Download characterizations of feebly totally open functions

Advances in Mathematics and Computer Science and their Applications
CHARACTERIZATIONS OF FEEBLY TOTALLY OPEN
FUNCTIONS
Raja Mohammad Latif
Department of Mathematics and Natural Sciences
Prince Mohammad Bin Fahd University
P.O. Box 1664, Al – Khobar 31952, Kingdom of Saudi Arabia
E-Mails: [email protected] & [email protected]
Mohammad Rafiq Raja
Department of Mathematics, University of Sargodha
Mandi Bahauddin Campus, Pakistan
E-Mails: [email protected] & [email protected]
Muhammad Razaq
Department of Mathematics, Horizon Degree and Commerce College,
Bhaun Road, Chakwal, Pakistan
E-Mail: [email protected]
Abstract: In this paper, feebly closed and feebly open sets are used to define and investigate a new
class of functions called, feebly totally open functions and also study some of their basic properties.
Relationships between this new class and other classes of existing known functions are established.
2010 Mathematical Subject Classification: 54C10
Keywords: Topological Space, feebly open set, feebly closed set, clopen set, feebly totally open
(closed) function, feebly totally continuous function.
ISBN: 978-1-61804-360-3
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Advances in Mathematics and Computer Science and their Applications
1. Introduction
closure of A, denoted by FCl(A), is the
intersection of all feebly closed sets containing
A. If (X, τ) is a space, then FO(X, τ) = {F⊆X :
F is feebly open set in X} is a topology on X,
τ⊆FO(X, FO(X, τ)) = FO(X, τ), and for each
U⊆X, sClτ (U ) = sClFO( X , τ ) (U ) [13]. If (X, τ) is
Throughout this paper by a space (X, τ) we
mean a topological space. If A is any subset of a
space X, then Cl(A) and Int(A) denote the
closure and the interior of A respectively.
Njastad [40] introduced the concept of an α-set
in (X, τ). A subset A of (X, τ) is called an α-set
if A⊆Int[Cl(Int(A))]. The notion of semi-open
set and pre-open set were introduced by Levine
[28] and Mashhour et al [32] respectively. A
subset A of (X, τ) is called a semi-open set
(respectively pre-open set) if A⊆Cl[(Int(A))]
(respectively A⊆Int[(Cl(A))]. The complement
of an α-set, a semi-open set and a pre-open set
are called α-closed, semi-closed and pre-closed
respectively. We denote the family of all α-sets
in (X, τ) by α(X, τ). Njastad [40] proved that
α(X, τ) is a topology on X. If A is a subset of
(X, τ), then the intersection of all semi-closed
sets containing A is called the seni-closure of A,
and is denoted by sCl(A). The largest semiopen set contained in A is denoted by sInt(A).
In 1963 Levine [28] showed that since the semiopen set of a topological space need not be
closed under finite intersection, then the
collection of all semi-open sets need not be a
topology on the base set. This result raised
questions about the collection of feebly open
sets for a given topological space, which led to
the following discoveries. In [10] & [35] the
semi-closure of sets was used to define and
investigate feebly open sets, feebly closed sets,
and the feebly closure of sets. Maheshwari and
Tapi [36] defined A to be a feebly open set in
(X, τ) if there is an open set U such that
U⊆A⊆sCl(U). Equivalently [26] A is an α-set if
and only if A is feebly open set if and only if
A⊆Int[Cl(Int(A))] and A is an α-closed set if
and only if A is feebly closed if and only if
Cl[Int(Cl(A))]⊆A if and only if sInt[Cl(A)]⊆A.
Evidently every feebly open set is semi-open as
well as pre-open set. The complement of feebly
open set is called a feebly closed set. The feebly
ISBN: 978-1-61804-360-3
a topological space, then FO(X, τ) is called the
feebly induced topology. In [13] & [15] several
properties of topological spaces were
investigated and characterized using the feebly
induced topology of a given space. Feebly
continuous is one of the important properties for
studying topological spaces. Several types of
feebly continuous functions occur in the
literature, many authors used the concepts of
feebly open and feebly closed sets to study
other concepts and gave several results about
that. In 1980, Jain [25] introduced totally
continuous functions. T.M. Nour [43]
introduced the concept of totally semicontinuous functions as a generalization of
totally continuous functions and several
properties of totally semi-continuous functions
were obtained. In [6] Benchalli and Neeli
introduced semi-totally continuous and semitotally open functions and investigated some
basic properties and characteristics of these
functions.
In this paper, feebly closed and feebly open sets
are used to define and investigate a new class of
functions called, feebly totally open functions.
We study some of their basic properties. We
will also establish some relationships between
this new class of functions and other classes of
existing known functions.
2. PRELIMINARIES
Definition 2.1. A topological space (X, τ) is
said to be:
1. semi-T₀ [33] if for each pair of distinct points
in X, there exists a semi-open set containing
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Advances in Mathematics and Computer Science and their Applications
totally continuous [25] if the inverse image of
every open subset of Y is a clopen subset of X.
3. strongly continuous [44] if the inverse image
of every subset of Y is a clopen subset of X.
4. totally semi-continuous [43] if the inverse
image of every open subset of Y is semi-clopen
in X.
5. strongly semi-continuous [43] if the
inverse image of every subset of Y is semiclopen in X. 6. pre-semi-open [43] if the image
of every semi-open set in X is semi-open in Y.
7. semi-totally continuous [6] if the inverse
image of every semi-open set in Y is clopen in
X. 8. [13] feebly-continuous ( f − continuous)
function if the inverse image of every closed
(open) set in Y is feebly closed ( f − open) set
one point but not the other. 2. sT1/2 [6] if every
semi-closed set of X is closed in X. 3. semi-T₁
[33] (resp. clopen T₁ [22]) if for each pair of
distinct points x and y of X, there exist semiopen (clopen) sets U and V containing x and y
respectively such that y∉U and x∉V. 4. semi-T₂
[33] & [26] (resp. ultra Hausdorff or UT₂ [45])
if every two distinct points of X can be
separated by disjoint semi-open (resp. clopen)
sets. 5. s-normal [34] (resp. ultra-normal [33 &
45]) if each pair of non-empty disjoint closed
sets can be separated by disjoint semi-open
(resp. clopen) sets. 6. s-regular [26] (resp. ultra
regular [33]) if for each closed set F of X and
each x ∉ F, there exist disjoint semi-open (resp.
clopen) sets U and V such that F⊆U and x∈V.
7. semi-normal [14] (resp. clopen normal [22])
if for each pair of disjoint semi-closed (resp.
clopen) sets U and V of X, there exist two
disjoint semi-open (resp. open) sets G and H
such that U⊆G and V⊆H. 8. semi-regular [12]
(resp. clopen regular [22]) if for each semiclosed (resp. clopen) set F of X and each x∉F,
there exist disjoint semi-open (resp. open) sets
U and V such that F⊆U and x∈V. 9. locally
indiscrete [41] if every open set of X is closed
in X. 10. s-connected [36] if X is not the union
of two nonempty disjoint semi-open subsets of
X. 11. feebly T₁ [26] if for each pair of distinct
points x, y in X, there exist feebly open sets U
and V containing x and y respectively such that
y∉U and x∉V. 12. [33] clopen-T₁ if for each
pair of distinct points x and y of X, there exist
clopen sets U and V containing x and y
respectively such that y∉U and x∉V. 13. [33]
ultra – normal if each pair of nonempty disjoint
closed sets can be separated by clopen sets.
in X. 9. [3] f * − continuous if the inverse image
of every feebly open (closed) set in Y is open
(closed) set in X. 10. [3] f ** − continuous if the
inverse image of every feebly open (closed) set
in Y is feebly open (closed) set in X. 11. [24]
feebly-closed ( f − closed) [ feebly-open ( f −
open)] function if the image of every closed
(open) set in X is feebly closed (feebly open)
set in Y. 12. semi-open [7] if f (U ) is semiopen in Y for each open set U in X. 13. semiclosed [8] if f (U ) is semi-closed in Y for each
closed set F in X.
Definition 2.3. Let f : (X, τ) → (Y, σ) be a
function. Then the graph function of f
defined by g ( x ) = ( x, f ( x ) ) for each x∈X.
Definition 2.4. [43] Let (X, τ) be a topological
space and x∈X. Then the set of all points y in X
such that x and y cannot be separated by semiseparation of X is said to be the quasi semicomponent of x. A quasi semi-component of a
point x in a space X means the intersection of
all semi-clopen sets containing x.
Definition 2.2. Let f : (X, τ) →(Y, σ) be a
function. Then f is said to be:
1. semi-continuous [28] if the inverse image of
each open subset of Y is semi-open in X. 2.
ISBN: 978-1-61804-360-3
is
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Advances in Mathematics and Computer Science and their Applications
3. FEEBLY TOTALLY OPEN FUNCTIONS
This implies, f ( F )= Y − V , which is clopen in
Definition 3.1. A function f : (X, τ) →(Y, σ) is
said to be feebly totally open (closed) if the
image of every feebly open (closed) set in X is
clopen in Y.
Y. Thus, the image of a feebly open set in X is
clopen in Y. Therefore f is a feebly totally
open function.
Theorem 3.5. For any bijective function f :
(X, τ) →(Y, σ), the following statements are
equivalent:
Theorem 3.2. Let f : (X, τ) →(Y, σ) be a
bijective function. Then f is feebly totally
open if and only if f is feebly totally closed
function.
(i) Inverse of f is feebly totally continuous.
(ii) f is feebly totally open.
Proof. Suppose that f is a feebly totally open
(closed) funtion. Let F be a feebly closed
(feebly open) set in X. Then X − F is feebly
open (feebly closed) in X. Since f is feebly
Proof. (i) ⇒ (ii): Let U be a feebly open set of
X. By assumption ( f −1 (U ) ) = f (U ) is clopen
−1
in Y . So f is feebly totally open.
totally open (closed), f ( X − F ) =Y − f ( F ) is
clopen in Y. This implies f ( F ) is clopen in Y.
(ii) ⇒ (i): Let V be a feebly open set in X.
Then
Theorem 3.3. A surjective function f : (X, τ)
→(Y, σ) is feebly totally open if and only if for
each subset B of Y and for each feebly closed
f (V ) is
( f (V ) )
−1
−1
clopen
in
Y.
That
is
= f (V ) is clopen in Y. Therefore
f −1 is feebly totally continuous.
set U containing f −1 ( B ) , there exists a clopen
Theorem 3.6. If f : (X, τ) → (Y, σ) is feebly
totally continuous surjection from a feebly
normal space X to a space Y, then Y is ultranormal
set V of Y such that B⊆V and f −1 (V ) ⊆ U .
Theorem 3.4. Suppose f : (X, τ) →(Y, σ) is a
surjective feebly totally open function and
B⊆Y. Let U be feebly closed set of X such that
is a
Proof. Let A and B be disjoint closed subsets
of Y. Since f : X → Y is feebly totally
clopen subset of Y containing B such that
continuous function, f −1 ( A ) and f −1 ( B ) are
Y − f ( X −U )
f −1 ( B ) ⊆U. Then V =
f −1 (V ) ⊆ U .
disjoint clopen hence disjoint closed sets in X.
Since X is feebly normal, there exist disjoint
feebly open sets U and V such that f −1 ( A ) ⊆U
On the other hand, suppose F is a feebly closed
set of X. Then f −1 (Y − f ( F ) ) ⊆ ( X − F ) and
and f −1 ( B ) ⊆V. Let G = f −1 ( A ) and H =
X − F is feebly open. By hypothesis, there
exists a clopen set V of Y such that
Y − f ( F )  ⊆ V ,
f
−1
(V ) ⊆ ( X − F ) .
which
f −1 ( B ) . Clearly A⊆G, B⊆H and f −1 ( G ) ⊆U,
f −1 ( H ) ⊆V. Then we have, f −1 ( G ) ∩ f −1 ( H )
implies
⊆U∩V = ϕ, which implies f −1 ( G ∩ H ) ⊆ ϕ,
Therefore F ⊆ X − f (V ) .
which implies G∩H = ϕ. Thus every pair of
nonempty disjoint closed sets in Y can be
Hence
(Y − V ) ⊆ f ( F ) ⊆
f  X − f −1 (V )  ⊆ (Y − V ) .
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Advances in Mathematics and Computer Science and their Applications
separated by disjoint clopen sets. Therefore Y is
ultra-normal.
Proof. Let V be a feebly open set in Z. Since g
is f ** − continuous, g −1 (V ) is feebly open set
in Y. Now since f is feebly totally continuous,
Theorem 3.7. Let f : (X, τ) → (Y, σ) and g :
(Y, σ) → (Z, δ) be feebly totally open functions,
then g  f : X → Z is feebly totally open.
f −1  g −1 (V )  = ( gο f )
Hence gο f
continuous.
Proof. Let V be any feebly open set in X. Since
f is feebly totally open, f (V) is clopen in Y.
Z. Hence g  f is feebly totally open.
Theorem 3.8. Let f : (X, τ) → (Y, σ) and g :
(Y, σ) → (Z, δ) be two functions such that
g  f : X → Z is feebly totally open function.
: X → Z is feebly totally
Conversely, let g  f : X → Z be feebly totally
continuous. Let W be feebly open set in Z.
Since g  f : X → Z is feebly totally
continuous,
( g  f ) (W ) =
−1
f −1  g −1 (W ) 
is
clopen in X. Since f is feebly totally open
bijection,
Proof. (i) Let V be a feebly open set in Y.
Then f −1 (V ) is feebly open set in X because
(
)
−1
=
f ( g  f ) (W )  f=
f −1  g −1 (W )  g −1 (W )


is feebly open set in Y. Thus the inverse image
of each feebly open set in Z is feebly open in Y.
( g  f ) is feebly
( g  f ) ( f −1 (V ) ) = g (V ) is clopen
f is f − continuous. Since
Hence g is f ** − continuous.
in Z. This shows that g is feebly totally open.
(ii) Since g is injective, we have,
f ( A ) = g −1 ( ( g  f )( A ) ) is true for every subset
ACKNOWLEDGEMENT
The first author is highly and gratefully
indebted to the Prince Mohammad Bin Fahd
University, Al – Khobar, Saudi Arabia, for
providing all necessary research facilities
during the preparation of this research paper.
A of X. Let U be any feebly open set in X.
Therefore ( g  f )(U ) is clopen and hence open
in Z. Since g is totally continuous,
g −1 ( ( g  f )(U ) ) = f (U ) is clopen in Y. This
shows that f is feebly totally open.
Theorem 3.9. If f : (X, τ) → (Y, σ) is feebly
totally continuous and g : (Y, τ) → (Z, η) is
f ** − continuous, then gο f : X → Z is feebly
totally continuous.
ISBN: 978-1-61804-360-3
is clopen in X.
Proof. Let g : Y → Z be f ** − continuous.
Then the proof follows from Theorem 3.9.
Then (i) If f is f ** − continuous and surjective,
then g is feebly totally open. (ii) If g is totally
continuous and injective, then f is feebly
totally open.
totally open
(V )
Theorem 3.10. Let f : (X, τ) → (Y, σ) be
feebly totally open bijection and g : (Y, σ) →
(Z, δ) be any function. Then g  f : X → Z is
feebly totally continuous if and only if g is
f ** − continuous.
Since each clopen set is feebly open set. So f
(V) is feebly open set in Y. Since g is feebly
totally open, g ( f (V ) ) = ( g  f )(V ) is clopen in
**
−1
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Advances in Mathematics and Computer Science and their Applications
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