅ which implies, U ∩ (A\{x}) 6= ∅.
Therefore, x is βc-limit point of A.
Theorem 2.3.31. Let τ1 and τ2 be topologies on X such that βCO(X, τ1 ) ⊆ βCO(X, τ2 ).
For any subset A of X, every βc-limit point of A with respect to τ2 is βc-limit point of A
with respect to τ1 .
Proof. Let x be βc-limit point of A with respect to τ2 . Then, for every βc-open set G in
τ2 such that x ∈ G we have, G ∩ (A\{x}) 6= ∅. But βCO(X, τ1 ) ⊆ βCO(X, τ2 ) so that,
in particular, G ∩ (A\{x}) 6= ∅ for every βc-open set G in τ1 such that x ∈ G. Hence, x
is βc-limit point of A with respect to τ1 .
Theorem 2.3.32. Let A and B be subsets of the space X with the βc-derived sets βcD(A) and βc-D(B) respectively. Then, the following properties hold:
(a) βc-D(∅) = ∅.
(b) If x ∈ βc-D(A), then x ∈ βc-D(A\{x}).
(c) If A ⊆ B, then βc-D(A) ⊆ βc-D(B).
(d) βc-D(A) ∪ βc-D(B) ⊆ βc-D(A ∪ B).
(e) βc-D(A ∩ B) ⊆ βc-D(A) ∩ βc-D(B).
Proof. (a) Let x ∈ βc-D(∅) and G be any βc-open set containing x. Then G ∩ ∅\{x} = ∅
which is a contradiction.
30
(b) Let x ∈ βc-D(A). Then for any βc-open set G containing x we have, G ∩ A\{x} =
6 ∅.
That is, for any βc-open set G containing x, G ∩ A contain points other than x and so
the intersection of G and A\{x} contain points and hence x ∈ βc-D(A\{x}).
(c) Let x ∈ βc-D(A). Then for any βc-open set G containing x, G ∩ A\{x} 6= ∅. But
A ⊆ B and so, G ∩ B\{x} =
6 ∅. Hence, x ∈ βc-D(B).
(d) Since A ⊆ A ∪ B and B ⊆ A ∪ B, by Part (c), βc-D(A) ⊆ βc-D(A ∪ B) and βc-D(B)
⊆ βc-D(A ∪ B). Hence, βc-D(A) ∪ βc-D(B) ⊆ βc-D(A ∪ B).
(e) Since A∩B ⊆ A and A∩B ⊆ B, by Part (c), βc-D(A∩B) ⊆ βc-D(A) and βc-D(A∩B)
⊆ βc-D(B). Hence, βc-D(A ∩ B) ⊆ βc-D(A) ∩ βc-D(B).
Remark 2.3.33. The converse of part (c) and the reverse inclusion of (d) and (e) need not
be true in general. Consider the following examples.
Example 2.3.34. [βc-D(A) ⊆ βc-D(B) ; A ⊆ B and βc-D(A ∪ B) * βc-D(A) ∪ βcD(B)]
Let X = {a, b, c} with the topology τ = {∅, X, {a}, {b}, {a, b}}. Then, if A = {a} and
B = {b}, then βc-D(A) = ∅ and βc-D(B) = ∅. So, βc-D(A) ⊆ βc-D(B) but A * B.
Moreover, A ∪ B = {a, b} and βc-D(A ∪ B) = {c}. Then, βc-D(A ∪ B) * βc-D(A)
∪βc-D(B) = ∅.
Example 2.3.35. [βc-D(A) ∩ βc-D(B) * βc-D(A ∩ B)]
In Example 2.3.34, Let A = {a, c} and {b, c}, then βc-D(A) = βc-D(B) = {a, b} which
implies, βc-D(A) ∩ βc-D(B) = {a, b}. But A ∩ B = {c} and βc-D(A ∩ B) = ∅.
Theorem 2.3.36. Let A and B be subsets of the space X with the βc-derived sets βcD(A) and βc-D(B) respectively. Then, the following properties hold:
(a) βc-D(βc-D(A))\A ⊆ βc-D(A).
(b) βc-D(A ∪ βc-D(A)) ⊆ A ∪ βc-D(A).
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Proof. (a) Let x ∈ βc-D(βc-D(A)) \A and U be βc-open set containing x, then U ∩
(βc-D(A)\{x}) 6= ∅. So, there exists y ∈ U ∩ (βc-D(A)\{x}) which means y ∈ U and
y ∈ βc-D(A)\{x}) which implies, U ∩ (A\{y}) 6= ∅. Let z ∈ U ∩ (A\{y}), then z 6= x
because z ∈ A and x ∈
/ A. Hence, U ∩ (A\{x}) 6= ∅. Therefore, x ∈ βc-D(A).
(b) Let x ∈ βc-D(A∪ βc-D(A)). Then, if x ∈ A, the result hold. So, let x ∈ βc-D(βcD(A) ∪ A) \A, then for βc-open set U containing x, U ∩ (A ∪ βc-D(A)) \{x} =
6 ∅. Thus,
U ∩ (A\{x}) 6= ∅ or U ∩ (βc-D(A)\{x}) 6= ∅. The first case implies, x ∈ βc-D(A) and
the second case implies, x ∈ βc-D(βc-D(A)). Since x ∈
/ A, it follows from (a) that, x ∈
βc-D(βc-D(A)) \A ⊆ βc-D(A). Therefore, (b) is valid.
Proposition 2.3.37. For any subset A of X, βc-D(A) ⊆ βc- cl(A).
Proof. Let x ∈ βc-D(A) then, for any βc-open set G containing x we have, G ∩ A\{x}
6= ∅ which implies, G ∩ A 6= ∅. Then by Theorem 2.3.8, x ∈ βc-cl(A). Hence, βc-D(A) ⊆
βc-cl(A).
Proposition 2.3.38. For any subset A of X, βc-cl(A) = A∪ βc-D(A).
Proof. Let x ∈ βc-cl(A) such that x ∈
/ A and let G be any βc-open set containing x.
Then, G ∩ A\{x} 6= ∅ which implies, x ∈ βc-D(A) and so, βc-cl(A) ⊆ A∪ βc-D(A).
Conversely, since A ⊆ βc-cl(A) and by Proposition 2.3.37, βc-D(A) ⊆ βc-cl(A), then
A ∪ βc-D(A) ⊆ βc-cl(A). Therefore, βc-cl(A) = A∪ βc-D(A).
Theorem 2.3.39. Let A be subset of a space X. Then, A is βc-closed set if and only if
it contains all of its βc-limit points.
Proof. Suppose that A is βc-closed set and x is βc-limit point such that x ∈
/ A. Then,
x ∈ X\A where X\A is βc-open set and so, there exists βc-open set G such that x ∈ G ⊆
X\A. Therefore, G ∩ A = ∅ and x ∈
/ βc-D(A) which is a contradiction. Conversely, let
32
A contains all βc-limit points, then for any x ∈ X\A we have, x is not βc-limit point of
A which implies, for any βc-open set G containing x, G ∩ A ⊆ {x}. But x ∈
/ A which
implies, G ∩ A = ∅ and so, G ⊆ X\A. This proves that X\A is βc-open set and hence,
A is βc-closed set.
Definition 2.3.40. For any subset A of X, the set
βc-F r(A) := βc-cl(A) \ βc-int(A)
is called the βc-frontier of A and denoted by βc-F r(A).
Proposition 2.3.41. For any subset A of a topological space X. The following statements
hold:
(1) βc-cl(A) = βc-int(A) ∪ βc-F r(A).
(2) βc-int(A) ∩ βc-F r(A) = ∅.
(3) βc-F r(A) = βc-cl(A) ∩ βc-cl(X\A).
Proof. (1) βc-int(A) ∪ βc-F r(A) = βc-int(A) ∪[βc-cl(A)\βc-int(A)] = [βc-int(A) ∪ βccl(A)] ∩ [βc-int(A) ∪ βc-int(A)c ] = βc-cl(A) ∩ X = βc-cl(A).
(2) βc-int(A) ∩ βc-F r(A) = βc-int(A)∩[βc-cl(A)\βc-int(A)] = βc-int(A)∩[βc-cl(A)∩βcint(A)c ] = ∅.
(3) βc-F r(A) = βc-cl(A)\βc-int(A) = βc-cl(A)∩βc-int(A)c = βc-cl(A)∩βc-cl(X\A).
Corollary 2.3.42. For any subset of the space X, βc-F r(A) is βc-closed set.
Proof. By part (3) of Proposition 2.3.41 we have, βc-F r(A) = βc-cl(A) ∩ βc-cl(X\A).
Since the intersection of two βc-closed sets is βc-closed set, βc-F r(A) is βc-closed set.
Remark 2.3.43. Let A and B be two subsets of a space X such that A ⊆ B. Then, their
is no relations between βc-F r(A) and βc-F r(B). Consider the following example:
33
Example 2.3.44. In Example 2.3.34, we have the following:
(a) If A = {c} and B = {a, c}. Then we have, βc-int(A) = ∅, βc-cl(A) = X, βc-int(B)
= {a, c} and βc-cl(B) = X which implies, βc-F r(A) = X and βc-F r(B) = {b}. Hence,
it follows that A ⊆ B but, βc-F r(A) * βc-F r(B).
(b) If A = {a} and B = {a, b}. Then we have, βc-int(A) = ∅, βc-cl(A) = {a}, βc-int(B)
= ∅ and βc-cl(B) = X which implies, βc-F r(A) = {a} and βc-F r(B) = X. Hence, it
follows that A ⊆ B but, βc-F r(B) * βc-F r(A).
In the following theorem we give new equivalent definitions for βc-open sets and βcclosed sets using the concept of βc-frontier.
Theorem 2.3.45. For a subset A of a topological space X. The following are hold:
(1) A is βc-open if and only if A ∩ βc-F r(A) = ∅.
(2) A is βc-closed if and only if βc-F r(A) ⊆ A.
(3) A is both βc-open and βc-closed set if and only if βc-F r(A) = ∅.
Proof. (1) Let A be βc-open set, then βc-int(A) = A. Then, by Proposition 2.3.41, we
have, βc-F r(A) ∩βc-int(A) = ∅ which implies, βc-F r(A) ∩A = ∅. Conversely, Assume
that A ∩βc-F r(A) = ∅. Then A ∩ [βc-cl(A) \ βc-int(A)] = A ∩ [βc-cl(A)] ∩ A\[βc-int(A)]
= A\βc-int(A) = ∅. Thus, A = βc-int(A). Therefore, A is βc-open set.
(2) Let A be βc-closed set, then A = βc-cl(A) and so, βc-F r(A) = βc-cl(A) \ βc-int(A)
= A\βc-int(A) and hence, βc-F r(A) ⊆ A. On the other hand, let βc-F r(A) ⊆ A, then
βc-cl(A) = βc-int(A) ∪ βc-F r(A) ⊆ βc-int(A) ∪ A = A ⊆ βc-cl(A). Therefore, A = βccl(A) and A is βc-closed set.
(3) Assume that A is both βc-open and βc-closed set. Then, βc-int(A) = A = βc-cl(A)
and by definition, we get βc-F r(A) = A\A = ∅. Conversely, let βc-F r(A) = ∅ which
means, βc-cl(A) \ βc-int(A) = ∅ and in other words, βc-int(A) = βc-cl(A). But from
34
properties of βc-interior and βc-closure we have, βc-int(A) ⊆ A ⊆ βc-cl(A). Hence, it
follows, βc-int(A) = A = βc-cl(A), which means that A is both βc-open and βc-closed
set.
2.4
The Topology Generated by βc-Open Sets
Although none of SO(X), P O(X), BO(X) and βO(X) is a topology on X, each of
these classes generates a topology in a natural way. Njastad showed that τα was the
topology generated by SO(X) and Andrijevic [3] showed that τγ and τb were the topologies
generated by P O(X) and BO(X) respectively and moreover he proved that τγ = τb =
τβ which generated by SP O(X) or βO(X). In this section we shall study a topology
generated by the class βCO(X) and we denoted it by τβc .
Definition 2.4.1. A subset A of a topological space (X, τ ) is called a βc-set if A ∩ B ∈
βCO(X) for all B ∈ βCO(X).
The class of all βc-sets in (X, τ ) will be denoted by τβc .
Theorem 2.4.2. τβc is a topology on X.
Proof. (1) Clearly ∅ and X ∈ τβc .
(2) Let {Aα : α ∈ ∆} ⊆ τβc . Then Aα ∩ B ∈ βCO(X) for all B ∈ βCO(X). Therefore,
S
S
( α∈∆ Aα ) ∩ B = α∈∆ {Aα ∩ B} where Aα ∩ B is βc-open set and since arbitrary
S
union of βc-open sets is βc-open, it follows that, α∈∆ {Aα ∩ B} is a βc-open set for each
S
S
B ∈ βCO(X) and hence, ( α∈∆ Aα ) ∩ B is βc-open set. Therefore, α∈∆ Aα ∈ τβc .
(3) Let C and D ∈ τβc . Then (C ∩ D) ∩ B = C ∩ (D ∩ B) ∈ βCO(X) for all B ∈ βCO(X)
and hence C ∩ D ∈ τβc . By (1), (2) and (3) we have τβc is a topology on X.
Theorem 2.4.3. τβc ⊆ βCO(X, τ ) for any τ .
35
Proof. Let A ∈ τβc , then for any βc-open set B in X we have, A ∩ B is a βc-open set. In
paretically, if we let B = X, then A ∩ X = A is βc-open set and A ∈ βCO(X, τ ). Hence,
τβc ⊆ βCO(X, τ ).
Remark 2.4.4. The converse of Theorem 2.4.3 need not be true in general. Consider the
following example.
Example 2.4.5. ( βCO(X, τ ) * τβc )
Let X = {a, b, c} with the topology τ = {∅, X, {a}, {b}, {a, b}}, then βCO(X, τ ) = {∅, X, {a, c}, {b, c}}
and τβc = {∅, X}. Note that {a, c} ∈ βCO(X, τ ) but {a, c} ∈
/ τβc because {a, c} ∩ {b, c}
= {c} ∈
/ βCO(X, τ ).
Theorem 2.4.6. A subset A of a topological space (X, τ ) is closed in (X, τβc ) if and only
if A ∪ B is βc-closed for any βc-closed set B in X.
Proof. A subset A is closed in (X, τβc ) if and only if Ac is open in (X, τβc ) if and only if
Ac ∩ C is βc-open for any βc-open set C if and only if ( Ac ∩ C)c = A ∪ C c is βc-closed
for any βc-closed set C c that is, A ∪ B is βc-closed for any βc-closed set B in X.
Theorem 2.4.7. Let (X, τ ) be a locally indiscrete space. Then τ ⊆ τβc .
Proof. Let (X, τ ) be a locally indiscrete topological space and consider A ∈ τ . Then by
Theorem 1.2.17, for any βc-open set B in X we have, A ∩ B is β-open set. But B is
S
βc-open so that, B = α∈∆ Fα where Fα is a closed set for each α ∈ ∆ which implies,
S
A∩B = α∈∆ (A∩Fα ) which union of closed sets because A is closed in a locally indiscrete
space. Hence, A ∩ B is βc-open set for any βc-open set B in X. So that, A ∈ τβc and τ
⊆ τβc .
Theorem 2.4.8. Let (X, τ ) be T1 space. Then τ ⊆ τβc .
36
Proof. Let X be a T1 topological space and consider A ∈ τ . Then for any βc-open set B
S
in X we have, A ∩ B is β-open set. But A ∩ B = x∈A∩B {x} where {x} is a closed set
in T1 space. So that, A ∩ B is βc-open set for any βc-open set B in X and so, A ∈ τβc .
Hence, τ ⊆ τβc for any T1 space.
Theorem 2.4.9. Let (X, τ ) be a regular topological space. Then τ ⊆ τβc .
Proof. Let X be a regular space and let A ∈ τ . Then for any βc-open set B in X
we have, A ∩ B is β-open set. Since X is regular, there is an open set Gx such that
x ∈ Gx ⊆ cl(Gx ) ⊆ A. Since B is βc-open set, there is a closed set Fx such that
x ∈ Fx ⊆ A. So that, for any x ∈ A ∩ B we have, x ∈ cl(Gx ) ∩ Fx ⊆ A ∩ B where
cl(Gx ) ∩ Fx is a closed set. Hence, A ∩ B is βc-open set for any βc-open set B in X. So,
A ∈ τβc and therefore, τ ⊆ τβc .
Theorem 2.4.10. For a space (X, τ ) and x ∈ X the following are equivalent:
(1) {x} ∈ βCO(X).
(2) {x} ∈ τβc .
Proof. (1 ⇒ 2) Let {x} ∈ βCO(X). Then for any βc-open set B in X we have, either
{x} ∩ B = ∅ which is βc-open set or {x} ∩ B = {x} which is a βc-open set. Hence, in
both cases, {x} ∩ B ∈ βCO(X). Therefore, {x} ∈ τβc .
(2 ⇒ 1) Let {x} ∈ τβc . Then by Theorem 2.4.3, τβc ⊆ βCO(X) and hence, {x} ∈
βCO(X).
Theorem 2.4.11. Let (X, τ ) be an extremally disconnected Alexandroff space. Then the
following are equivalent:
(1) A is a clopen set in (X, τ ).
(2) A ∈ τβc .
37
Proof. (1 ⇒ 2) Let A be a clopen set in (X, τ ). Then, for any βc-open set B in X we
have, A ∩ B is β-open and for any x ∈ A ∩ B there is a closed set F ⊆ B such that
x ∈ A ∩ F ⊆ A ∩ B where A ∩ F is a closed set. Hence, A ∩ B is βc-open and so, A ∈ τβc .
S
(2 ⇒ 1) Let A ∈ τβc . Then by Theorem 2.4.3, A ∈ βCO(X) and so, A = α∈∆ Fα
where Fα is closed for each α ∈ ∆. Since X is Alexandroff space, A is closed. But A
is β-open which implies, A ⊆ cl(int(cl(A))) = cl(int(A)) ⊆ cl(A) = A which implies,
A = cl(int(A)). Since X is extremally disconnected, cl(int(A)) is open and so A is open.
Therefore, A is a clopen set.
Corollary 2.4.12. In an extremally disconnected Alexandroff space (X, τ ), τβc ⊆ τ .
Proof. Let A ∈ τβc . Then by Theorem 2.4.11, A is a clpen set in X. Hence, A ∈ τ and
therefore, τβc ⊆ τ .
Remark 2.4.13. The converse of Corollary 2.4.12 need not be true in general. Consider
the following example.
Example 2.4.14. [τ * τβc ]
Let X = {a, b, c} with the topology τ = {∅, X, {a}, {b}, {a, b}}, then τβc = {∅, X}. Note
that {a, b} ∈ τ but {a, b} ∈
/ τβc .
Theorem 2.4.15. Let (X, τ ) be an extremally disconnected topological space. Then
RC(X) ⊆ τβc .
Proof. Let A be a regular closed set in an extremally disconnected space (X, τ ). Then
A = cl(int(A)) where cl(int(A)) is open in X and so A is open. So that, for any βc-open
set B we have, A ∩ B is β-open. Since B is βc-open, for any x ∈ A ∩ B there is a closed
set F ⊆ B such that x ∈ A ∩ F ⊆ A ∩ B where A ∩ F is closed because A is closed. Hence,
A ∩ B is βc-open set for any βc-open set B in X. Therefore, A ∈ τβc .
38
Theorem 2.4.16. Let (X, τ ) be a topological space. Then τθ ⊆ τβc .
Proof. Let A ∈ τθ , then A is open and for any βc-open set B in X we get, A ∩ B is
β-open. If x ∈ A ∩ B, then since A is θ-open there exists an open set G such that
x ∈ G ⊆ cl(G) ⊆ A. Since B is βc-open, there exists a closed set F such that x ∈ F ⊆ B.
So that, x ∈ cl(G) ∩ F ⊆ A ∩ B where cl(G) ∩ F is a closed set. So A ∩ B is βc-open set
for any βc-open set B in X. Therefore, A ∈ τβc and τθ ⊆ τβc .
Theorem 2.4.17. Let (X, τ ) be an extremally disconnected Alexandroff space. Then τθ
= τβc .
Proof. Let (X, τ ) be an extremally disconnected Alexandroff space and A ∈ τβc . Then by
Theorem 2.4.11, A is clopen set. But for any x ∈ A we have, x ∈ A ⊆ cl(A) ⊆ A which
implies that, A is θ-open set and hence, A ∈ τθ . Hence, τβc ⊆ τθ . But by Theorem 2.4.16,
we have, τθ ⊆ τβc . Therefore, τθ = τβc .
Corollary 2.4.18. Let (X, τ ) be an extremally disconnected Alexandroff regular space.
Then τβc = τ .
Proof. Let (X, τ ) be a regular space. Then by Theorem 1.2.10, τθ = τ . But X is extremally disconnected Alexandroff space which implies, by Theorem 2.4.17, τθ = τβc .
Theorem 2.4.19. Let (X, τ ) be an extremally disconnected topological space. Then, τδ
⊆ τβc .
Proof. Let X be an extremally disconnected space and A ∈ τδ . Then A is open and for
any βc-open set B in X we have, A ∩ B is β-open. If x ∈ A ∩ B then, since X is extremely
disconnected, there is an open set G such that x ∈ G ⊆ int(cl(G)) ⊆ A. Since B is
βc-open, there is a closed set F such that x ∈ F ⊆ B which implies, x ∈ int(cl(G)) ∩ F
39
⊆ A ∩ B where cl(G) is open in extremally disconnected space. So int(cl(G)) ∩ F =
cl(G) ∩ F which is a closed set. So that, A ∩ B is βc-open set for any βc-open set B in
X and so, A ∈ τβc . Therefore, τδ ⊆ τβc .
Theorem 2.4.20. Let (X, τ ) be an extremally disconnected Alexandroff space. Then τδ
= τβc .
Proof. Let X be an extremally disconnected Alexandroff space such that A ∈ τβc . Then
A is a clopen set and so for any x ∈ A we have, x ∈ A = int(cl(A)) ⊆ A which implies
that A is δ-open and hence, A ∈ τδ . So that, τβc ⊆ τδ . But by Theorem 2.4.19, we have,
τδ ⊆ τβc . Therefore, τδ = τβc .
40
Chapter 3
βc-Continuous Functions
In this chapter, we study a new class of continuous functions called βc-continuous functions. In addition, the relations among βc-continuous functions and other classes of
continuous functions will be considered. Moreover, the characterization of βc-continuous
functions will be studied. Finally, some properties of βc-continuous function will be investigated especially, the extension βc-continuous functions, the retraction βc-continuous
functions, the composition βc-continuous function, the graph of βc-continuous functions
and the βc-product functions.
3.1
Definitions and Characterizations
In this section, we define a new type of continuous functions called βc-continuous functions
and obtain some of their characterizations.
Definition 3.1.1. Let (X, τ ) and (Y, σ) be two topological spaces. The function
f : X → Y is called βc-continuous function at a point x ∈ X if for each open set V of Y
containing f (x), there exists a βc-open set U of X containing x such that f (U ) ⊆ V . If
41
f is βc-continuous at every point x of X, then it is called βc-continuous.
Proposition 3.1.2. A function f : X → Y is βc-continuous if and only if the inverse
image of every open set in Y is βc-open set in X.
Proof. (⇒) Let f be βc-continuous function and V be an open set in Y . If f −1 (V ) = ∅,
then f −1 (V ) is βc-open set in X as we need. If not, then for any x ∈ f −1 (V ), f (x) ∈ V
so by βc-continuity, there exists βc-open set U in X containing x such that f (U ) ⊆ V .
Hence, x ∈ U ⊆ f −1 (V ) and therefore, by Proposition 2.1.13, f −1 (V ) is βc-open set in
X.
(⇐) Assume that the inverse image of every open set in Y is βc-open set in X and let
V be an open set in Y containing f (x). Then, x ∈ f −1 (V ) which is βc-open set by our
assumption and f ( f −1 (V )) ⊆ V . Therefore, f is βc-continuous function
Remark 3.1.3. Their is no relation between the continuous function and βc-continuous
function. Consider the following examples.
Example 3.1.4. [ βc-continuous not continuous function ]
Let X = {a, b, c} with the topology τ = {∅, X, {a}, {b}, {a, b}} and Y = {1, 2, 3, 4} with
the topology σ = {∅, Y, {1}, {1, 2}, {1, 2, 3}}. Define f : X → Y where f (a) = 1, f (b) = 3
and f (c) = 1. Then, f is βc-continuous but not continuous function because f −1 ({1})
= {a, c} which is not open in X while {1} is open in Y
Example 3.1.5. [ continuous not βc-continuous function ]
Let X = {a, b, c} and define the topology τ = {∅, X, {a}, {b}, {a, b}}. Let f : X → X be
the identity map. Then, f is continuous function but not βc-continuous function because
{a} is an open set in X and f −1 ({a}) = {a} is not βc-open.
Corollary 3.1.6. Every βc-continuous function is β-continuous function.
42
Proof. This follows from the fact that every βc-open set is β-open set.
Remark 3.1.7. The converse of Corollary 3.1.6 is not always true. Consider the following
example.
Example 3.1.8. Let X = {a, b, c} and define the topology τ = {∅, X, {a}, {b}, {a, b}}. Let
f : X → X be the identity map. Then, f is β-continuous function but not βc-continuous
function because {a} is an open set in X and f −1 ({a}) = {a} is not βc-open.
Proposition 3.1.9. Let f : X → Y be a function such that X is T1 space. Then, f is
βc-continuous function if and only if f is β-continuous function.
Proof. Assume that f is β-continuous such that X is T1 space, then for any open set V of
Y we have, f −1 (V ) is β-open set and by Proposition 2.1.15, V is βc-open set. Therefore,
f is βc-continuous. Conversely, if f is βc-continuous, then by Corollary 3.1.6, f is βcontinuous function.
Proposition 3.1.10. A function f : X → Y is βc-continuous if and only if f is βcontinuous function and for each x ∈ X and each open set V of Y such that f (x) ∈ V ,
there exists a closed set F in X containing x such that f (F ) ⊆ V .
Proof. Let f : X → Y be βc-continuous function. Then, f is β-continuous and if x ∈ X
such that f (x) ⊆ V where V is an open set in Y , then by βc-continuity there exists
βc-open set U of X such that x ∈ U and f (U ) ⊆ V , but U is βc-open set which implies
there exists a closed set F in X such that x ∈ F ⊆ U . Therefore, f (F ) ⊆ f (U ) ⊆ V and
hence, f (F ) ⊆ V . Conversely, Let V be an open set in Y , then by assumption, f −1 (V )
is β-open set and for any x ∈ f −1 (V ) we have, f (x) ∈ V and so, there exists a closed set
F in X containing x such that f (F ) ⊆ V , which implies, x ∈ F ⊆ f −1 (V ). Therefore,
by Definition 2.1.1, f −1 (V ) is βc-open set in X and hence, by Proposition 3.1.2, f is
βc-continuous fu1ction.
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Proposition 3.1.11. If f : X → Y is a strongly θ-continuous function, then f is βccontinuous function.
Proof. Let f be a strongly θ-continuous function and consider an open set V ⊆ Y , then by
Definition 1.2.20, f −1 (V ) is θ-open set. But by Corollary 2.1.23, every θ-open is βc-open
set and hence, f −1 (V ) is βc-open set. Therefore, by Proposition 3.1.2, f is βc-continuous
function.
Remark 3.1.12. The converse of Proposition 3.1.11 need not be true. Consider the following example.
Example 3.1.13. Consider the function defined in Example 3.1.4, then f is βc-continuous
function but not strongly θ-continuous because {1} is an open set in Y but f −1 ({1})
= {a, c} which is not θ-open in X because there is no open set G in X such that
c ∈ G ⊆ cl(G) ⊆ {a, c}.
Proposition 3.1.14. Let f : X → Y be a contra continuous and β-continuous function,
then f is βc-continuous function.
Proof. Let f : X → Y be a contra continuous and β-continuous function and consider V
as open subset of Y . Then, by Definition 1.2.20, f −1 (V ) is closed and β-open set in X
which implies , f −1 (V ) is βc-open set. Hence, f is βc-continuous function.
Remark 3.1.15. The converse of Proposition 3.1.14 need not be true in general. Consider
the following example.
Example 3.1.16. Let X = R with the usual topology and consider f : R → R be the
identity map. Then, if V is open in R, then f −1 (V ) = V is open in R which implies that
V is β-open and also, V is a union of closed sets namely its singletons and hence, f −1 (V )
44
is βc-open and so, f is βc-open. On the other hand, f is not contra continuous because,
(0, 1) is open and f −1 ((0, 1)) = (0, 1) which is not closed in the usual topology.
Proposition 3.1.17. Let X be an Alexandroff space. Then, f : X → Y is a contra
continuous and β-continuous function if and only if f is βc-continuous function.
Proof. Let X be an Alexandroff space and let f : X → Y be βc-continuous function.
Then, for any open set V in Y we have, f −1 (V ) is βc-open set in X which implies f −1 (V )
is β-open and f −1 (V ) is a union of closed sets which is closed in an Alexandroff space X.
Therefore, f is β-continuous and contra continuous function. The converse part is got
directly from Proposition 3.1.14.
Proposition 3.1.18. Let f : X → Y be a perfectly continuous function, then f is βccontinuous function.
Proof. Let f : X → Y be a perfectly continuous function and consider V be an open
subset of Y . Then, by Definition 1.2.20, f −1 (V ) is a clopen set in X which implies ,
f −1 (V ) is open set and so, β-open set and f −1 (V ) is closed set. Hence, f −1 (V ) is βc-open
set in X and therefore, f is βc-continuous function.
Remark 3.1.19. The converse of Proposition 3.1.18 need not be true in general. Consider
the following example.
Example 3.1.20. Consider the function defined in Example 3.1.4, then f is βc-continuous
function but not perfectly continuous because {1} is an open set in Y but f −1 ({1}) = {a, c}
which is not open in X and hence, f −1 ({1}) is not clopen set in X.
Proposition 3.1.21. Let X be an extremally disconnected Alexandroff space. Then,
f : X → Y is perfectly continuous function if and only if f is βc-continuous function.
45
Proof. (⇒) Let f be perfectly continuous, then by Proposition 3.1.18, f is βc-continuous.
(⇐) Let f : X → Y be βc-continuous and V be an open set in Y . Then f −1 (V ) is βc-open
set in X. Since X is Alexandroff and f −1 (V ) is union of closed sets, f −1 (V ) is closed
set. So that f −1 (V ) is closed and β-open set which implies, f −1 (V ) ⊆ cl(int(cl(f −1 (V )))).
Therefore, f −1 (V ) ⊆ cl(int(f −1 (V ))) ⊆ cl(f −1 (V )) = f −1 (V ). Hence, f −1 (V ) = cl(int(f −1 (V ))).
But X is extremally disconnected, cl(int(f −1 (V ))) is open and hence f −1 (V ) is open and
also closed. Thus, by Definition 1.2.20, f is perfectly continuous function.
Proposition 3.1.22. Let f : X → Y be RC-continuous function, then f is βc-continuous
function.
Proof. Let f : X → Y be RC-continuous function and let V be an open subset in Y .
Then by Definition 1.2.20, f −1 (V ) is regular closed set in X and by Proposition 2.1.14,
we have, f −1 (V ) is βc-open set. Therefore, f is βc-continuous function.
Proposition 3.1.23. Let f : X → Y be a function such that X is a locally indiscrete
topological space. Then, f is semi-continuous if and only if f is βc-continuous function.
Proof. Assume that f is semi-continuous such that X is a locally indiscrete space, then for
any open set V of Y we have, f −1 (V ) is semi-open set and by Proposition 2.1.16, f −1 (V )
is βc-open set. Therefore, f is βc-continuous. Conversely, if f is βc-continuous, then for
S
any open set V of Y , f −1 (V ) is βc-open set which implies that, f −1 (V ) = x∈f −1 (V ) Fx
where Fx is closed set in X for each x ∈ f −1 (V ). But X is a locally indiscrete so that, Fx
is open for each x and therefore, f −1 (V ) is open set because it is union of open sets and
hence, f −1 (V ) is semi-open. So that, f is semi-continuous function.
Corollary 3.1.24. Let f : X → Y be a function such that X is a locally indiscrete topological space. Then, f is continuous function if and only if f is βc-continuous function.
46
Proof. Directly, by using Remark 2.1.17 and Proposition 3.1.23.
Proposition 3.1.25. Let f : X → Y be a function such that X is a regular space. Then,
if f is continuous function, then f is βc-continuous function.
Proof. Let f be continuous function such that X is regular space. Then for any open set
V of Y we have f −1 (V ) is open set in a regular space X and so, by Proposition 2.1.18, V
is βc-open set. Therefore, f is βc-continuous.
Theorem 3.1.26. Let (X, τ ) and (Y, σ) be two topological spaces and consider f : X → Y
be a function. Then, the following are equivalent:
(1). f is βc-continuous function.
(2). f −1 (V ) is βc-open set in X for any open set V of Y .
(3). f −1 (F ) is βc-closed set in X for any closed set F of Y .
(4). f (βc-cl(A)) ⊆ cl(f (A)) for any subset A of X.
(5). βc-cl(f −1 (B)) ⊆ f −1 (cl(B)) for any subset B of Y .
(6). f −1 (int(B)) ⊆ βc-int(f −1 (B)) for any subset B of Y .
(7). int(f (A)) ⊆ f (βc-int(A)) for any subset A of X.
Proof. (1) ⇒ (2) Follows from Proposition 3.1.2.
(2) ⇒ (3) Let F be any closed set F of Y , then Y \F is open set of Y and by part (2) we
have, f −1 (Y \F ) = X\f −1 (F ) is βc-open set in X and hence, f −1 (F ) is βc-closed set in
X.
(3) ⇒ (4) Let A be any subset of X. Then, f (A) ⊆ cl(f (A)) where cl(f (A)) is a closed
set in Y . Hence, by βc-continuity and using part (3) we have, f −1 (cl(f (A))) is βc-closed
set in X and A ⊆ f −1 (cl(f (A))). Therefore, βc-cl(A) ⊆ f −1 (cl(f (A))) and hence, f (βccl(A)) ⊆ cl(f (A)).
(4) ⇒ (5) Let B be any subset of Y . Then, f −1 (B) is a subset of X and by using part (4)
47
we have, f (βc-cl(f −1 (B))) ⊆ cl(f (f −1 (B))) ⊆ cl(B). Hence, βc-cl(f −1 (B)) ⊆ f −1 (cl(B)).
(5) ⇒ (6) Let B be a subset of Y , then using part (5) to Y \B we have, βc-cl(f −1 (Y \B)) ⊆
f −1 (cl(Y \B)) which implies, βc-cl(X\f −1 (B)) ⊆ f −1 (Y \int(B)) and so, X\βc-int(f −1 (B))
⊆ X\f −1 (int(B)). Hence, f −1 (int(B)) ⊆ βc-int(f −1 (B)).
(6) ⇒ (7) Let A be a subset of X, then f (A) is a subset of Y and by using part (6) we
have, f −1 (int(f (A))) ⊆ βc-int(f −1 (f (A))) ⊆ βc-int(A). Therefore, int(f (A)) ⊆ f (βcint(A)).
(7) ⇒ (1) Let x ∈ X and let V be any open set of Y containing f (x), then x ∈ f −1 (V )
and f −1 (V ) is a subset of X. By part (7), int(f (f −1 (V ))) ⊆ f (βc-int(f −1 (V ))) which
implies, int(V ) ⊆ f (βc-int(f −1 (V ))) but V is an open set so that, V ⊆ f (βc-int(f −1 (V )))
which implies, f −1 (V ) ⊆ βc-int(f −1 (V )). Therefore, f −1 (V ) is βc-open set in X which
contains x and f (f −1 (V )) ⊆ V . Hence, f is βc-continuous function.
Proposition 3.1.27. A function f : X → Y is βc-continuous if and only if βc-F r(f −1 (A))
⊆ f −1 (F r(A)) for every A ⊆ Y .
Proof. Assume that f is βc-continuous, then f −1 (F r(A)) = f −1 (cl(A)\int(A)) = f −1 (cl(A))
\ f −1 (int(A)) but using parts (5), (6) of Theorem 3.1.26, we have, f −1 (cl(A)) \ f −1 (int(A))
⊇ βc-cl(f −1 (A)) \ βc-int(f −1 (A)) = βc-F r(f −1 (A)). Hence, f −1 (F r(A)) ⊇ βc-F r(f −1 (A)).
Conversely, Let V be a closed set in Y , then βc-F r(f −1 (V )) ⊆ f −1 (F r(V )) but f −1 (F r(V ))
⊆ f −1 (cl(V )) = f −1 (V ). Hence, βc-F r(f −1 (V )) ⊆ f −1 (V ) and by Theorem 2.3.45, f −1 (V )
is βc-closed set in X. Therefore, f is βc-continuous function.
Proposition 3.1.28. A function f : X → Y is βc-continuous if and only if f (βc-D(A))
⊆ cl(f (A)) for every A ⊆ X.
Proof. Let f be βc-continuous function and let A ⊆ X, then cl(f (A)) is a closed set in
Y which implies, by βc-continuity, f −1 (cl(f (A))) is βc-closed set in X which containing
48
A. So that, A ⊆ f −1 (cl(f (A))) but by Theorem 2.3.37, βc-D(A) ⊆ βc-cl(A) ⊆ βccl(f −1 (cl(f (A)))) = f −1 (cl(f (A))). Therefore, f (βc-D(A)) ⊆ cl(f (A)).
On the other hand, let V be a closed subset of Y , then f −1 (V ) is a subset of X and
by assumption, f (βc-D(f −1 (V ))) ⊆ cl(f (f −1 (V ))) ⊆ cl(V ) = V . Hence, βc-D(f −1 (V ))
⊆ f −1 (V ) and so, by Proposition 2.3.39, f −1 (V ) is βc-closed in X. Therefore, f is βccontinuous.
Proposition 3.1.29. Let f : X → Y be a function and let B be a base for a topology
on Y . Then, f is βc-continuous if and only if f −1 (B) is βc-open subset of X for each
B ∈ B.
Proof. N ecessity. Suppose that f is βc-continuous. Then, since each B ∈ B is open
subset of Y so by Proposition 3.1.2, f −1 (B) is βc-open subset of X.
S
Suf f iciency. Let V be any open subset of Y . Then, V = {Bi : i ∈ I} where every Bi is
S
a member of B and I is a suitable index set. It follows that, f −1 (V ) = f −1 ( {Bi : i ∈ I})
S
= f −1 ({Bi : i ∈ I}). Since, f −1 (Bi ) is βc-open subset of X for each i ∈ I. Hence,
by Proposition 2.1.9, f −1 (V ) is βc-open set and therefore, by Proposition 3.1.2, f is
βc-continuous function.
3.2
Properties of βc-Continuous Functions
In this section, we study βc-extension functions, βc-retraction function, βc-composition
of functions and βc-graph function. Also, we define βc-irresolute and βc-open functions
and use it in composition functions.
Proposition 3.2.1. Let f : X → Y be βc-continuous function such that Y ⊆ Z. If Y is
an open subspace of the topological space Z, then f : X → Z is βc-continuous.
49
Proof. Let x ∈ X and let V be an open subset of Z such that f (x) ∈ V . Then V ∩ Y
is open subset of Y where f (x) ∈ Y for each x ∈ X which implies, f (x) ∈ V ∩ Y .
Since f : X → Y is βc-continuous, there exists βc-open set U in X containing x such
that f (U ) ⊆ V ∩ Y ⊆ V . Therefore, by Definition 3.1.1, f : X → Z is βc-continuous
function.
Proposition 3.2.2. Let f : X → Y be βc-continuous function such that A ⊆ X. If A
is a clopen subset of the topological space X, then f |A : A → Y is βc-continuous in the
subspace A.
Proof. Let V be an open subset of Y . Then by Proposition 3.1.2, f −1 (V ) is βc-open subset
in X. Since A is clopen set in X, then by Corollary 2.1.30, (f |A)−1 (V ) = f −1 (V ) ∩ A is
βc-open set in the subspace of A. Therefore, f |A : X → Y is βc-continuous.
Proposition 3.2.3. Let f : X → Y be a function such that for each x ∈ X, there exists
a clopen set A of X where x ∈ A and f |A : A → Y is βc-continuous function, then f is
βc-continuous function.
Proof. Assume that for each x ∈ X, there exists a clopen set A of X where x ∈ A and
f |A : A → Y is βc-continuous function. Let V be an open subset of Y containing f (x),
then there exists βc-open set U in A containing x such that f |A(U ) ⊆ V . Since A is
clopen set in X. Then by Proposition 2.1.31, U is βc-open set in X and hence, f (U ) ⊆ V .
Therefore, f is βc-continuous function.
Proposition 3.2.4. Let f : X → Y be a function such that X = A ∪ B where both A
and B are clopen sets. If f |A : A → Y and f |B : B → Y are βc-continuous functions,
then f is βc-continuous function.
Proof. Let V be any open set of Y . Then f −1 (V ) = (f |A)−1 (V ) ∪ (f |B)−1 (V ). But f |A
and f |B are βc-continuous functions so by Proposition 3.1.2, (f |A)−1 (V ) and (f |B)−1 (V )
50
are βc-open sets in A and B respectively. Since A and B are clopen sets in X, by
Proposition 2.1.31, (f |A)−1 (V ) and (f |B)−1 (V ) are βc-open sets in X. Since the union
of βc-open sets is βc-open, f −1 (V ) is βc-open in X. Therefore, by Proposition 3.1.2, f is
βc-continuous function.
Proposition 3.2.5. Let f, g : X → Y be two functions such that Y is Hausdroff. If f is
βc-continuous and g is perfectly continuous, then the set E = {x ∈ X : f (x) = g(x)} is
βc-closed in X.
Proof. Let x ∈
/ E. Then f (x) 6= g(x) but Y is Hausdroff, so there exist disjoint open sets
V1 and V2 of Y such that f (x) ∈ V1 and g(x) ∈ V2 . Since f is βc-continuous, there exists
βc-open set U1 of X containing x such that f (U1 ) ⊆ V1 . Since g is perfectly continuous,
there exists a clopen set U2 of X containing x such that g(U2 ) ⊆ V2 . But U1 ∩ U2 is
βc-open set of X containing x and U1 ∩ U2 ∩E = ∅. So that, U1 ∩ U2 ⊆ X\E and hence,
X\E is βc-open. Therefore, E is βc-closed set.
Corollary 3.2.6. Let f, g : X → Y be two functions such that Y is Urysohn space. If f
is βc-continuous and g is perfectly continuous, then the set E = {x ∈ X : f (x) = g(x)}
is βc-closed in X.
Proof. Directly from Remark 1.1.29 and Proposition 3.2.5.
Proposition 3.2.7. Let f : X1 → Y and g : X2 → Y be two βc-continuous functions
such that Y is Hausdroff, then the set E = {(x1 , x2 ) ∈ X1 × X2 : f (x1 ) = g(x2 )} is
βc-closed in X1 × X2 .
Proof. Let (x1 , x2 ) ∈
/ E. Then f (x1 ) 6= g(x2 ). Since Y is Hausdorff, there exist disjoint
open sets V1 and V2 of Y such that f (x1 ) ∈ V1 and g(x2 ) ∈ V2 . Since f and g are βccontinuous functions, there exists βc-open sets U1 and U2 of X1 and X2 containing x1
51
and x2 such that f (U1 ) ⊆ V1 and f (U2 ) ⊆ V2 respectively. So that, by Proposition 2.1.32,
(x1 , x2 ) ∈ U1 × U2 where U1 × U2 is βc-open in X1 × X2 and U1 × U2 ∩E = ∅. Therefore,
U1 × U2 ⊆ X1 × X2 \E and hence X1 × X2 \E is βc-open set and E is βc-closed set in
X1 × X2 .
Corollary 3.2.8. Let f : X1 → Y and g : X2 → Y be two βc-continuous functions
such that Y is Urysohn space, then the set E = {(x1 , x2 ) ∈ X1 × X2 : f (x1 ) = g(x2 )} is
βc-closed in X1 × X2 .
Proof. Directly from Remark 1.1.29 and Proposition 3.2.7.
Definition 3.2.9. A function f : X → Y is called:
(i) βc-irresolute if for every βc-open subset G of Y , f −1 (G) is βc-open in X.
(ii) βc-open if for every βc-open subset H of X, f (H) is βc-open in Y .
Proposition 3.2.10. Let f : X → Y and g : Y → Z be two functions, then the following
properties hold.
(1) If f is βc-continuous and g is continuous, then g ◦ f is βc-continuous.
(2) If f is continuous and g is perfectly continuous, then g ◦ f is βc-continuous.
(3) If f is βc-irresolute and g is βc-continuous, then g ◦ f is βc-continuous.
Proof. (1) Let V be open set in Z. Then g −1 (V ) is open in Y by continuity of g. Since
f is βc-continuous, f −1 (g −1 (V )) is βc-open in X and hence, (g ◦ f )−1 (V ) = f −1 (g −1 (V ))
is βc-open in X. Therefore, g ◦ f is βc-continuous.
(2) Let V be open set in Z. Then g −1 (V ) is clopen in Y by perfect continuity of g. Since
f is continuous, f −1 (g −1 (V )) is clopen in X and so βc-open by Proposition 2.2.9. Hence,
(g ◦ f )−1 (V ) = f −1 (g −1 (V )) is βc-open in X. Therefore, g ◦ f is βc-continuous.
(3) Let V be open set in Z. Since g is βc-continuous, g −1 (V ) is βc-open in Y . Since f
52
is βc-irresolute, f −1 (g −1 (V )) is βc-open in X and hence, (g ◦ f )−1 (V ) = f −1 (g −1 (V )) is
βc-open in X. Therefore, g ◦ f is βc-continuous.
Proposition 3.2.11. Let f : X → Y and g : Y → Z be two functions. If f is βc-open
and surjective and g ◦ f is βc-continuous, then g is βc-continuous.
Proof. Let V be open set in Z. Since g ◦ f is βc-continuous, (g ◦ f )−1 (V ) = f −1 (g −1 (V ))
is βc-open set in X. Since f is βc-open and surjective, then f (f −1 (g −1 (V ))) = g −1 (V ) is
βc-open set in y. Hence, g is βc-continuous.
Corollary 3.2.12. Let f : X → Y be βc-open, βc-irresolute and surjective and g : Y → Z
be functions. Then, g ◦ f : X → Z is βc-continuous if and only if g is βc-continuous.
Proof. Follows directly by Part (3) of Proposition 3.2.10 and Proposition 3.2.11.
Theorem 3.2.13. Let f : X → Y be a function and let G : X → X × Y be the graph
function of f defined by G(x) = (x, f (x)) where G(f ) = {(x, f (x)) : x ∈ X}. Then, G is
βc-continuous if and only if f is βc-continuous.
Proof. (⇒) Assume that G is βc-continuous and let V be an open set in Y , then X × V
is open set in X × Y . Since G is βc-continuous, G−1 (X × V ) is βc-open in X. But
G−1 (X × V ) = {x ∈ X : G(x) = (x, f (x)) ∈ X × V } = {x ∈ X : f (x) ∈ V } = f −1 (V ) is
βc-open set in X. Thus, by Proposition 3.1.2, f is βc-continuous.
(⇐) Assume that f is βc-continuous and let x ∈ X such that W is an open subset of
X × Y containing G(x). Since by Theorem 1.1.26, {x} × Y is homeomorphic to Y and
W ∩ ({x} × Y ) is open in the subspace {x} × Y containing G(x), {y ∈ Y : (x, y) ∈ W }
is open subset of Y . Since f is βc-continuous, f −1 ({y : (x, y) ∈ W }) is βc-open set in
S
X. But f −1 ({y : (x, y) ∈ W }) = {f −1 (y) : (x, y) ∈ W } is βc-open set in X and x ∈
S −1
{f (y) : (x, y) ∈ W } ⊆ G−1 (W ) . Hence, G−1 (W ) is βc-open set in X and therefore,
by Proposition 3.1.2, G is βc-continuous function.
53
Proposition 3.2.14. Let f : X → Y be βc-continuous function such that Y is Hausdorff,
then G(f ) = {(x, f (x)) : x ∈ X} is βc-closed set in X × Y .
Proof. Let (x, y) ∈
/ G(f ), then y 6= f (x). Since Y is Hausdorff, there exist disjoint open
sets V and W such that f (x) ∈ W and y ∈ V . Since f is βc-continuous, there exists
βc-open set U in X containing x such that f (U ) ⊆ W . Therefore, (x, y) ∈ U ×V ⊆ X ×Y
where U × V is βc-open set because U is βc-open and V is open in Hausdorff and hence,
by Proposition 2.1.15, V is βc-open. Hence, (X × Y )\G(f ) is βc-open set and therefore,
G(f ) is βc-closes set.
54
Chapter 4
βc-Separation Axioms
In this chapter, we introduce and study βc-g.closed sets. Also, new types of separation
axioms is given in terms of βc-open sets, βc-closed sets and βc-g.closed sets.
4.1
βc-Generalized Closed Sets
In this section, we introduce the concepts of βc-generalized closed sets in topological
spaces and study some of its fundamental properties.
Definition 4.1.1. A subset A of a topological space (X, τ ) is called βc-generalized closed
set ( briefly, βc-g.closed ) if βc-cl(A) ⊆ U whenever A ⊆ U and U is a βc-open set in
(X, τ ). A subset A of X is βc-generalized open set if its complement X\A is βc-generalized
closed set in X.
The family of all βc-g.closed ( βc-g.open ) in a topological space (X, τ ) will be denoted
by βCGC(X) (βCGO(X)).
Proposition 4.1.2. In a topological space (X, τ ), every βc-closed set is βc-g.closed.
55
Proof. Let A be a βc-closed set and let U be βc-open set such that A ⊆ U . Then, βc-cl(A)
= A ⊆ U . Hence, A is βc-g.closed.
Remark 4.1.3. The converse of Proposition 4.1.2 need not be true in general. Consider
the following example.
Example 4.1.4. Let X = {a, b, c} with the topology τ = {∅, X, {a}, {b}, {a, b}, {a, c}}.
Then βCO(X) = {∅, X, {b}, {a, c}}. If A = {a}, then {a, c} and X are the βc-open sets
contain A, βc-cl(A) = {a, c}. Hence, βc-cl(A) ⊆ {a, c} and βc-cl(A) ⊆ X. Therefore, A
is βc-g.closed but not βc-closed set.
Proposition 4.1.5. Let (X, τ ) be a topological space with a subset A. If A is βc-open set
and βc-g.closed set, then A is βc-closed set.
Proof. Let A be βc-open set and βc-g.closed set. Since A is βc-open and A ⊆ A, βc-cl(A)
⊆ A. But A ⊆ βc-cl(A). Hence, A = βc-cl(A) and A is βc-closed set.
Proposition 4.1.6. Let (X, τ ) be a topological space with subsets A and B. If A is
βc-g.closed set and B is βc-closed set, then A ∩ B is βc-g.closed set.
Proof. Let A be βc-g.closed set and B be βc-closed set. Then, if U is βc-open set such
that A ∩ B ⊆ U and if we let G = X\B, then A ⊆ U ∪ G where U ∪ G is βc-open set.
Since A is βc-g.closed set, then βc-cl(A) ⊆ U ∪ G and so, βc-cl(A ∩ B) ⊆ βc-cl(A) ∩
βc-cl(B) = βc-cl(A) ∩ B ⊆ (U ∪ G) ∩ B = (U ∩ B) ∪ (G ∩ B) ⊆ U . Therefore, A ∩ B
is βc-g.closed set.
Proposition 4.1.7. Let (X, τ ) be a topological space with subsets A and B. If A is
βc-g.closed set such that A ⊆ B ⊆ βc-cl(A), then B is βc-g.closed set.
56
Proof. Let A be βc-g.closed set such that A ⊆ B ⊆ βc-cl(A). If U is βc-open set such
that B ⊆ U , then A ⊆ U but A is βc-g.closed which implies, βc-cl(A) ⊆ βc-cl(B) ⊆
βc-cl(A) ⊆ U . That is βc-cl(B) ⊆ U and hence, B is βc-g.closed set.
Proposition 4.1.8. For each x ∈ X, either {x} is βc-closed or {x}c is βc-g.closed set
in X.
Proof. If {x} is not βc-closed, then {x}c is not βc-open set and so the only βc-open set
containing {x}c is X itself. Therefore, βc-cl({x}c ) ⊆ X and hence, {x}c is βc-g.closed set
in X.
Theorem 4.1.9. Let (X, τ ) be a topological space with a subset A. Then, A is βc-g.closed
if and only if for any x ∈ βc-cl(A) we have, βc-cl({x}) ∩ A 6= ∅.
Proof. (⇒) Let A be βc-g.closed and assume there exists x ∈ βc-cl(A) such that βccl({x}) ∩ A = ∅. Since βc-cl({x}) is βc-closed set, then X\ βc-cl({x}) is βc-open set and
A ⊆ X\ βc-cl({x}). But A is βc-g.closed, so that βc-cl(A) ⊆ X\ βc-cl({x}) and so, x ∈
/
βc-cl({x}) which is a contradiction. So that, βc-cl({x}) ∩ A 6= ∅ for any x ∈ βc-cl(A).
(⇐) Let U is βc-open set such that A ⊆ U and assume for any x ∈ βc-cl(A) we have βccl({x}) ∩ A 6= ∅. So that, there exists z ∈ A ⊆ U and z ∈ βc-cl({x}). Then, by Theorem
2.3.8, U ∩ {x} =
6 ∅ and hence, x ∈ U which implies that, βc-cl(A) ⊆ U . Therefore, A is
βc-g.closed set.
Theorem 4.1.10. Let (X, τ ) be a topological space with a subset A. Then, A is βcg.closed if and only if βc-cl(A) \A does not contain any non empty βc-closed set.
Proof. (⇒) Let A be βc-g.closed in X and assume there exists a non empty βc-closed set
B such that B ⊆ βc-cl(A) \A, then B ⊆ X\A and so, A ⊆ X\B where A is βc-g.closed
and X\B is βc-open set. Therefore, βc-cl(A) ⊆ X\B which implies, B ⊆ X\ βc-cl(A).
57
Hence, B ⊆ βc-cl(A) ∩ (X\ βc-cl(A)) = ∅ and so B = ∅ which a contradiction. So that,
βc-cl(A) \A does not contain any non empty βc-closed set.
(⇐) If βc-cl(A) \A does not contain any non empty βc-closed set and if U is βc-open set
such that A ⊆ U and βc-cl(A) * U . Then, βc-cl(A) ∩ X\U 6= ∅ and βc-cl(A) ∩ X\U ⊆
βc-cl(A) ∩ X\A = βc-cl(A) \A. Therefore, βc-cl(A) ∩ X\U ⊆ βc-cl(A) \A and βc-cl(A)
∩ X\U is a non empty βc-closed set which a contradiction. Hence, βc-cl(A) ⊆ U and so,
A is βc-g.closed set.
Proposition 4.1.11. Let (X, τ ) be a topological space. Then, the following are equivalent:
1. Every subset of X is βc-g.closed set.
2. βCO(X) = βCC(X).
Proof. (1 ⇒ 2) Let (X, τ ) be a topological space with every subset of X is βc-g.closed
set. Then, if A ∈ βCO(X) we have, A is βc-g.closed set and so, βc-cl(A) ⊆ A which
implies, βc-cl(A) = A. Therefore, A ∈ βCC(X). Conversely, if B ∈ βCC(X), then X\B
∈ βCO(X) and so, X\B ∈ βCC(X) which implies B ∈ βCO(X). Therefore, βCO(X)
= βCC(X).
(2 ⇒ 1) Let βCO(X) = βCC(X). Then, for any A ⊆ U where U is βc-open set, U is
βc-closed set and hence, βc-cl(A) ⊆ U . Therefore, A is βc-g.closed set.
Definition 4.1.12. Let (X, τ ) be a topological space with a subset A. Then, the intersection of all βc-open subsets of X containing A is called the βc-kernel of A and its
denoted by βc-ker(A).
Theorem 4.1.13. A subset A of a space (X, τ ) is βc-g.closed if and only if βc-cl(A) ⊆
βc-ker(A).
Proof. (⇒) Suppose that A be βc-g.closed set in X. and Let x ∈ βc-cl(A) such that
x∈
/ βc-ker(A). Then, there exists βc-open set V such that A ⊆ V and x ∈
/ V . But A is
58
βc-g.closed which implies, βc-cl(A) ⊆ V and hence, x ∈
/ βc-cl(A) which is a contradiction.
So that, x ∈ βc-ker(A) and βc-cl(A) ⊆ βc-ker(A).
(⇐) Let βc-cl(A) ⊆ βc-ker(A) and let U be βc-open set such that A ⊆ U . Then, βc-cl(A)
⊆ βc-ker(A) ⊆ U . Therefore, A is βc-g.closed set.
4.2
New Types of Separation Axioms
Definition 4.2.1. A topological space (X, τ ) is said to be:
(1.) βc-T0 if for every distinct points x and y of X, there is βc-open set containing one of
them but not the other.
(2.) βc-T 1 if every βc-g.closed set in X is βc-closed set in X.
2
(3.) βc-T1 if for every distinct points x and y of X, there is βc-open set U in X containing
x but not y and βc-open set V in X containing y but not x.
(4.) βc-T2 if for every distinct points x and y of X, there is βc-open set U and V in X
containing x and y respectively such that U ∩ V = ∅.
Remark 4.2.2. We will see later, that βc-T2 ⇒ βc-T1 ⇒ βc-T 1 ⇒ T0 . The converse is not
2
true, consider the following examples.
Example 4.2.3. [βc-T0 space not βc-T 1 space ]
2
Consider X = {a, b, c} with a topological space τ = {∅, X{a}, {b}, {a, b}}. Then βCO(X, τ ) =
{∅, X, {a, c}, {b, c}}. Then X is βc-T0 but not βc-T 1 because {c} is βc-g.closed but is not
2
βc-closed.
Example 4.2.4. [ βc-T1 space not βc-T2 space ]
Let X be any infinite set with the cof inite topology ( in which the closed sets are the finite
sets and X). Since X\{x} is βc-open, X is βc-T1 . But no non empty βc-open sets are
disjoint, so X can not be βc-T2 .
59
Proposition 4.2.5. A topological space (X, τ ) is βc-T0 if and only if for each pair of
distinct points x and y of X, βc-cl({x}) 6= βc-cl({y}).
Proof. Assume that (X, τ ) is βc-T0 and consider x, y be two distinct points of X. Then
there is βc-open set U in X containing x but not y which implies, y ∈ X\U where X\U
is βc-closed set. Hence, βc-cl({y}) ⊆ X\U and x ∈
/ βc-cl({y}) and so βc-cl({x}) 6= βccl({y}). Conversely, Let x, y ∈ X such that x 6= y and βc-cl({x}) 6= βc-cl({y}). Then
either βc-cl({x}) * βc-cl({y}) or βc-cl({y}) * βc-cl({x}). Take βc-cl({x}) * βc-cl({y}).
Then, X\βc-cl({y}) is βc-open set where y ∈
/ X\βc-cl({y}) and x ∈ X\βc-cl({y}) because
if x ∈
/ X\βc-cl({y}), then x ∈ βc-cl({y}) where βc-cl({y}) is βc-closed set and hence, βccl({x}) ⊆ βc-cl({y}) which contradicts with our assumption. Therefore, X is βc-T0 .
Proposition 4.2.6. A topological space (X, τ ) is βc-T 1 if and only if for each x ∈ X,
2
{x} is either βc-closed or {x} is βc-open set.
Proof. (⇒) Suppose that {x} is not βc-closed, then by Proposition 4.1.8, X\{x} is βcg.closed. Since X is βc-T 1 , X\{x} is βc-closed. Thus, {x} is βc-open set.
2
(⇐) Let A be βc-g.closed in X and let x ∈ βc-cl(A). By hypothesis, {x} is βc-closed or
βc-open. If {x} is βc-closed, then x ∈
/ A implies, x ∈ βc-cl(A)\A, which is a contradiction
by Theorem 4.1.10. Hence, x ∈ A and βc-cl(A) = A. On the second case, if {x} is βcopen where x ∈ βc-cl(A), {x} ∩ A 6= ∅. Hence, x ∈ A and A is βc-closed. Thus, in both
two cases, A is βc-closed and therefore, X is βc-T 1 .
2
Corollary 4.2.7. A topological space X is βc-T 1 if and only if βCGC(X) = βCC(X).
2
Proof. Let X be βc-T 1 , then every βc-g.closed is βc-closed, that is, βCGC(X) ⊆ βCC(X),
2
but by Proposition 4.1.2, βCC(X) ⊆ βCGC(X). Thus, βCGC(X) = βCC(X). Conversely, Let βCGC(X) = βCC(X), then if A is βc-g.closed which implies, A is βc-closed.
Hence, X is βc-T 1 .
2
60
Corollary 4.2.8. A topological space X is βc-T 1 if and only if every βc-g.open set in X
2
is βc-open set in X.
Proof. Directly, from Corollary 4.2.7.
Corollary 4.2.9. If a topological space (X, τ ) is βc-T 1 , then X is βc-T0 .
2
Proof. Let X be βc-T 1 and let x 6= y in X. Then, by Proposition 4.2.6, {x} is either
2
βc-closed or βc-open. If {x} is βc-closed, then U = X\{x} is βc-open set contains y
and not x. Hence, X is βc-T0 . In the second case, if {x} is βc-open, then x ∈ {x} and
y∈
/ {x}. Therefore, X is βc-T0 .
Proposition 4.2.10. A topological space (X, τ ) is βc-T1 if and only if for each x ∈ X,
{x} is βc-closed set.
Proof. (⇒) Let x ∈ X. Since X is βc-T1 , for each y ∈ X\{x} there exists βc-open set Vy
such that y ∈ Vy and x ∈
/ Vy . So, {x} ∩ Vy = ∅ and so, Vy ⊆ X\{x}. Hence, we get that,
for any y ∈ X\{x} there exists βc-open set Vy such that y ∈ Vy ⊆ X\{x}. Hence, X\{x}
is βc-open set and so, {x} is βc-closed set.
(⇐) Let x 6= y in X. Then, since {y} is βc-closed, U = X\{y} is βc-open in X contains
x and not y. Similarly, V = X\{x} is βc-open set contains y and not x. Thus, X is βc-T1
space.
Proposition 4.2.11. A topological space (X, τ ) is βc-T1 if and only if βc-D({x}) = ∅
for each x ∈ X.
Proof. Let X be βc-T1 and assume that βc-D({x}) 6= ∅ for some x ∈ X, then there exists
y ∈ βc-D({x}) and y 6= x. Since X is βc-T1 , there exists βc-open set U such that y ∈ U
and x ∈
/ U which implies, U ∩ {x} = ∅ and hence, y ∈
/ βc-D({x}) which a contradiction.
61
Thus, βc-D({x}) = ∅ for each x ∈ X. Conversely, Let βc-D({x}) = ∅ for each x ∈ X,
then βc-cl({x}) = {x} ∪ βc-D({x}) = {x} which implies, {x} is βc-closed and hence, by
Proposition 4.2.10, X is βc-T1 .
Proposition 4.2.12. Let (X, τ ) be the discrete topology, then X is βc-T1 .
Proof. Let (X, τ ) be the discrete topology, then by Proposition 2.2.10, βCO(X, τ ) forms
the discrete topology. So that, for any x ∈ X, {x} is βc-closed set and hence, by Proposition 4.2.10, X is T1 space.
Remark 4.2.13. The converse of Proposition 4.2.12, if X is βc-T1 , then X is the discrete
topology need not be true in general. Consider the following example.
Example 4.2.14. Consider the the infinite set X with cof inite topology in Example 4.2.5,
then X is βc-T1 where X is not the discrete topology.
Proposition 4.2.15. The following statements are equivalent for a topological space(X, τ ):
(1). X is βc-T2 .
(2). For each y ∈ X\{x}, there exists βc-open set U such that x ∈ U and y ∈
/ βc-cl(U ).
T
(3). For each x ∈ X, {βc-cl(U ) : U ∈ βCO(X) and x ∈ U } = {x}.
Proof. (1 ⇒ 2) Since X is βc-T2 , there exist disjoint βc-open sets U and V containing
x and y respectively. So, U ⊆ X\V where X\V is βc-closed set. Therefore, βc-cl(U )
⊆ X\V and hence, y ∈
/ βc-cl(U ).
(2 ⇒ 3) Assume there exists y distinct from x in X such that y ∈
T
{βc-cl(U ) : U ∈
βCO(X) and x ∈ U }, then y 6= x such that y ∈ βc-cl(U ) for every βc-open set U
containing x, which is a contradiction.
(3 ⇒ 1) Let y 6= x in X, then there exists βc-open set U such that x ∈ U and y ∈
/ βc-cl(U ).
Let V = X\βc-cl(U ), then y ∈ V and U ∩ V = ∅. Thus, X is βc-T2 .
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Theorem 4.2.16. A topological space (X, τ ) is βc-Tk space, k = {0, 1, 2} if for each pair
of distinct points x, y of X, there exists βc-continuous function f from X into Tk space
Y such that f (x) 6= f (y).
Proof. We prove only for k = 2 and the other are similar. Let x, y be any pair of distinct
points points in X. So by hypothesis, there exists βc-continuous function f from X into
T2 space Y such that f (x) 6= f (y). Thus, there exist disjoint open sets Hx and Hy such
that f (x) ∈ Hx and f (y) ∈ Hy . Since f is βc-continuous, f −1 (Hx ) and f −1 (Hy ) are
disjoint βc-open sets in X containing x and y respectively. Hence, X is βc-T2 space.
Proposition 4.2.17. Let f : X → Y be βc-irresolute injective function and Y is βc-T2
space, then X is βc-T2 space, and repeat this for separation axiom βc-T1 .
Proof. Let x1 , x2 be any two distinct points in X. Since f is injective, there exist distinct
points y1 , y2 in Y such that y1 = f (x1 ) and y2 = f (x2 ). Since Y is βc-T2 , there exist
disjoint βc-open sets U and V such that y1 ∈ U and y2 ∈ V which implies, x1 ∈ f −1 (U )
and x2 ∈ f −1 (V ). Since f is βc-irresolute injective, f −1 (U ) and f −1 (V ) are disjoint βcopen sets in X. Thus, for two distinct points x1 , x2 in X there exist disjoint βc-open sets
f −1 (U ) and f −1 (V ) containing x1 , x2 respectively. Hence, X is βc-T2 space.
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Conclusion
In this thesis, we studied the new class βc-open sets in topological space. The location of
βc-open sets was introduced. Moreover, The βc-operators were investigated. Also, a new
class of continuous function with characterizations were introduced. Finally, the definition
of βc-g.closed sets and new separation axioms were studied. This thesis will open a new
way for other researcher to study the applications of βc-open sets in Alexandroff topology.
Also, the relations among the classes of continuous functions.
64
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