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King Fahd University of Petroleum & Minerals
DEPARTMENT OF MATHEMATICAL SCIENCES
Technical Report Series
TR 352
May 2006
Contra-Gamma-Continuous Mappings in Topological
Spaces
Raja Mohammad Latif
DHAHRAN 31261 ● SAUDI ARABIA ● www.kfupm.edu.sa/math/ ● E-mail: [email protected]
Contra-Gamma-Continuous Mappings in
Topological Spacs
Raja Mohammad Latif
Department of Mathematical Sciences
King Fahd University of Petroleum and Minerals
Dhahran 31261 Saudi Arabia
[email protected]
Abstract
The notion of semi-convergence of …lters was introduced by Latif (1999) who
investigated some characterizations related to semi-open continuous functions.
In the spirit of Latif (1999) ; Min (2002) used the idea of semi-convergence of
…lters to introduce a new class of sets, called
closure,
interior and
open sets, and the notions of
continuity and investigated some properties. In
this paper, we apply the notion of
open sets in topological spaces to present
and study certain properties and characterizations of contra
as a new generalization of contra
continuity
continuity [Dontchev; 1996] :
—————————————————————————————
2000 Mathematics Subject Classi…cation: 54A05; 54C08:
Key Words and Phrases: Topological Space,
closed;
compact; strongly S
open set,
closed, contra
1
closed set, contra
continuity:
1. Introduction
The notion of
open set (originally called
sets) in topological spaces was
introduced by Min [M in; 2002]. For these sets, in [Latif; (T R#331) 2005] we introduced the notions of
derived,
border;
f rontier, and
exterior of a set
and showed that some of their properties are analogous to those for open sets. Also,
we gave some additional properties of
closure and
interior of a set. We
also continued to explore further properties and characterizations of
irresolute and
open mappings. In [Latif; (T R#332) 2005] we also intro-
duced and studied properties and characterizations of
pre
continuous;
closed; pre
open and
closed mappings.
In 1996; Dontchev [Dontchev; 1996] introduced a new class of functions called
contra-continuous functions. Recently, Dontchev and Noiri [1999] introduced and
studied, among others, a new weaker form of this class of functions called contrasemicontinuous functions. They also introduced the notion of RC-continuity [Dontchev
and N oiri; 1999] which is weaker than contra-continuity and stronger than
continuity [T ong; 1998] : Jafari and Noiri [Jaf ari and N oiri; 1999] introduced and
studied a new class of functions called contra-supper-continuous functions which lies
between classes of RC-continuous functions and contra-continuous functions.
This paper is devoted to introduce and investigate a new class of functions called
contra- -continuous functions which is weaker than contra-continuous functions.
2. Preliminaries
Throughout this paper, (X; ) (simply X) always mean topological space on which
no separation axioms are assumed unless explicitly stated. Let S be a subset of
X: The closure (resp., interior) of S will be denoted by Cl (S) (resp., Int (S)): A
subset S of X is called a semi-open set [Levine; 1963] (resp.,
1965]) if S
Cl [Int (S)] ( resp:; S
semi-open set (resp.,
open set [N jastad;
Int [Cl (Int (S))]): The complement of a
open set) is called semi-closed set (resp.,
2
closed set).
The family of all semi-open sets (resp.,
will be denoted by SO (X) (resp.,
open sets) in a topological space (X; )
). The complement of a semi-opem set S is
called a semi-closed set. The intersection of all semi-closed sets containing A is
called the semi-closure of A; denoted by Cls (A) : A subset M (x) of a space X is
called a semi-neighbourhood of a point x 2 X if there exists a semi-open set S
such that x 2 S
M (x) : In [Latif; 1999] Latif introduced the notion of semi-
convergence of …lters and investigated some characterizations related to semi-open
continuous function. Now, we recall the concept of semi-convergence of …lters. Let
S (x) = fA 2 SO (X) : x 2 Ag and let Sx = fA
is f inite and \
that
For any …lter
X : there exists
S (x) such
Ag: Then, Sx is called the semi-neighbourhood …lter at x:
on X, we say that
semi-converges to x if and only if
is …ner than
the semi-neighbourhood …lter at x:
A subset U of X is called a
to x and x 2 U; U 2
open set if whenever a …lter
: The complement of a
set. The intersection of all
We denote the family of all
that
open set is called a
closed sets containing A is called the
A; denoted by Cl (A) : A subset A is also
semi-converges
closed
closure of
closed if and only if A = Cl (A) :
open sets of (X; ) by
: It is shown in [M in; 2002]
is a topology on X: In a topological space (X; ); it is always true that
SO (X)
:
3. Contra -Gamma-Continuous Mappings
The purpose of this section is to explore properties and characterizations of contra
continuous mappings.
De…nition 3.1. A function f : (X; ) ! (Y; ) is called contra
f
1
(V ) is
continuous if
closed in X for each open set V of Y:
De…nition 3.2. [Dontchev; 1996] : A function f : (X; ) ! (Y; ) is called contra
continuous if f
1
(V ) is closed in X for each open set V of Y:
3
De…nition 3.3. [Dontchev & N oiri; 1999] : A function f : (X; )
called contra
1
semicontinuous if f
! (Y; ) is
(V ) is semi-closed in X for each open
set V of Y:
Remark 3.4. Every contra
continuous function is contra
continuous but
not conversely as the following example shows.
Example 3.5. Let X = fa; b; cg ;
= fX; ; fagg and
= fX; ; fbg ; fcg ; fb; cgg:
Then the identity function f : (X; ) ! (X; ) is contra
not contra
continuous but
continuous.
De…nition 3.6. [M rsevic; 1986] : Let A be a subset of a space (X; ) : The set
\ fU 2
U g is called the kernel of A and is denoted by Ker (A) :
:A
Lemma 3.7. The following properties hold for subsets A; B of a space X :
(1) x 2 Ker (A) if and only if A \ F 6=
for any closed subset F of X such
that x 2 F:
(2) A
Ker (A) and A = Ker (A) if A is open in X:
(3) A
B; then Ker (A)
Ker (B) :
Theorem 3.8. The following are equivalent for a function f : (X; ) ! (Y; ) :
(1) f is contra
continuous;
(2) for every closed subset F of Y; f
1
(F ) 2
;
(3) for each x 2 X and each closed subset F of Y containing f (x) ; there exists
U2
such that x 2 U and f (U )
(4) f [Cl (A)]
(5) Cl [f
1
F;
Ker [f (A)] for every subset A of X;
(B)]
f
1
[Ker (B)] for every subset B of Y:
Proof The implications (1) () (2) and (2) =) (3) are obvious.
4
1
(3) ) (2) : Let F be any closed subset of Y and x 2 f
and there exists Ux 2
f
1
such that x 2 Ux and f (Ux )
1
(F ) = [ fUx jx 2 f
(F )g 2
(F ) : Then f (x) 2 F
F: Therefore, we obtain
:
(2) =) (4) : Let A be any subset of X: Suppose that y 2
= Ker [f (A)] : Then
by Lemma 3:7 there exists F a closed subset of Y such that y 2 F and f (A) \
1
F = : Thus, we have A \ f
obtain f [Cl (A)] \ F =
(F ) =
and Cl (A) \ f
1
(F ) = : Therefore, we
and y 2
= f [Cl (f (A))] : This implies that f [Cl (A)]
Ker [f (A)] :
(4) =) (5) : Let B be any subset of Y . By (4) and Lemma 3:7 we have f [Cl (f
Ker (B) and Cl [f
1
(B)]
f
1
1
(V ) is
1
(V )]
f
1
[Ker (V )] = f
1
(V ). This
closed in X:
Theorem 3.9. A function f : (X; ) ! (Y; ) is contra
only if f : (X;
(B))]
[Ker (B)] : (5) =) (1) : Let V be any open set
of Y: Then, by Lemma 3:7 we have Cl [f
shows that f
1
) ! (Y; ) is contra
continuous if and
continuous:
De…nition 3.10. A subset A of a topological space is called a
set if it is the
intersection of open sets.
Theorem 3.11. A function f : (X; ) ! (Y; ) is contra
only if the inverse images of
continuous if and
sets are closed.
De…nition 3.12 A function f : (X; ) ! (Y; ) is said to be ( ; s) open if f (U ) 2
SO (Y ) for every U 2
:
Theorem 3.13. Prove that a mapping f : (X; ) ! (Y; ) is ( ; s)
only if for each x 2 X; and U 2
set V
open if and
such that x 2 U; there exists a semi
Y containing f (x) such that V
Proof. Follows directly from De…nition 3:12:
5
f (U ) :
open
Theorem 3.14. Let f : (X; ) ! (Y; ) be ( ; s)
a
closed set containing f
containing W such that f
Proof. Let H = Y
1
1
(H)
(X
1
[Cls (Ints (Cls (B)))]
Proof. Cl [f
1
(B)] is
exists semi
f
1
closed set H
1
(W )
F; we have f (X
! (Y; ) be ( ; s)
Cl [f
1
1
F)
(Y
1
(H) = X f
W):
[f (X
F )]
closed B
1
Y such that f
H
f
open and let B
1
Y: Then
(B)] :
closed in X containing f
[Cls (Ints (Cls (B)))]
(B) : By Theorem 3:14, there
1
(H)
[Cls (Ints (Cls (H)))]
Cl [f
f
1
1
(H)
(B)] : Thus,
Cl [f
Theorem 3.16. Prove that a function f : (X; ) ! (Y; ) is ( ; s)
only if f [Int (A)]
Ints [f (A)] ; for all A
Proof. Necessity. Let A
x 2 Ux
Y
F ) = F:
Corollary 3.15. Let f : (X; )
f
X is
F:
Since f is ( ; s) open; then H is semi closed and f
X
Y and F
(W ) ; there exists a semi
F ) : Since f
f (X
open. If W
X:
Ints [f (A)] :
: Then by hypothesis, f [Int (U )]
Int (U ) = U as U is
open: Also Ints [f (U )]
Ints [f (U )] : Thus f (U ) is semi
Ints [f (U )] : Since
f (U ) : Hence f (U ) =
open in Y: So f is ( ; s)
open:
Theorem 3.17. Prove that a function f : (X; ) ! (Y; ) is ( ; s)
only if Int [f
1
(B)]
Proof. Necessity. Let B
open; f [Int (f
1
f
1
[Ints (B)] ; for all B
Y: Since Int [f
(B))] is semi
such that
f (A) and by hypothesis, f (Ux ) 2 SO (Y ) :
Hence f (x) 2 Ints [f (A)] : Thus f [Int (A)]
Su¢ ciency. Let U 2
(B)] :
open if and
X: Let x 2 Int (A) : Then there exists Ux 2
A: So f (x) 2 f (Ux )
1
1
(B)] is
open if and
Y:
open in X and f is ( ; s)
open in Y: Also we have f [Int (f
6
1
(B))]
f [f
f
1
1
1
B: Hence, f [Int (f
(B)]
Ints (B) : Therefore Int [f
(B))]
1
(B)]
[Ints (B)] :
Su¢ ciency.Let A
X: Then f (A)
1
(f (A))]
1
f
Y: Hence by hypothesis, we obtain
[Ints (f (A))] : Thus f [Int (A)]
Int (A)
Int [f
for all A
X: Hence, by Theorem 3:16; f is ( ; s)
Ints [f (A)] ;
open:
We remark that the equality does not hold in the preceding two theorems as the
following example shows.
Example 3.18. Let X = Y = f1; 2g : Suppose
and
be the antidiscrete topology on X
be the discrete topology on Y: Then
A = f1g : Then
= Int [f
1
and SO (Y ) = : Let f = Id:;
=
= f [Int (A)] 6= Ints [f (A)] = f1g : Let B = f1g
(B)] 6= f
1
Y: Then
[Ints (B)] = f1g :
Theorem 3.19. Let f : (X; )
! (Y; ) be a mapping. Then a necessary and
su¢ cient condition for f to be ( ; s) open is that f
1
[Cls (B)]
Cl [f
1
(B)]
for every subset B of Y:
Proof. Necessity. Assume f is ( ; s)
f (x) 2 Cls (B) : Let U 2
is a semi
Su¢ ciency. Let B
1
(Y
Hence X
1
Y: Let x 2 f
[Cls (B)] : Then
such that x 2 U: Since f is ( ; s) open; then f (U )
open set in Y: Therefore, B \ f (U ) 6= : Then U \ f
Hence x 2 Cl [f
Cl [f
open: Let B
1
(B)] : We conclude that f
Y: Then (Y
f
1
1
10 [Latif; 1993] ; Int [f
follows that f is ( ; s)
(B)]
(B)]
f
[Cls (B)]
Cl [f
Y: By hypothesis, f
B)
B)] : This implies X Cl [f
Cl [X
1
1
[Y
f
1
1
(Y
Cls (Y
B)]
X f
1
1
(B) 6= :
1
(B)] :
[Cls (Y
1
[Cls (Y
B)]
B)] :
B)] : By applying Theorem
[Ints (B)] : Now from Theorem 3:17; it
open:
De…nition 3.20. A …lter base
is said to be
7
convergent (resp: c
convergent)
to a point x in X if for any U 2
such that x 2 U (resp: closed set U
such that x 2 U ); there exists B 2
such that B
X
U:
Theorem 3.21. A function f : (X; ) ! (Y; ) is contra
continuous if and
only if for each point x 2 X and each …lter base
converging to x;
the …lter base f ( ) is c
convergent to f (x) :
Proof. Necessity. Let x 2 X and
to x: Since f is contra
be any …lter base in X such that
converges
continuous; then for any closed subset V of Y such
that f (x) 2 V; there exists U 2
such that x 2 U and f (U )
converging to x; there exists a B 2
f (B)
in X
such that B
V and therefore the …lter base f ( ) is c
V: Since
is
U: This means that
convergent to f (x) :
Su¢ ciency. Let x 2 X and V be a closed subset of Y such that f (x) 2 V: Let
= fU
X:U 2
and x 2 U g : Then
to x: Thus, there exists U 2
is a …lter base which
such that f (U )
converges
V:
4. Contra-Gamma-Closed Graph
In this section we de…ne contra
closed graph and study some of its charac-
teristics.
De…nition 4.1. The graph G (f ) = f(x; f (x)) jx 2 Xg of a function f : (X; ) !
(Y; ) is said to be contra
there exists U 2
and we have (U
closed if for each (x; y) 2 (X
Y)
such that x 2 U and V a closed subset of Y such that y 2 V
V ) \ G (f ) = :
Lemma 4.2. The graph G (f ) of a function f : (X; ) ! (Y; ) is contra
in X
G (f ) ;
Y if and only if for each (x; y) 2 (X
Y)
closed
G (f ) ; there exists U 2
such that x 2 U and V a closed subset of Y such that y 2 V and we have
f (U ) \ V = :
8
Theorem 4.3. If f : (X; ) ! (Y; ) is contra
then G (f ) is contra
Proof. Let (x; y) 2 (X
closed in X
continuous and Y is Urysohn,
Y:
G (f ) : Then y 6= f (x) and there exist open sets
Y)
V , W such that f (x) 2 V; y 2 W and Cl (V ) \ Cl (W ) =
contra
continuous, there exists U 2
: Since f is
such that x 2 U and f (U )
Cl (V ) : Therefore, we obtain f (U ) \ Cl (W ) = : This shows that G (f ) is
contra
closed:
De…nition 4.4. [M in; 2002] : A function f : (X; )
continuous if f
1
(V ) 2
for every V 2 :
Theorem 4.5. If f : (X; ) ! (Y; ) is
contra
closed in X
Proof. Let (x; y) 2 (X
! (Y; ) is said to be
continuous and Y is T1 ; then G (f ) is
Y:
G (f ) : Then f (x) 6= y and there exists an open
Y)
set V of Y such that f (x) 2 V and y 2
= V: Since f is
exists a
f (U )\(Y
open neighbourhood of x such that f (U )
V)=
and (Y
continuous, there
V: Therefore, we obtain
V ) is a closed subset of Y such that y 2 (Y
This shows that G (f ) is contra
closed in X
V ):
Y:
De…nition 4.6. [Dontchev; 1996] : A space X is said to be strongly S
closed if
every closed cover of X has a …nite subcover. A subset A of a space X is said
to be strongly S
closed if the subspace A is strongly S
Theorem 4.7. If f : (X; )
! (Y; ) has a contra
inverse image of a strongly S
closed set K of Y is
Proof. Assume that K is a strongly S
closed:
closed graph, then the
closed in X:
closed set of Y and x 2
=f
k 2 K; (x; k) 2
= G (f ) : By Lemma 4:2; there exist Uk a
1
(K) : For each
open neighbourhood
of x and Vk a closed subset of Y with k 2 Vk such that f (Uk ) \ Vk = : Since
9
fK \ Vk jk 2 Kg is a closed cover of the subspace K; there exists a …nite subset
K1
K such that K
[ fVk jk 2 K1 g : Then U = \ fUk jk 2 K1 g is a
neighbourhood of x and f (U ) \ K = : Therefore U \ f
x2
= Cl [f
1
(K)] : This shows that f
Theorem 4.8. Let Y be a strongly S
(Y; ) has a contra
1
(K) is
(K) =
and hence
closed in X:
closed space. If a function f : (X; ) !
closed graph, then f is contra
Proof. Suppose that Y is strongly S
continuous:
closed and G (f ) is contra
we show that an open set of Y is strongly S
Y and fH :
1
open
closed: First,
closed: Let V be an open set of
2 rg be a cover of V by closed sets H of V: For each
2 r;
there exists a closed set K of X such that H = K \ V: Then, the family
fK j 2 rg [ f(Y
V )g is a closed cover of Y: Since Y is strongly S
there exists a …nite subset r0
closed ,
r such that Y = ([ fK j 2 r0 g) [ (Y
V ):
Therefore we obtain V = [ fK j 2 r0 g : This shows that V is strongly S
closed: For any open set V; by Theorem 4:7 f
contra
1
(V ) is
closed in X and f is
continuous:
5. Covering Properties
In this section we study the properties of compact and strongly S
under the contra
closed spaces
continuous functions.
Theorem 5.1. Let f : (X; ) ! (Y; ) be a contra
K be a compact subset of (X;
continuous function. Let
) : Then f (K) is strongly S
closed in Y:
Proof. Let fH j 2 rg be any cover of f (K) by closed sets of the subspace f (K) :
For each
2 r; there exists a closed set K of Y such that H = K \ f (K) :
For each x 2 K; there exists
3:8; there exists Ux a
(x) 2 r such that f (x) 2 K
(x)
and by Theorem
open neighbourhood of x such that f (Ux )
Since the family fUx jx 2 Kg is a cover of K by
10
K
(x) :
open sets of X; there
exists a …nite subset K0 of K such that K
obtain f (K)
f (K) = [ H
[ fUx jx 2 K0 g : Therefore we
[ ff (Ux ) jx 2 K0 g which is a subset of [ K
(x) jx
2 K0 and hence f (K) is strongly S
Corollary 5.2. If f : (X; ) ! (Y; ) is a contra
(X;
) is compact, then Y is strongly S
Theorem 5.3. If f : (X; )
(X;
) is a strongly S
2 K0 : Thus,
closed:
continuous surjection and
closed:
! (Y; ) is contra
continuous surjection and
closed space, then (Y; ) is compact.
Proof. Let fV j 2 rg be any open cover of Y: Then ff
of X: Since f is contra
1
continuous; f
2 r: This implies that ff
S
(x) jx
1
1
(V ) j 2 rg is a cover
(V ) is
(V ) j 2 rg is a
closed space X: We have X = [ ff
1
closed in X for each
closed cover of the strongly
(V ) j 2 r0 g for some …nite r0 of r:
Since f is surjective, Y = [ fV j 2 r0 g : This shows that (Y; ) is compact.
6. Connected Spaces
In this section we study the properties of connected spaces under the contra
continuous functions.
Theorem 6.1. If f : (X; )
(X;
! (Y; ) is contra
continuous surjection and
) is connected, then (Y; ) is connected.
Proof. Suppose Y is not connected. There exist nonempty disjoint open sets V1 and
V2 such that Y = V1 [ V2 : Therefore, V1 and V2 are clopen in Y: Since f is
contra
continuous; f
in X and hence
disjoint and X = f
1
(V1 ) and f
1
clopen in X: Moreover, f
1
(V1 ) [ f
1
(V2 ) are
1
(V1 ) and f
(V2 ) : This shows that (X;
closed and
1
open
(V2 ) are nonempty
) is not connected.
De…nition 6.2. [Steen and Seebach; 1970] : A topological space (X; ) is said to be
hyperconnected if the closure of every nonempty open set is the entire set X: It
is well-known that every hyperconnected space is connected but not conversely.
11
Remark 6.3. In example 3:5; (X; ) is hyperconnected and f : (X; ) ! (X; ) is a
continuous surjection, but (X; ) is not hyperconnected. This shows
contra
that contra
continuous surjections do not preserve hyperconnectedness.
De…nition 6.4. [Levine; 1961] : A function f : (X; ) ! (Y; ) is said to be weakly
continuous if for each point x 2 X and each open set V of Y containing f (x) ;
there exists an open set U containing x such that f (U )
Cl (V ) :
Theorem 6.5. [N oiri; 1974] : If f : (X; ) ! (Y; ) is a weakly continuous surjection and X is connected, then Y is connected.
Remark 6.6. Contra
continuity and weak continuity are independent of each
other. In Example 3:5; the function f is contra
continuous but not weakly
continuous. The following example shows that not every weakly continuous
function is contra
continuous:
Example 6.7. Let X = fa; b; c; dg and
= f ; X; fbg ; fcg ; fb; cg ; fa; bg ; fa; b; cg ;
fb; c; dgg: De…ne a function f : (X; ) ! (X; ) as follows: f (a) = c; f (b) = d;
f (c) = b and f (d) = a: Then f is weakly continuous [N eubrunnova; 1980] :
However, f is not contra
and f
1
continuous since fag is a closed set of (X; )
(fag) = fdg is not
open in (X; ) :
Acknowledgement
The author is highly and gratefully indebted to the King Fahd University of Petroleum and Minerals, Dhahran, Saudi Arabia, for providing necessary research facilities
during the preparation of this paper.
References
1. J. Dontchev, Contra
continuous f unctions and strongly S
Internat. J. Math. Sci., 19 (1996) ; 303
12
310:
closed spaces,
2. J. Dontchev, and T. Noiri, Contra
nonica, 10 (1999) ; 159
semicontinuous f unctions; Math. Pan-
168:
3. S. Jafari, and T. Noiri, Contra
Sci. Budapest 42 (1999) ; 27
sup er
continuous f unctions; Annales Univ.
34:
4. R.M. Latif, On Characterizations of Mappings; Soochow Journal of Mathematics, Vol. 19; No. 4(Oct: 1993); 475
495:
5. R.M. Latif, Semi-convergence of Filters and Nets; Mathematical Journal of
Okayama University, Vol. 41(1999); 103
109:
6. R.M. Latif, Characterizations and Applications of Gamma-Open Sets, Department of Mathematical Sciences, KFUPM., Technical Report No. 331 (May,
2005).
7. R.M. Latif, Characterizations of Mappings in Gamma-Open Sets, Department
of Mathematical Sciences, KFUPM., Technical Report No. 332 (May, 2005).
8. N. Levine, A decomposition of continuity in topo log ical spaces; Amer. Math.
Monthly, 68 (1961) ; 44
46:
9. N. Levine, Semi-open sets and semi-continuity in Topological Spaces, Amer.
Math. Monthly, 70 (1963) ; 36
41:
10. S. N. Maheshwari and S.S. Thakur, On
R, 13 (1985) ; 341
cmpactspaces, Bull. Inst. Sinica
347:
11. A.S. Mashhour, I.A. Hasanein, and S.N. El-Deeb,
mappings, Acta Math. Hungar., 41 (1983) ; no. 3
12. Won Keun Min,
sets and
Sci., 31 (2002) ; no. 3; 177
continuous and
4; 213
open
218:
continuous functions, Int. J. Math. Math.
181:
13
13. M. Mrsevic, On pairwise R and pairwise R1 bitop log ical spaces; Bull. Math.
Soc. Sci. Math. R. S. Roumanie, 30 (1986) ; 141
148:
14. O. Njastad, On some classes of nearly open sets, Paci…c J. Math., 15(1965);
961
970:
15. T. Noiri, On weakly continuous mappings; Proc. Amer. Math. Soc., 46 (1974) ;
120
124:
16. T. Noiri and B. Ahmad, On semi
Math. J., 25 : 2(1985); 123
17. T. Noiri, W eakly
no. 3; 483
weakly continuous mappings, Kyungpook
126:
ontinuous f unctions, Int. J. Math. Math. Sci. 10 (1987) ;
490:
18. T. Noiri, Pr operties of
compact spaces; Suppl. Rend. Cire Mat. functions,
Int. J. Math. Math. Sci. 10 (1987) ; no. 3; 483
19. I.L. Reilly and M.K. Vamanmurthy, On
Acta Math. Hung. 45 (1
2) (1985) ; 27
490:
continuity in Topological Spaces,
32:
20. L.A. Steen, and J.A. Seebach, Counter Examples in T opo log y; Holt. Reihart
and Winston, New York. (1970) :
21. T. Thompson, S closed spaces; Proc. Amer. Math. Soc., 60 (1976) ; 335 338:
22. J. Tong, On decompasition of continuity in topological spaces, Acta Math.
Hungar. 54 (1998) ; 51
23. N.V. Velicko, H
78 (1968) ; 103
55:
closed topo log ical spaces; Amer. Math. Soc. Transl.,
118:
24. S. Willard, General Topology, Addison-Wesley Publishing Company (1974) :
14