Download g#-SEMI-CLOSED SETS IN TOPOLOGICAL SPACES

g#-SEMI-CLOSED
SETS
IN TOPOLOGICAL SPACES
M.K.R.S.VEERA KUMAR
J.K.C.COLLEGE, GUNTUR-522 006, A.P. I N D I A
1991 AMS Classification: 54 A 05, 54 D 10
Key words and Phrases: g-open set, g#-semi-closed sets, #Tb, Tb#,, Tb# # and
Tb
#
spaces.
In this paper, we introduce and study a new class of sets, namely g#-semi-closed sets for topological
spaces. Applying g#-semi-closed sets, we introduce and study four new classes of spaces, namely #Tb spaces,
Tb# spaces, Tb# # spaces and Tb# spaces. The class of Tb# spaces (resp. #Tb spaces) is the dual of the class of #Tb
spaces (resp. Tb# spaces) to the class of Tb spaces (resp. T1/2 spaces). We also introduce and study g#scontinuous and g#s-irresolute maps.
Levine introduced and studied semi-open sets1[11] and generalized closed sets2[10]
in 1963 and 1970 respectively. S.P.Arya and T.Nour3[3] defined generalized semi-closed sets
(briefly gs-closed sets) in 1990 for obtaining some characterizations of s-normal spaces. In
this paper, we introduce and study g#-semi-closed sets. This new class is properly placed in
between the class of semi-closed sets and gs-closed sets.
Njåstad4[16] and Abd El-Monsef et. al 5[1] introduced -sets (called as -closed
sets) and semi-preopen sets respectively. Semi-preopen sets are also known as -sets6[ 2].
Maki et.al. introduced generalized -closed sets (briefly g-closed sets)7[13] and generalized closed sets (briefly g-closed sets)8[12] in 1993 and 1994 respectively.
The class of g#-semi-closed sets is independent from the classes of g-closed sets, gclosed sets, g-closed sets and preclosed sets.
Levine2[10] and Devi et. al 9[8] introduced T1/2 spaces and Tb spaces respectively.
As applications of g#-semi-closed sets, we introduce and study #Tb spaces, Tb# spaces, Tb# #
spaces and Tb# spaces. We proved that the class of Tb# spaces( resp. #Tb spaces) is the dual of
the class of #Tb spaces (resp. Tb# spaces) to the class of Tb spaces (resp. T1/2 spaces).
Recently Devi et. al. 10[6] introduced Tb spaces. We observe that the class of #Tb
spaces properly fits in between the class of T1/2 spaces and the class of Tb spaces. We also
introduce and study g#s-continuous maps and g#s-irresolute maps.
§2.PREREQUISITES
Throughout this paper (X,), (Y, ) and (Z, ) represent non-empty topological
spaces on which no separation axioms are assumed unless otherwise mentioned. For a
subset A of a space (X, ), cl(A), int(A) and C(A) denote the closure of A, the interior of A
and the complement of A in X respectively.
We recall the following definitions, which will be used often throughout this paper.
DEFINITION 2.01 - A subset A of a space (X, ) is called
(1) a preopen set11[14] if A  int(cl(A)) and a preclosed set if cl(int(A))  A.
(2) a semi-open set1[11] if A  cl(int(A)) and a semi-closed set if int(cl(A))  A.
(3) an -open set4[16] if A  int(cl(int(A))) and a -closed set if cl(int(cl(A)))  A.
(4) a semi-preopen set6[2] (=-open5[1]) if A  cl(int(cl(A))) and a semi-preclosed set 6[2]
(= -closed 5[1]) if int(cl(int(A)))  A.
The semi-closure (resp. -closure) of a subset A of (X, ) is denoted by scl(A) (resp.
cl(A) and spcl(A))and is the intersection of all semi-closed (resp. -closed and semipreclosed) sets containing A.
DEFINITION 2.02 – A subset A of a space (X, ) is called
(1) a generalized closed (briefly g-closed) set2[10] if cl(A)  U whenever A  U and U is
open in (X, ).
(2) a generalized semi-closed (briefly gs-closed) set3[3] if scl(A)  U whenever A  U and U
is open in (X, ).
(3) a generalized semi-preclosed (briefly gsp-closed) set12[9] if spcl(A)  U whenever A  U
and U is open in (X, ).
(4) an -generalized closed (briefly g-closed) set8[12] if cl(A)  U whenever A  U and
U is open in (X, ).
(5) a generalized -closed (briefly g-closed) set7[13] if cl(A)  U whenever A  U and U
is -open in (X, ).
DEFINITION 2.03 – A function f : (X, )  (Y, ) is called
(1) semi-continuous1[11] if f –1(V) is semi-open in (X, ) for every open set V of (Y, ).
(2) pre-continuous11[14] if f –1(V) is pre-closed in (X, ) for every closed set V of (Y, ).
(3) -continuous12[15] if f –1(V) is -closed in (X, ) for every closed set V of (Y, ).
(4)  -continuous5[1] if f –1(V) is semi-preopen in (X, ) for every open set V of (Y, ).
(5) g-continuous13[4] if f –1(V) is g-closed in (X, ) for every closed set V of (Y, ).
(6) gs-continuous14[7] if f –1(V) is gs-closed in (X, ) for every closed set V of (Y, ).
(7) g-continuous2[10] if f –1(V) is g-closed in (X, ) for every closed set V of (Y, ).
(8) g-continuous7[13] if f –1(V) is g-closed in (X, ) for every closed set V of (Y, ).
(9) gsp-continuous16[9] if f –1(V) is gsp-closed in (X, ) for every closed set V of (Y, ).
(10) g-irresolute10[6] if f –1(V) is g-closed in (X, ) for every g-closed set V of (Y, ).
(11) pre-semi-open15[5] if f(U) is semi-open in (Y, ) for every semi-open set U in (X, ).
DEFINITION 2.04 – A topological space (X, ) is said to be
(1) a T1/2 space2[10] if every g-closed set in it is closed.
(2) a Tb space8[8] if every gs-closed set in it is closed.
(3) an Tb space10[6] if every g-closed set in it is closed.
§3. BASIC PROPERTIES OF g#-SEMI-CLOSED SETS
We introduce the following definition :
DEFINITION 3.01 – A subset A of a space (X, ) is called a g#-semi-closed
set(written as g#s-closed) if scl(A)  U whenever A  U and U is a g-open set of (X, ).
THEOREM 3.01 – Every semi-closed (resp.-closed, g#s-closed) set is a g#s-closed
(resp.g#s-closed, gs-closed) set.
PROOF: Follows from the definitions.
The following examples show that a g#s-closed (resp.gs-closed) set need not be a
semi-closed (resp. g#s-closed) set.
EXAMPLE 3.01 – Let X = {a, b, c},  = {, X, {a, b}} and A = {a, c}. A is not
even a semi-closed set of (X, ). However A is a g#s-closed set.
EXAMPLE 3.02 – Let X = {a, b, c},  = {, X, {a}, {a, c}} and B = {a, b}. B is not a
gs-closed set but not a g#s-closed set of (X, ).
Thus the class of g#s-closed sets properly contains the class of closed sets, the class of
-closed sets and the class of semi-closed sets. Now we show that the class of g#s-closed sets
is properly contained in the class of gs-closed sets and the class of gsp-closed sets.
THEOREM 3.02 –Every closed (resp. g#s-closed) set is g#s-closed (resp.gs-closed)
set and hence gsp-closed. The converses are not true.
PROOF: The first assertion follows from the fact that every closed (resp.gs-closed) set is a
semi-closed (resp.gsp-closed) set. The set A in the Example 3.01 is is g#s-closed set but not a
closed set. The set B in the Example 3.02 is a gs-closed set and hence a gsp-closed set but not
a g#s-closed set.
So the class of g#s-closed (resp.gsp-closed) sets properly contains the class of closed
(resp.g#s-closed) sets. The class of g#s-closed sets properly contains the class of -closed sets
in view of the above Theorem since every -closed set is a semi-closed set.
g#s-closedness is independent from preclosedness, semi-preclosedness, gclosedness, g-closedness and g-closedness as we see the following examples.
EXAMPLE 3.03 –Let X = {a, b, c},  = {, X, {a}, {b}, {a, b}} and C = {a}. C is a
g#s-closed set. But C is not even an g-closed set of (X, ). Moreover C is not a preclosed set.
EXAMPLE 3.04 – Let X = {a, b, c},  = {, X, {a}, {b, c}} and D = {b}. D is a
preclosed and a g-closed set. Hence D is also a g-closed and hence an g-closed set of
(X, ). But D is not a g#s-closed set.
THEOREM 3.03 – If A is an g-open and g#s-closed set of (X, ), then A is a semiclosed set.
PROOF: Obvious.
THEOREM 3.04 – If A is a g#s-closed set of (X, ), then scl(A)-A does not contain
any non-empty g-closed set.
PROOF: Let F be a g-closed set contained in scl(A)-A. Then A  X-F and X-F is a g-open
set of (X, ). Since A is a g#s-closed set of (X, ), then scl(A)  X-F. Now F  X-scl(A).
Then F  (X-scl(A))  (scl(A)-A))  (X-scl(A))  scl(A) = . Hence F = .
THEOREM 3.05 – If A is a g#s-closed set of (X, ) and A  B  scl(A), then B is
also a g#s-closed set.
PROOF: Let U be an g-open set of (X, ) such that B  U. Then A  U. Since A is g#sclosed, then scl(A)  U. Since B  scl(A), then scl(B)  scl(scl(A)). Thus scl(B)  U.
Therefore B is a g#s-closed set of (X, ).
3.01 – The following diagram shows the relationships of g#s-closed sets with other sets.
closed
g-closed
-closed
g-closed
g-closed
g #s-closed
semi-closed
gs-closed
semi-preclosed
gsp-closed
, where A
(resp.A and
preclosed
B (resp.A
B) represents A implies B but not conversely
B are independent).
§ 4.
APPLICATIONS
OF
g#s-CLOSED SETS
We introduced the following definition :
DEFINITION 4.01 –A space (X, ) is called an Tb# space if every g#s-closed set in
it is semi-closed.
THEOREM 4.01 – Every T1/2 space and an
Tb
#
space but not conversely.
PROOF: Let (X, ) be a T1/2 space and A be a g#s-closed set of (X, ). By the Theorem 3.01,
A is a gs-closed set. Since (X, ) is T1/2, then by the Theorem 6.5 of 9[8], A is a semi-closed
set of (X, ). Therefore (X, ) is an
Tb
#
space. The space (X, ) in the example 3.04 is not a
T1/2 space since {b} is neither open nor closed. However (X, ) is an
Thus the class of
Tb
#
Tb
#
space.
spaces properly contains the class of T1/2 spaces. Next we
show
that the class of
Tb
#
spaces properly contains the class of
Tb
spaces and the class of Tb
spaces.
THEOREM 4.02 –Every Tb (resp.
Tb )
space is an
Tb
#
space but not conversely.
PROOF: The first assertion follows from the Remark 6.10(i)9[8] and the Theorem 5.3(i)10[6]
and the above Theorem 4.01. The space in the example 3.04 is neither Tb nor an Tb since {b}
is both gs-closed and an g-closed set but not a closed set. Already we know that (X, ) is an
Tb
#
space.
Now we characterize Tb# spaces.
THEOREM 4.03 – For a space (X, ), the following conditions are equivalent:
(1) (X, ) is an Tb# space.
(2) Every singleton of X is either g-closed or semi-open.
PROOF: (1)  (2) Let x  X and suppose that {x} is not g-closed of (X,). Then X-{x} is a
g#s-closed set since X is the only g-open set containing X-{x}. By (1), X-{x} is a semiclosed set of (X, ) or equivalently {x} is semi-open.
(2)  (1) Let A be a g#s-closed set of (X, ). Clearly A  scl(A). Let x  X. By (2), {x} is
either g-closed or semi-open.
Case (i) Suppose{x} is g-closed. If x  A, then x scl(A)-A contains the g-closed set
{x} and A is g#s-closed set. In view of the Theorem 3.04, we arrived to a contradiction.
Thus x  A.
Case(ii) Suppose {x} is semi-open. Since x  scl(A), then {x}  A  . So x  A.
Thus in any case x  A. So scl(A)  A.
Therefore A = scl(A) or equivalently A is semi-closed.
Therefore (X, ) is an
Tb
#
-space.
We introduce the following definition:
DEFINITION 4.02 – A space (X, ) is called a Tb# space if every g#s-closed set in it
is closed.
THEOREM 4.04 - Every Tb# space is an
Tb
#
space but not conversely.
PROOF: The first assertion follows from the fact that every closed set is a semi-closed set.
The space (X, ) in the example 3.01 supports the second assertion.
Thus the class of
show
Tb
#
spaces properly contains the class of Tb# spaces. Next we
that the class of Tb# spaces properly contains the class of Tb spaces.
THEOREM 4.05 - Every Tb space is a Tb# space but not conversely.
PROOF: Let (X, ) is a Tb space and A be a g#s-closed set. By the Theorem 3.01, A is gsclosed of (X, ). Since (X, ) is a Tb space, then A is closed. Therefore (X, ) is a Tb# space.
The space (X, ) in the Example 3.04 is a Tb# space but not a Tb space.
THEOREM 4.06 – If (X, ) is a Tb# space, then every singleton of X is either gclosed or open. The converse is not true.
PROOF: Let x  X and suppose that {x} is not g-closed. Then X-{x} is not g-open. Then
X is the only g-open set containing X-{x}. So X-{x} is g#s-closed. Since (X, ) is a Tb#
space, then X-{x} is closed or equivalently {x} is open.
Consider the space (X, ) in the example 3.03. {a}and {b} are open sets and {c} is an gclosed set. But (X, ) is not a Tb# space since {a} is a g#s-closed set but not a closed set of
(X,).
REMARK 4.01 - Tb#ness is independent from Tbness and T1/2 ness.
PROOF: The space (X, ) in the Example 3.03 is an Tb space and hence a T1/2 space. But
(X, ) is not a Tb# space. The space (X, ) in the Example 3.04 is a Tb# space. But (X, ) is not
even a T1/2 space.
We introduce the following definition :
DEFINITION 4.03 – A space (X, ) is called a Tb# # space if every g#s-closed set in it
is -closed.
THEOREM 4.07 – Every Tb# (Tb) space is a Tb# # space but not conversely.
PROOF: Let (X, ) be a Tb# space and A be a g#s-closed set of (X, ). Since (X, ) is
Tb#space, then A is closed. Since every closed set is an -closed set, then A is -closed.
Therefore (X, ) is a Tb# # space. By the Theorem 4.05, (X, ) is also a Tb# # space if (X, ) is a
Tb space.
The space in the Example 3.02 is a Tb# # space but it is neither a Tb# space nor a Tb space since
{c} is a g#-closed set of (X, ) but it is neither a closed set nor a gs-closed set.
Thus the class of Tb# # spaces properly contains the class of Tb# spaces and hence the
class of Tb spaces.
THEOREM 4.08 – If (X, ) is a Tb# # space, then every singleton of X is either gclosed or -open. The converse is not true.
PROOF: Suppose (X, ) is a Tb# # space. Suppose that {x} is not g-closed for some x  X.
Then X-{x} is not g-open. Then X is the only g-open set containing X-{x}. So X-{x} is
g# s-closed of (X, ). Since (X, ) is a Tb# # space, then X-{x} is -closed or equivalently
{x}-open. The space in the Example 3.03 supports that the converse is not true.
We introduce the following definition:
DEFINITION 4.04 - A space (X, ) is called a #Tb space if every gs-closed set in it is
gs#-closed.
THEOREM 4.09 – Every T1/2 space is a #Tb space but not conversely.
PROOF: Let (X, ) be a T1/2 space and A be a gs-closed set of (X, ). Since (X, ) is T1/2,
then by the Theorem 6.59[8], A is semi-closed in (X, ). By the Theorem 3.01, A is g#sclosed. Therefore (X, ) is a #Tb space. The space in the example 3.03 is a #Tb space but not a
T1/2 space.
Therefore the class of #Tb spaces properly contains the class of T1/2 spaces. Now we
show that the class of Tb spaces and the class of Tb spaces are properly contained in the class
of #Tb spaces.
THEOREM 4.10 – Every Tb (Tb) space is a #Tb space but not conversely.
PROOF: The first part follows from the above Theorem 4.09 and the Remark 6.109[8] and
the Theorem 5.3 10[6]. The space (X, ) in the example 3.01 is not even an Tb space.
However (X, ) is a #Tb space.
Now we show that the class of Tb# spaces is the dual of the class of
#
Tb spaces to the
class of Tb spaces and the that the class of
to
#
Tb spaces is the dual of the class of Tb# spaces
the class of T1/2 spaces.
THEOREM 4.11 – A space (X, ) is a Tb space if and only if (X, ) is Tb# and #Tb.
PROOF: Necessity- Follows from the Theorems 4.05 and 4.10.
Sufficiency- Suppose (X, ) is both Tb# and #Tb. Let A be a gs-closed set of (X, ). Since
(X, ) is #Tb, then A is a g#s-closed set of (X, ). Since (X, ) is also a Tb#, then A is a closed
set of (X, ). Therefore (X, ) is a Tb space.
THEOREM 4.12 – A space (X, ) is T1/2 if and only if it is #Tb and Tb#.
PROOF: Necessity- Follows from the Theorems 4.01 and 4.09.
Sufficiency- Suppose (X, ) is both #Tb and
Tb
#
. Let A be a gs-closed set of (X, ). Since
(X, ) is #Tb, then A is a g#s-closed set of (X, ). Since (X, ) is also an
Tb
#
space, then A is
a semi-closed set of (X, ). Then by the Theorem 6.59[8], (X, ) is a T1/2 space.
4.01 – The following diagram shows the inter relationships between the separation
axioms discussed in this section.
Tb
 Tb
#
 Tb
Tb#
T1/2
##
Tb
#
Tb

Tb
Tb# + #Tb
, where A
A always
B (resp. A
T1/2

#
Tb +
#
Tb
B) represents A implies B but B need not imply
(resp. A and B are independent).
§5. g#s-CONTINUOUS MAPS AND g#s-IRRESOLUTE MAPS
We introduce the following definition:
DEFINITION 5.01 – A function f : (X, )  (Y, ) is called g#s-continuous if f-1(V)
is a g#s-closed set of (X, ) for every closed set V of (Y, ).
THEOREM 5.01 – Every continuous (resp. -continuous and semi-continuous) map
is g#s-continuous.
PROOF: Follows from the Theorem 3.01.
The following example supports that the converse of the above Theorem is not true in
general.
EXAMPLE 5.01 – Let X = Y = {a, b, c},  = {, X, {a, b}} and  = {, Y, {a}}.
Define f : (X, )  (Y, ) by f(a) = b, f(b) = a and f(c) = c. f is not continuous, in fact, it is
not even semi-continuous since {b, c} is a closed set of (Y, ) but f -1({b, c}) = {a, c} is not a
semi-closed set of (X, ). However f is g#s-continuous.
Thus the class of g#s-continuous maps properly contains the class of continuous maps,
the class of -continuous maps and the class of semi-continuous maps. Next we show that the
class of g#s-continuous maps is properly contained in the class of gs-continuous maps and
hence in the class of gsp-continuous maps.
THEOREM 5.02 – Every g#s-continuous map is gs-continuous and hence gspcontinuous.
PROOF:: Follows from the Theorem 3.01 and the Theorem 3.02.
The function in the following example is gs-continuous and hence gsp-continuous but
not g#s-continuous.
EXAMPLE 5.02 – Let X = {a, b, c} = Y,  = {, X, {a}, {a, c}} and  = {, Y, {a}}.
Define g : (X, )  (Y,) by g(a) = b, g(b) = c and g(c) = a. g is not a g#s-continuous map
Since {b, c} is a closed set of (Y, ) but g -1({b, c}) = {a, b} is not a g#s-closed set of (X, ).
However g is a gs-continuous map and hence gsp-continuous.
g#s-continuity is independent from pre-continuity, g-continuity, g-continuity and gcontinuity.
EXAMPLE 5.03 – Let X = {a, b, c} =Y,  = {, X, {a}, {b}, {a, b}} and  = {, Y,
{a}, {b, c}}. Let h : (X, )  (Y,) be the identity map. {a} is a closed set of (Y, ) but
h -1({a}) = {a} is neither a preclosed set nor an g-closed set. Therefore h is neither precontinuous nor g-continuous. Since {a} is not an g-closed set of (X, ), then {a} is neither
g-closed nor g-closed. Therefore h is neither g-continuous nor g-continuous. However h is
g#s-continuous.
EXAMPLE 5.04 – Let X= {a, b, c},  = {, X, {a}, {b, c}}. Define  : (X, ) 
(X, ) by (a) = b, (b) = a and (c) = c.  is not g#s-continuous since {a} is a closed set of
(X, ) but  -1({a}) = {b} is not a g#s-closed set of (X, ). However  is pre-continuous, gcontinuous and hence g-continuous and g-continuous.
The composition of two g#s-continuous maps need not be g#s-continuous as it can be
seen from the following Example.
EXAMPLE 5.05 – Let X = Y = Z = {a, b, c},  = {, X, {a}, {b, c}},  = {, Y,
{a, b}} and  = {, Z, {a}}. Define  : (X, )  (Y, ) by (a) = c, (b) = a and (c) = b.
Define  : (Y, )  (Z, ) by  (a) = b,  (b) = a and (c) = c. Clearly both  and  are g#scontinuous maps. {b, c} is a closed set of (Z, ) but ( o ) -1({b, c}) =  -1( -1({b, c})) =
 -1({a, c}) = {a, b} is not a g#s-closed set of (X, ). Therefore  o  : (X, )  (Z, ) is not a
g#s-continuous map.
We introduce the following definition :
DEFINITION 5.02 – A function f : (X, )  (Y, ) is called g#s-irresolute if f -1(V)
is a g#s-closed set of (X, ) for every g#s-closed set V of (Y, ).
THEOREM 5.03 – Every g#s-irresolute map is g#s-continuous but not conversely.
PROOF: Let f : (X, )  (Y, ) be a g#s-irresolute map. Let V be a closed set of (Y, ). Then
by the Theorem 3.02, V is g#s-closed. Since f is g#s-irresolute, then f -1(V) is a g#s-closed set
of (X, ). Therefore f is g#s-continuous.
The function  : (X, )  (Y, ) in the Example 5.05 is not g#s-irresolute since {a, c}
is a g#s-closed set of (Y, ) but  -1({a, c}) = {a, b}is not a g#s-closed set of (X, ). Already
we have seen that  is g*s-continuous.
Thus the class of g#s-continuous maps properly contains the class of g#s-irresolute
maps.
THEOREM 5.04 – Let f : (X, )  (Y, ) and g : (Y, )  (Z, ) be any two
functions. Then
(i) f o g : (X, )  (Z, ) is g#s-continuous if f is g#s-irresolute and g is g#s-continuous.
(ii) f o g : (X, )  (Z, ) is g#s-irresolute if both f and g are g#s-irresolute.
(iii)f o g : (X, )  (Z, ) is g#s-continuous if f is g#s-continuous and g is continuous.
PROOF: Omitted.
DEFINITION 5.03 – A function f : (X, )  (Y, ) is called
pre-g-closed if f(U)
is a g-closed set of (Y, ) for every g-closed set U of (X, ).
Note that the above definition is different from the definition 3.1(ii) of14 [7].
THEOREM 5.05 - If the domain of a surjective, pre-g-closed and pre-semi-open
map is an Tb# space, then so is the range (=codomain).
PROOF: Let f : (X, )  (Y, ) be a surjective, pre-g-closed and pre-semi-open map,
where (X, ) is an Tb# space. Let y  Y. Since f is a surjection, then y = f(x) for some x  X.
Since (X, ) is an Tb# space, then by the Theorem 4.03, {x} is either g-closed or semi-open.
If {x} is g-closed, then {y} = f({x}) is g-closed since f is a pre-g-closed map. If {x} is
semi-open, then {y} = f({x}) is semi-open since f is a pre-semi-open map. Thus {y} is either
an g-closed or semi-open set of (Y, ). By the Theorem 4.03 again, (Y, ) is an
space.
Tb
#
THEORME 5.06 – Let f : (X, )  (Y, ) be a g#s-continuous map. If (X, ), the
domain of f is an
Tb
#
space, then f is semi-continuous.
PROOF: Let V be a closed set of (Y, ). Then f -1(V) is a g#s-closed set of (X, ) since f is
g#s-continuous. Since (X, ) is an
Tb
#
space, then f -1(V) is a semi-closed set of (X, ).
Therefore f is semi-continuous.
THEOREM 5.07 - Let f : (X, )  (Y, ) be a g#s-continuous map. If (X, ), the
domain of f is a Tb# # space, then f is an -continuous map.
PROOF: Similar to as that of the above Theorem 5.06.
THEOREM 5.08 - Let f : (X, )  (Y, ) be a g#s-continuous map. If (X, ), the
domain of f is a Tb# space, then f is continuous.
PROOF: Similar to as that of the Theorem 5.06.
THEOREM 5.09 - Let f : (X, )  (Y, ) be a gs-continuous map. If (X, ), the
domain of f is a #Tb space, then f is g#s-continuous.
PROOF: Similar to as that of the Theorem 5.06.
THEOREM 5.10 – Let f : (X, )  (Y, ) be an g-irresolute and a pre-semi-closed
map. Then f(A) is a g#s-closed set of (Y, ) for every g#s-closed set A of (X, ).
PROOF: Let A be a g#s-closed set of (X, ). Let U be an g-open set of (Y, ) such that f(A)
 U. Since f is g-irresolute, then f -1(U) is an g-open set of (X, ). Since A  f -1(U) and A
is a g#s-closed set of (X, ), then scl(A)  f -1(A). This implies f(scl(A))  U. Since f is a presemi-closed map, then f(scl(A))  scl(f(scl(A))) = f(scl(A))  U. Thus scl(f(A))  U.
Therefore f(A) is a g#s-closed set of (Y, ).
REFERENCES:
[1] . M.E.Abd El-Monsef, S.N.El-Deeb and R.A.Mahmoud, -open sets and
-continuous
mappings, Bull. Fac. Sci. Assiut Univ., 12(1983), 77-90.
[2] .D.Andrijevic, Semi-preopen sets, Mat. Vesnik, 38(1)(1986), 24-32.
[3] .S.P.Arya and T.Nour, Characterizations of s-normal spaces, Indian J. Pure. Appl.
Math.,
12(8)(1990), 717-719.
[4] .K.Balachandran, P.Sundaram and H.Maki, On generalized continuous maps in
topological spaces, Mem. Fac. Sci. Kochi Univ. Ser.A. Math., 12(1991), 5-13.
[5] .S.G.Crossley and S.K.Hildebrand, Semi-topological properties, Fund. Math.,
74(1972), 233-254.
[6] .R.Devi, K.Balachandran and H.Maki, Generalized -closed maps and -generalized
closed maps, Indian J. Pure. Appl. Math., 29(1)(1998), 37-49.
[7] .R.Devi, K.Balachandran and H.Maki, Semi-generalized homeomorphisms and
generalized semi-homeomorphisms in topological spaces, Indian J. Pure. Appl.
Math.,
26(3)(1995), 271-284.
[8] .R.Devi, H.Maki and K.Balachandran, Semi-generalized closed maps and generalized
semi-closed maps, Mem. Fac. Sci. Kochi Univ. Ser.A., Math., 14(1993), 41-54.
[9] .J.Dontchev, On generalizing semi-preopen sets, Mem. Fac. Sci. Kochi Univ. Ser.A.
Math., 16(1995), 35-48.
[10] .N.Levine, Generalized closed sets in topology, Rend. Circ. Mat. Palermo,
19(2)(1970), 89-96.
[11] .N.Levine, Semi-open sets and semi-continuity in topological spaces, Amer. Math.
Monthly, 70(1963), 36-41.
[12] .H.Maki, R.Devi and K.Balachandran, Associated topologies of generalized -closed
sets and -generalized closed sets, Mem. Fac. Sci. Kochi Univ. Ser.A. Math.,
15(1994),
51-63.
[13] .H.Maki, R.Devi and K.Balachandran, Generalized -closed sets in topology, Bull.
Fukuoka Univ. Ed. Part III, 42(1993), 13- 21.
[14] .A.S.Mashhour, M.E.Abd El-Monsef and S.N.El-Deeb, On pre-continuous and weak
pre-continuous mappings, Proc. Math. and Phys. Soc. Egypt, 53(1982), 47-53.
[15] .A.S.Mashhour, I.A.Hasanein and S.N.El-Deeb, -continuous and -open mappings,
Acta Math. Hung., 41(3-4)(1983), 213-218.
[16] .O.Njåstad, On some classes of nearly open sets, Pacific J. Math., 15(1965), 961-970.
M.K.R.S.VEERA KUMAR
J.K.C. COLLEGE,
GUNTUR-522 006,
ANDHRA PRADESH
I N D I A
REFERENCES:
1. N.Levine, Semi-open sets and semi-continuity in topological spaces, Amer. Math.
Monthly,
70(1963), 36-41.
2. N.Levine, Generalized closed sets in topology, Rend. Circ. Mat. Palermo,
19(2)(1970),
89-96.
3. S.P.Arya and T.Nour, Characterizations of s-normal spaces, Indian J. Pure. Appl. Math.,
12(8)(1990), 717-719.
4. O.Njåstad, On some classes of nearly open sets, Pacific J. Math., 15(1965), 961-970.
5. M.E.Abd El-Monsef, S.N.El-Deeb and R.A.Mahmoud, -open sets and -continuous
mappings, Bull. Fac. Sci. Assiut Univ., 12(1983), 77-90.
6. D.Andrijevic, Semi-preopen sets, Mat. Vesnik, 38(1)(1986), 24-32.
7. H.Maki, R.Devi and K.Balachandran, Generalized -closed sets in topology, Bull.
Fukuoka Univ. Ed. Part III, 42(1993), 13-21.
8. H.Maki, R.Devi and K.Balachandran, Associated topologies of generalized -closed
sets
and -generalized closed sets, Mem. Fac. Sci. Kochi Univ. Ser.A. Math., 15(1994),
5163.
9. R. Devi, H.Maki and K.Balachandran, Semi-generalized closed maps and generalized
semi-closed maps, Mem. Fac. Sci. Kochi Univ. Ser.A., Math., 14(1993), 41-54.
10. R.Devi, K.Balachandran and H.Maki, Generalized -closed maps and -generalized
closed maps, Indian J. Pure. Appl. Math., 29(1)(1998), 37-49.
11. A.S.Mashhour, M.E.Abd El-Monsef and S.N.El-Deeb, On pre-continuous and weak precontinuous mappings, Proc. Math. and Phys. Soc. Egypt, 53(1982), 47-53.
12. J.Dontchev, On generalizing semi-preopen sets, Mem. Fac. Sci. Kochi Univ. Ser.A.
Math., 16(1995), 35-48.
13. K.Balachandran, P.Sundaram and H.Maki, On generalized continuous maps in
topological spaces, Mem. Fac. Sci. Kochi Univ. Ser.A., Math., 12(1991), 5-13.
14. R.Devi, K.Balachandran and H.Maki, Semi-generalized homeomorphisms and
generalized semi-homeomorphisms in topological spaces, Indian J. Pure. Appl. Math.,
26(3)(1995), 271-284
15. S.G.Crossley and S.K.Hildebrand, Semi-topological properties, Fund. Math.,
74(1972),
233-254.
16 .A.S.Mashhour, I.A.Hasanein and S.N.El-Deeb, -continuous and -open mappings,
Acta Math. Hung., 41(3-4)(1983), 213-218.
M.K.R.S.VEERA KUMAR
J.K.C. COLLEGE,
GUNTUR-522 006,
ANDHRA PRADESH