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Asian Journal of Current Engineering and Maths 3: 2 March – April (2014) 29 - 32.
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ASIAN JOURNAL OF CURRENT ENGINEERING AND MATHS
Journal homepage: http://www.innovativejournal.in/index.php/ajcem
ON GENERALIZED E-CLOSED SETS AND E-CONTINUOUS FUNCTIONS
A.P. Dhana Balan*, C.Santhi.
Department of Mathematics, Alagappa Govt. Arts college, Karaikudi-630003, Tamil Nadu, India.
ARTICLE INFO
ABSTRACT
Corresponding Author
A.P. Dhana Balan
Department of Mathematics,
Alagappa Govt. Arts college,
Karaikudi-630003, Tamil Nadu,
India.
In this paper, the authors introduce and investigate some new classes of sets
and some new classes of continuity namely generalized e-closed sets, δgeneralized e-closed sets and g-e-continuous, super-e-continuous, perfectly econtinuous functions and give several characterizations of such functions.
Key words: δ-e-open, δ-e-closed,
perfectly e-continuous, super econtinuous.
1. INTRODUCTION
In 1963, Levine[9] introduced and investigated
the semi open sets and semi continuous functions. In
1987, Bhattacharyya and Lahiri [1] used semi open sets to
define and investigate the notion of semi generalized
closed sets. In topology weak forms of open sets play an
important role in the generalization of various forms of
continuity. Using various forms of open sets, many authors
have introduced and studied various types of continuity.
The importance of continuity and generalized
continuity is significant in various areas of mathematics
and related sciences. One of them, which has been in
recent years of interest to general topologists, is its
decomposition. The decomposition of continuity has been
studied by many authors. The purpose of this note is to
present some new decomposition of continuity and study
the classes of sets in connection with studying the further
properties of g-e-closed, δ-generalized e-closed functions
due to [5,6,7]
All topological spaces considered in this paper
lack any separation axioms unless explicitly stated. The
topology of a space X is denoted by τ and (X, τ) will be
replaced by X if there is no chance for confusion. For a
subset A of X, the closure of A and the interior of A in X are
denoted by cl(A) and int(A) respectively. The end or the
omission of a proof will be denoted by.
A subset A of a space (X, τ) is called semiopen [9]
(resp.preopen[10], δ-semiopen[14],
δ-preopen[15]) if
A ⊂ cl(int(A)) (resp. A ⊂ int(cl(A)), A ⊂ cl(δ -int(A)), A ⊂
A subset A of a space X is said to be
int(δ -cl(A))).
regular open(resp. regular closed) if A = int(cl(A)) (resp. A
= cl(int(A))).
The δ-interior [16] of a set A of X is the union of all
regular open sets of X contained in A and is denoted by δint(A). The subset A is called δ-open if A = intδ(A), ie a set A
is δ-open if it is the union of regular open sets. The
complement of δ-open set is called δ-closed. Alternatively
a set A ⊂ X is δ-closed if A = clδ(A), where clδ(A) = {x∈X :
cl(U)∩A≠φ,U∈τ and x∈U}. The family of all δ-open sets
©2014, AJCEM, All Right Reserved.
29
forms a topology on X. In any space, a singleton is δ-open if
and only if it is regular open. The family of regular open
sets forms a base for a smallest topology τs called the semi
regularization. A detailed study of the relationship
between τ and τs is made in Mrsevic, Reilly and
Vamanamurthy[11].
Definition 1.1 [5] A subset A of a topological space X is
said to be
(i) e-open if A ⊂ cl(δ-int(A))∪ int(δ-cl(A));
(ii) e-closed if cl(δ-int(A)) ∩ int(δ-cl(A)) ⊂ A.
Example 1.2 (i) Let X = {a,b,c} and let τ = {φ,
X,{a},{b},{a,b}}.Then the set {b,c} is e-open.
(ii) Let X = {a,b,c,d} and let τ = {φ,
X,{a},{c},{a,b},{a,c},{a,b,c},{a,c,d}}. Then the set {b,c} is eopen, but the set {a,d} is not e-open.
The complement of an e-open set is said to be e-closed[5].
The intersection of all e-closed sets containing A in X is
called the e-closure of A and is denoted by e-cl(A). The
union of all e-open sets contained in A in X is called the einterior of A and is denoted by e-int(A). The collection of all
e-open sets of X is denoted by eo(X) or eo(X, τ).and the
collection of all e-closed sets of X is denoted by ec(X).
Properties 1.3 Let A and B be subsets of a space X,
(i) A is e-closed in X iff A = e-cl(A).
(ii) e-cl(A) ⊂ e-cl(B) whenever A ⊂ B.
(iii) e-cl(A) is e-closed in X.
(iv) e-cl(e-cl(A) = e-cl(A)
(v) e-cl(A) = {x∈X : U∩A ≠ φ for every e-open set U
containing x}.
Properties 1.4 (i) The union of any family of e-open sets
in X is an e-open.
(ii) The intersection of any family of e-closed sets is an eclosed set.
Lemma 1.5 If A is a semi open subset of a space X then ecl(A) = δ-cl(A).
proof: It suffices to show that δ-cl(A) ⊂ e-cl(A). Suppose
x∈ e-cl(A). Then there exists an e-open set U containing x
such that U∩A = φ which implies that U∩int(A) = φ and so
Dhana et.al/On Generalized e-Closed Sets and e-Continuous Functions
is U∩e-int(A). Then e-int(e-cl(U))∩A = φ, which implies
that x ∉ δ-cl(A).
Recall that a space X is said to be extremally disconnected
(E.D)[2]if the closure of every open set A in X is open in X.
and X is Hausdorff if distinct points in X have disjoint
neighbourhoods.
Remark 1.6 In an E.D space,
(i)δ-closure of every semi open set in X is open.
(ii) Semi closure of every semi open set in X is open.
(iii) closure of every semi open set in X is open.
Definition 1.7 A subset A of X is (i)e-clopen if it is both eopen and e-closed and is denoted by e-co(A), (ii)e-door if it
is either e-open or e-closed.
Recall that a subset A of X is e-regular open if A = e-int(ecl(A)).
Definition 1.8 The δ-e-interior of a subset A of X is the
union of all e-regular open sets of X contained in A and is
denoted by e-intδ(A). The subset A is called δ-e-open if A=
e-intδ(A). ie, a set is δ-e-open if it is the union of e-regular
open sets. The complement of a δ-e-open set is called δ-eclosed.
The family of all δ-e-open sets form a topology on X and is
denoted by δ-eo(X). As the intersection of two e-regular
open sets is e-regular open, the collection of all e-regular
open sets forms a base for a coarser topology τs than the
original one τ. The family τs is the so called e-semiregularization of τ. The following lemma is well-known.
Lemma 1.9 δ-eo(X) = τs.
Definition 1.10 A subset A of a space X is called:
(i) a generalized e-closed set (abbr. g-e-closed) if e-cl(A) ⊆
U whenever A ⊆ U and U is open.
(ii) a semi-generalized e-closed set (abbr. sg-e-closed) if se-cl(A) ⊆ U, whenever A ⊆ U and U is semi-open.
(iii) a generalized α-e-closed set (abbr. gα-e-closed) if α-ecl(A) ⊆ U, whenever A ⊆ U and U is semi-open.
(iv) a generalized semi-e-closed set (abbr. gs-e-closed) if se-cl(A) ⊆ U, whenever A ⊆ U and U is open.
(v) a α-generalized e-closed set (abbr. α-g-e-closed) if α-ecl(A) ⊆ U, whenever A ⊆ U and U is open.
Result 1.11 (i) In an E.D space X, the δ-closure of every s.o
set in X is open.
(ii) cl(A) = s.cl(A) for every s.o set A in an E.D space X.
Definition 1.12 Let f : X→Y is called
(i) e-open if f(U) is e-open in Y, for every e-open set U of X.
(ii) δ-e-open if f(U) is δ-e-open in Y for every δ-e-open set
U of X.
(iii) δ-e-closed if f(V) is δ-e-closed in Y for every δ-e-closed
set V of X.
2. The δ - generalized e-closed sets.
Definition 2.1 A subset A of a topological space X is called
δ-generalized e-closed
(abbr. δ-g-e-closed) if e-clδ(A) ⊆ U, whenever A ⊆ U and U
is open in X.
The class of δ-generalized e-closed sets is properly placed
between the class of δ-e-closed and g-e-closed sets. Our
next result will show that.
Theorem 2.2 Let X be a topological space.
(i) Every δ-e-closed set is δ-generalized e-closed.
(ii) Every δ-generalized e-closed set in X is g-e-closed in X.
The reverse claims in the theorem above are not usually
true. First we give an example of a δ-generalized e-closed
set which is not δ-e-closed.
Example 2.3 Let X = {a,b,c} and let τ = {φ,X,{a,b}}.Let A =
{a,c}. Since the only e-open super set of A is X, then A is
30
clearly δ-generalized e-closed. But it is easy to see that A is
not δ-e-closed.
Next we show that even a e-closed set in a space X need
not be δ-generalized e-closed in X.
Example 2.4 Let X be the real line and let τ be the point
generated topology on X. ie, the non empty open sets are
those containing a fixed point, say the zero point. Then the
set P of all irrationals is e-closed in (X,τ), and thus g-eclosed. Since (X,τs) is the indiscrete space, P is g-e-closed
in (X,τs) but clearly not δ-generalized e-closed in (X,τ).
None of the following implications is reversible.
δ-e-closed ⇒ δ-g-e-closed
⇓
⇓
e-closed ⇒ g-e-closed
Definition 2.5 A space X is said to be semi regular, if for
every closed set F and a point x∉F, there exist disjoint
semiopen sets A and B such that x∈A and F ⊂ B. The spaces
in which the concepts of g-closed and closed sets coincide
are called T1/2 space. In T1/2 space every singleton is either
open or closed and so is a door space.
Theorem 2.6 Let A be a subset of a semi regular space
(X,τ).
(i) A is δ-generalized e-closed if and only if A is g-e-closed.
(ii) If, in addition, (X,τ) is T1/2, then A is δ-generalized eclosed if and only if A is e-closed.
The previous observations lead to the problem of finding
the space (X,τ) in which the g-e-closed sets of (X,τs) are δgeneralized e-closed in (X,τ). While we have not been able
to completely resolve this problem, we offer partial
solutions. For that reason we will call the spaces with T1/2
semi regularization almost weekly Hausdorff. Recall that a
space is called weekly Hausdorff if its semi regularization
is T1. The point excluded topology on any infinite set gives
an example of an almost weekly Hausdorff space, which is
not weakly Hausdorff.
Theorem 2.7 In an almost weekly Hausdorff space (X,τ),
the g-e-closed sets of (X,τs) are δ-e-closed in (X,τ) and thus
δ-generalized e-closed set in (X,τ).
Proof: Let A ⊆ X be g-e-closed subset of (X,τs). Let x ∈
clδ(A). If {x} is δ-e-open, then x ∈ A. If not, then X-{x} is δ-eopen, since X is almost weekly Hausdorff. Assume that x ∉
A. since A is g-e-closed in (X,τs), then e-clδ(A) ⊂ X-{x}. ie, x ∉
e-clδ(A). By contradiction x ∈ A. Thus e-clδ(A) = A or
equivalently A is δ-e-closed and hence δ-generalized eclosed in (X,τ).
Recall that a topological space (X,τ) is called an R1 space
[4] if every two different points with distinct closures have
disjoint neighbourhoods.
Theorem 2.8 Let A be a subset of an R1 topological space
(X,τ). If A is a δ-generalized e-closed set, then A is a
generalized e-closed set. In R1 space, the concepts of eclosure and δ-e-closure coincide for compact sets.
Corollary 2.9 In Hausdorff space, a finite set is g-e-closed
if and only if it is δ-g-e-closed.
Theorem 2.10 Let A be a pre open subset of a topological
space (X,τ). If A is δ-g-e-closed, then A is g-e-closed.
Recall that a partition space [12] is a topological space
where every open set is closed.
Corollary 2.11 Let A be a subset of the partition space
(X,τ). If A is δ-g-e-closed, then A is
g-e-closed.
Proof: A topological space is a partition space if and only if
every subset is pre open. Thus the claim follows straight
from Theorem: 2.10.
Concerning partition spaces, consider the following
characterization via δ-generalized e-closed sets.
Dhana et.al/On Generalized e-Closed Sets and e-Continuous Functions
Theorem 2.12 For a topological space (X,τ), the following
conditions are equivalent:
(i) X is a partition space,
(ii) Every subset of X is δ-generalized e-closed.
Proof: (i) ⇒ (ii) Let A ⊆ U and U is open, and A is an
arbitrary subset of X. Since X is a partition space, U is eclopen. Thus e-clδ(A) ⊆ e-clδ(U) = U.
(ii) ⇒ (i) If U ⊆ X is e-open, then by (ii), e-clδ(A) ⊆ U or
equivalently U is δ-e-closed and hence e-closed.
Theorem 2.13 (i) Finite union of δ-generalized e-closed
sets is always a δ-generalized e-closed set.
(ii) Countable union of δ-g-e-closed sets need not be a δ-ge-closed set.
(iii) Finite intersection of δ-g-e-closed sets may fail to be a
δ-g-e-closed set.
Proof: (i) Let A,B ⊆ X be a δ-g-e-closed set, and let A∪B ⊆
U with U is e-open.
Then e-clδ(A∪B) = e-clδ(A) ∪ e-clδ(B) ⊆ U. ie, A∪B is δ-g-eclosed.
(ii) Let X be the real line with the usual topology. Since X is
semi regular, then by Theorem 2.6, every singleton in X is
δ-g-e-closed. Let N be the set of all positive integers. Take
A = ∪n∈N{1/n}. Clearly A is countable union of δgeneralized e-closed sets but A is not δ-generalized eclosed, since A ⊆ (0,∞) but 0∈ e-clδ(A).
(iii) Let X = {a,b,c,d,e} and let τ = {φ, X,{c},{a,b},{a,b,c}}. Let
A = {a,c,d}, and B = {b,c,e}. A and B are δ-generalized eclosed sets, since X is their only e-open superset. But
C = {c} = A∩B is not δ-generalized e-closed, since C ⊆ {c}∈τ
and e-clδ(C) = {c,d,e} ⊈ {c}.□
Theorem 2.14 The intersection of a δ-generalized eclosed set and a δ-e-closed set is always δ-generalized eclosed.
Proof: Let A be δ-generalized e-closed and let F be δ-eclosed. If U is an e-open set with A∩F ⊆ U, then A ⊆ U ∪ (XF ) and so e-clδ(A) ⊆ U ∪ (X-F ). Now, e-clδ(A∩F) ⊆ e-clδ(A)
∩ F ⊆ U and so A∩F is δ-generalized e-closed. □
3. Some results on e-Continuous functions
Definition 3.1 A function f : (X,τ)→(Y,σ) is said to be (i) econtinuous [6], if f -1(V) is e-open in X for every open set V
of Y.
(ii) strongly e-continuous ,
if f -1(V) is open in X for every e-open set V of Y. (iii) quasi
θ-continuous[8], if f -1(V) is θ-open in X for every θ-open
set V of Y. (iv) e-irresolute [5,7], if f -1(V) is e-open in X for
every e-open set V of Y.
(v) strongly
θ-continuous[13], if f -1(V) is θ-open in X for every open set
V of Y. (vi) faintly e-continuous[3], if f -1(V) is e-open in X
for every θ-open set V of Y.
Definition 3.2 A function f : (X,τ)→(Y,σ) is said to be (i) ge-continuous if f -1(V) is g-e-closed in X for every closed set
V
of
Y.
(ii) Super e-continuous if f -1(V) is δ-e-open in X for every
open set V of Y. (iii) perfectly e-continuous if f -1(V) is eclopen in X for every open set V of Y. (iv) δ-e-continuous if
f -1(V) is δ-e-open in X for every δ-open set V of Y.
Theorem 3.3 Let f : X→Y be e-continuous and g: Y→Z
strongly e-continuous. Then g ° f : X→Z is e-irresolute.
Proof: Let V be e-open in Z. Since g is strongly econtinuous, g -1(V) is open in Y. Since f is e-continuous, f 1(g -1(V)) is e-open in X. Then f -1(g -1(V)) is e-irresolute. But
f -1(g -1(V)) = (g ° f )-1(V), so g ° f is e-irresolute.
Theorem 3.4 Let f : X→Y be e-continuous and g: Y→Z
strongly θ-continuous. Then g ° f : X→Z is e-continuous.
Proof: Let V be open in Z. Since g is strongly θ-continuous,
g -1(V) is θ-open in Y. Since θ-open⇒open, we have g -1(V)
is open in Y. Since f is e-continuous, f -1(g -1(V)) is e-open in
X. And hence (g ° f )-1(V) = f -1(g -1(V)) is e-open in X.
Therefore g ° f is e-continuous.
Theorem 3.5 The following statements hold for functions
f : X→Y and g: Y→Z
(i) If f is e-irresolute and g is e-continuous, then g ° f : X→Z
is e-continuous.
(ii) If f is strongly e-continuous and g is e-continuous, then
g ° f is continuous.
(iii) If f is faintly e-continuous and g is strongly θcontinuous, then g ° f is e-continuous.
(iv) If f is e-continuous and g is quasi θ-continuous, then gf
is
faintly
e-continuous.
(v) If f is δ-e-continuous and g is δ-continuous, then g ° f is
super e-continuous.
Lemma 3.6 Let X, Y, and Z be three spaces, and f : X→Y
and g: Y→Z be two functions, and
g ° f : X→Z is e-open.
Then if g is e-irresolute and injective, then f is an e-open
function. Proof: Let W be e-open in X. Then g ° f (W) is eopen in Z. Because g ° f is e-open. since g is e-irresolute and
injective, f(W) = g -1(g ° f (W)) is e-open in Y. ie, e-open set
W in X is mapped with e-open set in Y, and so f is an e-open
function.
Theorem 3.7 If f : X→Y is e-continuous and Y is Hausdorff,
then {(x1,x2)/ f(x1) = f(x2)} is a e-closed subset of X x X.
proof: Let A = {(x1,x2)/ f(x1) = f(x2)}. Suppose (x1,x2)∉A.
Then f(x1) and f(x2) are distinct and hence have disjoint
open sets U and V in Y. Since f is e-continuous, f -1(U) and f
-1(V) are
e-open sets of x1 and x2 respectively. Then f 1(U) x f -1(V) is a e-open set of (x ,x ). Clearly this e-open set
1 2
cannot intersect A, and so A is e-closed. □
Theorem 3.8 If f is an e-open map of X into Y and the set
X x X, then Y is
{(x1,x2)/ f(x1) = f(x2)} is closed in
Hausdorff.
Proof: Suppose f(x1) and f(x2) are distinct points of Y. Then
(x1,x2)∉A = {(x1,x2)/ f(x1) = f(x2)}, so there are e-open sets U
of x1 and V of x2 such that (U x V)∩A = φ. Then since f is eopen, f(U)and f(V) are e-open sets of f(x1) and f(x2)
respectively, and f(U)∩f(V) = φ, otherwise (U x V)∩A ≠ φ.
Corollary 3.9 If f is a e-continuous e-open map of X onto Y,
then Y is Hausdorff if and only if {(x1,x2)/ f(x1) = f(x2)} is a
e-closed subset of X x X. proof: Follows by combining
Theorem 3.7 and 3.8.
Theorem 3.10 If f and g :X→Y are e-continuous and Y is
Hausdorff, then {x/f(x) = g(x)} is closed in X.
proof: Let A = {x/f(x)=g(x)}. If (xλ) is a net in A and (xλ)→x,
then by e-continuity we have both f(xλ)→f(x) and
g(xλ)→g(x) in Y. Since f(xλ) = g(xλ) for each λ and limits are
unique in Y, we must have f(x) = g(x). Thus x∈A and A is
closed. □
Corollary 3.11 If If f and g :X→Y are e-continuous, Y is
Hausdorff, and f and g agree on a dense set D in X, then f =
g.
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