Download PRE-SEMI-CLOSED SETS

PRE-SEMI-CLOSED SETS
M.K.R.S. VEERA KUMAR
J.K.C. COLLEGE, GUNTUR-522 006, A.P., I N D I A
1991 AMS Classification:
Key words & Phrases:
§1.INTRODUCTION
A new class of sets namely, pre-semi-closed sets is introduced for topological
spaces. This class is properly placed in between the class of semi-preclosed sets and the class
of generalized semi-preclosed sets. Applying pre-semi-closed sets, we introduce and study four
new spaces, namely, pre-semi-T1/2 spaces, semi-pre-T1/3 spaces, pre-semi-Tb spaces and
pre-semi-T3/4 spaces. We proved that the dual of the class of pre-semi-T1/2 spaces to the
class of semi-pre-T1/2 spaces is the class of semi-pre-T1/3 spaces. We also introduce and
study pre-semi-continuous maps and pre-semi-irresolute maps.
Levine [12], Mashhour et. al. [16], Njåstad [18] and Abd El-Monsef et. al. [1]
introduced
semi-open sets, preopen sets, -sets and -sets respectively. Andrijevic[2]
called -sets as semi-preopen sets. The complement of a semi-open (resp. preopen,
-open, semi-preopen) set is called a semi-closed(resp.preclosed, -closed, semi-preclosed) set.
Levine [11] introduced generalized closed (briefly g-closed) sets in 1970. Maki et. al.[15]
and Bhattacharya and Lahiri [5] introduced and studied g–closed sets and sg-closed
sets respectively. Maki et. al. [14] introduced g-closed sets. S.P.Arya and T.Nour
[3] defined gs-closed sets in 1994. Dontchev [8] introduced gsp-closed sets by
generalizing semi-preopen sets.
In this paper a new class of sets namely, pre-semi-closed sets is introduced and
studied. This class properly contains the class of semi-preclosed sets and is properly
contained in the class of gsp-clsoed sets.
Bhattacharya and Lahiri [5] and Dontchev [8] introduced and studied semi-T1/2
spaces and semi-pre-T1/2 spaces applying sg-closed sets and gsp-closed sets respectively. As
applications of pre-semi-closed sets, we introduce and study four new spaces namely,
pre-semi-T1/2 spaces, semi-pre-T1/3 spaces, pre-semi-Tb spaces and pre-semi-T3/4 spaces. We
proved that the class of pre-semi-T1/2 spaces properly contains the class of semi-pre-T1/2
spaces, the class of semi-pre-Tb spaces and the class of pre-semi-T3/4 spaces. The class
of semi-pre-T1/3 spaces properly contains the class of semi-pre-T1/2 spaces. The class of
semi-pre-T1/2 spaces is independent from the class of pre-semi-Tb spaces, the class of
pre-semi-Tb spaces. The class of pre-semi-T1/2 spaces is independent from the class of
semi-pre-T1/3 spaces and the class of semi-T1/2 spaces. The class of pre-semi-Tb spaces is
properly contained in the class of semi-T1/2 spaces. We also proved that the dual of the
class of pre-semi-T1/2 spaces to the class of semi-pre-T1/2 spaces is the class of semi-pre-T1/3
spaces.
Further
we
introduce
and
study pre-semi-continuous maps
and
pre-semi-irresolute maps.
§ 2.PRELIMINARIES
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.
Let us recall the following definitions, which are useful in the sequel.
DEFINITION 2.01 - A subset A of a space (X, ) is called
(1) a semi-open set[12] if A  cl(int(A)) and a semi-closed set if int(cl(A))  A.
(2) a preopen set[16] if A  int(cl(A)) and a preclosed set if cl(int(A))  A.
(3) an -open set[18] if A  int(cl(int(A))) and a -closed set if cl(int(cl(A)))  A.
(4) a semi-preopen set[2] (= -open[1]) if A  cl(int(cl(A))) and a semi-preclosed set[2]
(= -closed [1]) if int(cl(int(A)))  A.
(5) a regular open set if int(cl(A)) = A and a regular closed set if cl(int(A)) = A.
The semi-closure (resp. -closure, semi-preclosure) of a subset A of (X, ) is the
intersection of all semi-closed (resp. –closed, semi-preclosed) sets that contain A and is
denoted by scl(A) (resp. cl(A), spcl(A)).
The following definitions are useful in the sequel.
DEFINITION 2.02 – A subset A of a space (X, ) is called
(1) a generalized closed (briefly g-closed) set [11] if cl(A)  U whenever A  U and
U is open in (X, ).
(2) a semi-generalized closed set (briefly sg-closed) [5] if scl(A)  U whenever A  U
and U is semi-open in (X, ). The complement of a sg-closed set is called a sg-open
set.
(3) a generalized semi-closed set (briefly gs-closed) [3] if scl(A)  U whenever A  U
and U is open in (X, ).
(4) an -generalized closed set (briefly g-closed) [14] if cl(A)  U whenever A  U
and U is open in (X, ).
(5) an generalized -closed set (briefly g-closed) [15] if cl(A)  U whenever A  U
and U is -open in (X, ).
(6) a generalized semi-preclosed (briefly gsp-closed) set[8] if spcl(A)  U whenever A
 U and U is open in (X, ).
DEFINITION 2.03 – A function f : (X, )  (Y, ) is said to be
(1) semi-continuous[12] if f –1(V) is semi-open in (X, ) for every open set V of (Y, ).
(2) pre-continuous[16] if f –1(V) is pre-closed in (X, ) for every closed set V of (Y, ).
(3) -continuous[17] if f –1(V) is -closed in (X, ) for every closed set V of (Y, ).
(4)  -continuous[1] if f –1(V) is semi-preopen in (X, ) for every open set V of (Y, ).
(5) g-continuous[4] if f –1(V) is g-closed in (X, ) for every closed set V of (Y, ).
(6) sg-continuous[19] if f –1(V) is sg-closed in (X, ) for every closed set V of (Y, ).
(7) gs-continuous[6] if f –1(V) is gs-closed in (X, ) for every closed set V of (Y, ).
–1
(8) g-continuous[15] if f (V) is g-closed in (X, ) for every closed set V of (Y, ).
(9) g-continuous[9] if f –1(V) is g-closed in (X, ) for every closed set V of (Y, ).
(10) gsp-continuous[8] if f –1(V) is gsp-closed in (X, ) for every closed set V of (Y, ).
(11) gc-irresolute[4] if f –1(V) is g-open in (X, ) for every g-open set V of (Y, ).
(12) pre- -closed [13]if f(U) is semi-preclosed in (Y, ) for every semi-preclosed set V
of (X, ).
DEFINITION 2.04 – A space (X, ) is called a
(1) T1/2 space[11] if every g-closed set is closed.
(2) semi-T1/2 space[5] if every sg-closed set is semi-closed.
(3) Preregular T1/2 [9] if every gpr-closed set is preclosed.
(4) semi-pre-T1/2 [8] if every generalized semi-preclosed set is semi-preclosed.
§ 3. PROPERTIES OF PRE-SEMI-CLOSED SETS
We introduce the following definition:
DEFINITION 3.01 – A subset A of (X, ) is called pre-semi-closed if spcl(A)  U
whenever A  U and U is g-open in (X, ).
THEOREM 3.02 – Every semi-preclosed set is a pre-semi-closed set.
PROOF: Follows from the fact that spcl(A) = A for any semi-preclosed set.
The following example shows that the implication in the above Theorem is not
reversible.
EXAMPLE 3.03 – Let X = {a, b, c} and  = {, X, {a}, {a, c}}. Let A = {a, b}. A is a
pre-semi-closed set. But A is not a semi-preclosed set.
Thus the class of pre-semi-closed sets
properly
contains the class of
semi-preclosed sets.
THEOREM 3.04 – Every closed (resp. –closed, semi-closed, preclosed, g
–closed and sg-cllsoed) set is a pre-semi-closed set but converses are not true.
PROOF: Follows from the above Example 3.03 and the fact that every closed (resp.
–closed, semi-closed, preclosed, g –closed, sg-closed) set is -closed (resp.semi-closed,
semi-preclosed, semi-preclosed, semi-preclosed, semi-preclosed) set.
Thus the class of pre-semi-closed sets properly contains the classes of closed sets,
-closed sets, semi-closed sets, g –closed sets and sg-closed sets.
REMARK 3.05 – Pre-semi-closed ness is independent from g-closed ness,
g-closed ness and gs-closed ness as it can be seen by the following examples.
EXAMPLE 3.06 – Let X = {a, b, c} and  = {, X, {a, b}}. Let B = {a}. B is
pre-semi-closed but not even a gs-closed set of (X, ).
EXAMPLE 3.07 – Let X = {a, b, c} and  = {, X, {a}}. Let A = {a, b}. A is a g-closed
set and hence an g-closed set and a gs-closed set. A is not a pre-semi-closed set since {a, b}
is a g-open set such that A  U but spcl(A) = X  U = {a, b}.
THEOREM 3.08 – Every pre-semi-closed set is a gsp-closed set but
not conversely.
PROOF: The first assertion follows from the fact that every open set is a g-open set. The set A
in the Example 3.07 is a gsp-closed set. Already we observed that A is not a
pre-semi-closed set.
Thus the class of pre-semi-closed sets is properly contained in the class of
gsp-closed sets.
REMARK 3.09 – Union of
two pre-semi-closed sets need not be
pre-semi-closed.
PROOF: Let (X, ) be as in the Example 3.06. Let A = {a} and B = {b}. A and B are
pre-semi-closed sets. But A  B = {a, b} is not a pre-semi-closed set.
THEOREM 3.10 – If A is g-open and pre-semi-closed, then A is semi-preclosed.
PROOF: Omitted.
THEOREM 3.11 – For a subset A of a space (X, ) the following conditions are
equivalent:
(1) A is open and gs-closed.
(2) A is open and gsp-closed.
(3) A is open and sg-closed.
(4)A is open and pre-semi-closed.
(5) A is regular open.
PROOF: (1)  (5) is nothing but the Theorem 3.9 of [8]. (2)  (5) is nothing but the
Theorem 3.11 of [8]. (5)  (3) and (5)  (4) follows from the fact that regular open sets are
semi-closed, hence sg-closed and pre-semi-closed respectively.
(4)  (2) and
(3)  (4) follows from the Theorem 3.08 and the Theorem 2.4(i) of [7] respectively.
THEOREM 3.12 – If A is a pre-semi-closed set of (X, ) such that A  B 
spcl(A), then B is also a pre-semi-closed set of (X, ).
PROOF: Let U be a g-open set of (X, ) such that B  U. Then A  U. Since A is
pre-semi-closed, then spcl(A)  U. Now spcl(B)  spcl(spcl(A)) = spcl(A)  U. Therefore
B is also a pre-semi-closed set.
THEOREM 3.13 – Let A be a pre-semi-closed subset of (X, ). Then spcl(A)-A does
not contain any non-empty g-closed set.
PROOF: Let F be a g-closed set of (X, ) such that F  spcl(A)-A. Then A  X-F. Since A
is pre-semi-closed and X-F is g-open, then spcl(A)  X-F. This implies F  X-spcl(A). So F 
(X-spcl(A))  (spcl(A)-A)  (X-spcl(A))  spcl(A) = . Thus F = .
3.14. The following diagram gives the relationships of pre-semi-closed sets with some
other sets.
closed
g-closed
g-closed
-closed
gs-closed
g-closed
semi-closed
sg-closed
semi-preclosed
gsp-closed
pre-semi-closed
, where A
independent).
preclosed
B (resp. A
B ) represents A implies B (resp. A and B are
§4. APPLICATIONS OF PRE-SEMI-CLOSED SETS
As applications of pre-semi-closed sets, four spaces namely, pre-semi-T1/2 spaces,
semi-pre-T1/3 spaces, pre-semi-Tb spaces and pre-semi-T3/4 spaces are introduced.
We introduce the following definition:
DEFINITION 4.01 – A space (X, ) is called a pre-semi-T1/2 space if every
pre-semi-closed set in it is semi-preclosed.
Dontchev [8] introduced semi-pre-T1/2 spaces.
THEOREM 4.02 – Every semi-pre-T1/2 space is a pre-semi-T1/2 space.
PROOF: Follows from the Theorem 3.02.
The converse of the above Theorem is not true in general as it can be seen from the
following example.
EXAMPLE 4.03 – Let X ={a, b, c} and  = { , X, {a}}. (X, ) is not a semi-pre-T1/2
space since {a, b} is a gsp-closed set but not a semi-preclosed set of (X, ). However (X, ) is a
pre-semi-T1/2 space.
Thus the class of pre-semi-T1/2 spaces properly contains the class of semi-pre-T1/2
spaces.
Now we have a characterization of pre-semi-T1/2 spaces.
THEOREM 4.04 – For a space (X, ) the following conditions are equivalent:
(1) X is a pre-semi-T1/2 space.
(2) Every singleton of X is g-closed or semi-preopen.
(3) Every singleton of X is g-closed or preopen.
(1)  (2) Suppose {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} and hence X-{x} is trivially a pre-semi-closed set of (X, ). By
(1), X-{x} is a semi-preclosed set or equivalently {x} is semi-preopen.
(2)  (3) Suppose {x} is not preopen for some x  X. Since every singleton is either
preopen or nowhere dense(Lemma 2 of [10]), then {x} is a nowhere dense set. Hence {x} 
cl(int(cl({x}))) = . Therefore {x} is not semi-preopen. By(2), {x} is a g-closed set of (X, ).
(3)  (1) Let A be a pre-semi-closed set of (X, ).Trivially A  spcl(A). Let x  spcl(A). By(3),
{x} is either g-closed or preopen.
Case (i) – Suppose {x} is g-closed. If x  A, then spcl(A)-A contains a non-empty
g-closed set {x}. This is not possible according to the Theorem 3.13. Therefore x 
A.
Case(ii) – Suppose {x} is g-closed. Then {x} is semi-preopen. Since x spcl(A), then {x}
 A  . Therefore x  A.
So in any case, spcl(A)  A. Thus A = spcl(A) or equivalently A is a semi-preclosed set of
(X, ).
Hence (X, ) is a pre-semi-T1/2 space.
Gnanambal [9] introduced preregular T1/2 spaces as an application of generalized
preregular closed sets in 1997. He observed that T1/2 ness and preregular T1/2 ness are
independent of each other. He proved that every preregular T1/2 space is a semi-pre-T1/2
space.
THEOREM 4.05 – Every preregular T1/2 (T1/2 ) space is a pre-semi-T1/2 space but not
conversely.
PROOF: Follows from the Corollary 5.8 of [9], above Theorem 3.02 and the Theorem 4.2 of [8].
The space (X, ) in the example 4.03 is pre-semi-T1/2 space. But (X, ) is neither T1/2 nor
preregular T1/2.
REMARK 4.06 – Pre-semi-T1/2 ness is independent from semi-T1/2 ness as we see
the next two examples.
EXAMPLE 4.07 – Let X = {a, b, c} and  = {, X, {a}, {b, c}}. (X, ) is not a
semi-T1/2 space since {b} is a sg-closed set but not a semi-closed set. However (X, ) is a
pre-semi-T1/2 space since every subset of X is both pre-semi-closed and semi-preclosed.
EXAMPLE 4.08 – Let X = {a, b, c} and  = {, X, {a}, {a, c}}. (X, ) is not a
pre-semi-T1/2 space since {a, b} is a pre-semi-closed set but not a semi-preclosed set.
However (X, ) is a semi-T1/2 space.
Now we introduce the following definition:
DEFINITION 4.09 – A space (X, ) is called a semi-pre-T1/3 space if every gsp-closed
set in it is pre-semi-closed.
THEOREM 4.10 – Every semi-pre-T1/2 space is a semi-pre-T1/3 space but not conversely.
PROOF: Let (X, ) be a semi-pre-T1/3 space. Let A be a gsp-closed set of (X, ). Since (X,
) is semi-pre-T1/2, then A is semi-preclosed. By the Theorem 3.02, A is a pre-semi-closed
set of (X, ). Therefore (X, ) is a semi-pre-T1/3 space. Already we observed that the space
(X, ) in the example 4.08 is not a semi-pre-T1/2 space. However (X, ) is a
semi-pre-T1/3 space.
Thus the class of semi-pre-T1/3 spaces properly contains the class of semi-pre-T1/2
spaces.
Now we show that the dual of the class of semi-pre-T1/3 spaces to the class of
semi-pre-T1/2 spaces is the class of pre-semi-T1/2 spaces.
THEOREM 4.11 – A space (X, ) is a semi-pre-T1/2 if and only if it is semi-pre-T1/3 and
pre-semi-T1/2.
PROOF: Necessity – Follows from the Theorems 4.02 and 4.10.
Sufficiency – Let A be a gsp-closed set of (X, ). Since (X, ) is a semi-pre-T1/3 space,
then A is pre-semi-closed. Since (X, ) is pre-semi-T1/2 space, then A is semi-preclosed set
of (X, ). Therefore (X, ) is a semi-pre-T1/2 space.
REMARK 4.12 – Semi-pre-T1/2 ness and pre-semi-T1/2 ness are independent of each
other as we see the next two examples.
EXAMPLE 4.13 – Let X = {a, b, c} and  = {, X, {a}}. (X, ) is pre-semi-T1/2 . But
(X, ) is not a semi-pre-T1/3 space since {a, b} is a gsp-closed set but not a pre-semi-closed set.
THEOREM 4.14 – Every preregular T1/2 space is semi-pre-T1/3 space but not conversely.
PROOF: Follows from the Theorem 4.10 and the Corollary 5.8 of [9]. The space (X, ) in the
example 4.08 is a semi-pre-T1/3 but not a preregular T1/2 space.
Thus the class of semi-pre-T1/3 spaces properly contains the class of preregular T1/2
spaces.
DEFINITION 4.15 – A subset A of a space (X, ) is called pre-semi-open if
C(A) is pre-semi-closed.
Now we have a characterization of semi-pre-T1/3 spaces.
THEOREM 4.16 – If (X, ) is a semi-pre-T1/3 space, then for each x  X, {x} is either
g-closed or pre-semiopen.
PROOF: Suppose {x} is not a g-closed set of a semi-pre-T1/3 space (X, ). So {x} is not a closed
set. Then X is the only open set containing X-{x}.Therefore X-{x} is gsp-closed. Since (X, )
is a semi-pre-T1/3 space, X-{x}is a pre-semi-closed set or equivalently {x} is pre-semi-open.
REMARK 4.17 – The converse of the above Theorem is not true as it can be seen from
the following example.
EXAMPLE 4.18 – Let X = {a, b, c} and  = {, X, {a}}. Every singleton of
(X, ) is either g-closed or pre-semi-open. (X, ) is not a semi-pre-T1/3 space since {a, b} is a
gsp-closed set but not a pre-semi-closed set of (X, ).
We introduce the following definition:
DEFINITION 4.19 – A space (X, ) is called a pre-semi-Tb space if every
pre-semi-closed set in it is semi-closed.
THEOREM 4.20 – Every pre-semi-Tb space is a semi-T1/2 space but not conversely.
PROOF: Let A be a sg-closed set of a semi-Tb space (X, ). By the Theorem 3.04, we
know that A is a pre-semi-closed set. Since (X, ) is a pre-semi-Tb space, then A is a
semi-closed set. Therefore (X, ) is a semi-T1/2 space. The space (X, ) in the example
4.03 is a semi-T1/2 space. (X, ) is not a pre-semi-Tb space since {a, b} is a
pre-semi-closed set but not semi-closed set.
Thus the class of semi-T1/2 spaces properly contains the class of pre-semi-Tb
spaces.
REMARK 4.21 – Pre-semi-Tb ness is independent from semi-pre-T1/3 ness and
semi-pre-T1/2 ness as we see the following examples.
EXAMPLE 4.22 – The space (X, ) in the example 4.07 is a semi-pre-T1/2 space and
hence a semi-pre-T1/3 space. (X, ) is not a pre-semi-Tb space since {b} is a pre-semi-closed set
but not a semi-closed set.
EXAMPLE 4.23 – The space (X, ) in the example 4.18 is a pre-semi-Tb space. But
(X, ) is not even a semi-pre-T1/3 space.
THEOREM 4.24 – If (X, ) is a pre-semi-Tb space, then for each x  X, {x} is
either g-closed or semi-open.
Proof: Suppose {x} is not a g-closed set of pre-semi-Tb space (X, ). Then X is the
only g-open set containing X-{x} and hence X-{x} is a pre-semi-closed set. Since (X, ) is a
pre-semi-Tb space, then X-{x} is semi-closed or equivalently {x} is semi-open.
REMARK 4.25 – The converse of the above Theorem is not true as it can be seen from
the following example.
EXAMPLE 4.26 – Let (X, ) be as in the example 4.07. Each singleton of X is g-closed.
{b} is a pre-semi-closed set but not a semi-closed set of (X, ). Therefore (X, ) is not a
pre-semi-Tb space.
We introduce the following definition:
DEFINITION 4.27 – A space (X, ) is called a pre-semi-T3/4 space if every
pre-semi-closed set is preclosed.
THEOREM 4.28 – Every pre-semi-T3/4 space is a pre-semi-T1/2 space.
PROOF: Follows from the fact that every preclosed set is a semi-preclosed set.
The following example shows that the converse of the above Theorem is not true.
EXAMPLE 4.29 – Let X = {a, b, c} and  = {, X, {a}, {b}, {a, b}}. {a} is a
pre-semi-closed set but not a preclosed set of (X, ). So (X, ) is not a pre-semi-T3/4
space. However (X, ) is a pre-semi-T1/2 space.
Therefore the class of pre-semi-T1/2 spaces properly contains the class of
pre-semi-T3/4 spaces.
Now we have a characterization of pre-semi-T3/4 spaces.
THEOREM 4.30 – If (X, ) is a pre-semi-T3/4 space, then for each x  X, {x} is either
g-closed or preopen.
PROOF: Suppose {x} is not g-closed for some x  X. Then X-{x} is not g-open. X is the only
g-open set containing X-{x} and hence X-{x} is trivially a pre-semi-closed set of
(X, ). Since (X, ) is a pre-semi-T3/4 space, then X-{x} is a preclosed set or equivalently {x}
is preopen.
REMARK 4.31 – The converse of the above Theorem is not true as it can be seen from
the following example.
EXAMPLE 4.32 – Let (X, ) be as in the example 4.29. (X, ) is not a pre-semi-T3/4
space. However {a} and {b} are preopen and {c} is a g-closed set of (X, ).
REMARK 4.33 – Pre-semi-T3/4 ness is independent from semi-pre-T1/3 ness,
semi-pre-T1/2 ness and pre-semi-Tb ness as we see the following examples.
EXAMPLE 4.34 – Let (X, ) be as in the example 4.29. Already we observed
that (X, ) is not a pre-semi-T3/4 space. (X, ) is a semi-pre-T1/2 space and hence a
semi-pre-T1/3 space. (X, ) is also a pre-semi-Tb space.
EXAMPLE 4.35 – Let (X, ) be as in the example 4.18. (X, ) is not even a semi-pre-T1/3
space since {a, b} is a gsp-closed set but not a pre-semi-closed set. However
(X, ) is a pre-semi-T3/4 space.
EXAMPLE 4.36 – Let (X, ) be as in the example 4.07. (X, ) is not a pre-semi-Tb
space since {b} is a pre-semi-closed set but not a semi-closed set. However (X, ) is a
pre-semi-T3/4 space.
4.37 – Thus we have the following diagram:
preregular T1/2
semi-pre-T1/3
semi-pre-T1/2
pre-semi-Tb
pre-semi-T1/2
T1/2
pre-semi-T3/4
semi-T1/2
, where A
B (resp. A
B)represents A implies B (resp. A and B are
independent of each other).
§ 5. PRE-SEMI-CONTINUOUS MAPS AND PRE-SEMI-IRRESOLUTE MAPS
DEFINITION 5.01 – A function f
:
(X, )

(Y, )
is
called
–1
pre-semi-continuous if f (V) is a pre-semi-closed set of (X, ) for every closed set V of (Y, ).
THEOREM 5.02 – Every -continuous map is pre-semi-continuous.
PROOF: Follows from the Theorem 3.02.
The converse of the above Theorem is not true as it can be seen from the
following example,
EXAMPLE 5.03 - Let X = {a, b, c} = Y,  = {, X, {a}, {a, c}} and  = {, Y,
{a}, {b}, {a, b}}. Define f : (X, )  (Y, ) by f(a) = b, f(b) = c and f(c) = a. {b, c} is a
–1
closed set of (Y, ) but f ({b, c}) = {a, b} is not a semi-preclosed set of (X, ). So f is not
-continuous. However f is a pre-semi-continuous map.
Thus the class of pre-semi-continuous maps properly contains the class of
semi-pre-continuous maps.
THEOREM 5.04 – Every continuous (resp. –continuous, semi-continuous,
pre-continuous, g–continuous and sg-continuous) map is pre-semi-continuous map but not
conversely.
PROOF: The first assertion follows from the Theorem 3.04. The function f in the above
example 5.03 is pre-semi-continuous. Since f is not even a -continuous, then f is neither
semi-continuous nor pre-continuous. So f is neither -continuous nor sg-continuous.
Moreover f is not g-continuous. Finally it is evident that f is not continuous.
Therefore the class of pre-semi-continuous maps properly contains the classes of
-continuous maps, semi-continuous maps, pre-continuous maps, g–continuous maps and
the class of sg-continuous maps.
THEOREM 5.05 – Every pre-semi-continuous map is a gsp-continuous map.
PROOF: Follows from the fact that every pre-semi-closed set is gsp-clsoed set.
The following example shows that the converse of the above Theorem is not true in
general.
EXAMPLE 5.06 – Let X = {a, b c},  = {, X, {a}}. Define g : (X, )  (X, ) by g(a) =
b, g(b) = c and g(c) = a. {b, c} is a closed set of (X, ) but g –1({b, c}) = {a, b} is not a
pre-semi-closed set. So g is not a pre-semi-continuous map. However g is a gsp-continuous map.
Thus the class of semi-pre-continuous maps is properly contained in the class of
gsp-continuous maps.
REMARK 5.07 – Pre-semi-continuity is independent from g-continuity, g-continuity
and gs-continuity as we see the following examples.
EXAMPLE 5.08 – Let X = {a, b, c} and  = {, X, {a, b}}. Define h : (X, )  (X, )
by h(a) = c, h(b) = b and h(c) = a. {c} is a closed set of (X, ) but h –1({c}) = {a} is a
pre-semi-closed set but not even a gs-closed set of (X, ). So h is not even a gs-continuous map.
However h is a pre-semi-continuous map.
EXAMPLE 5.09 – Let X,  and g be as in the example 5.06. g : (X, )  (X, ) is not
pre-semi-continuous map. g is g-continuous and hence g-continuous and gs-continuous too.
Thus the class of pre-semi-continuous maps is independent from the classes of
g-continuous maps, g-continuous maps and gs-continuous maps.
The composition of two pre-semi-continuous maps need not be pre-semi-continuous as it
can be seen from the following example.
EXAMPLE 5.10 - Let X = {a, b, c} = Y = Z,  = {, X, {a}, {a, c}},  = {, Y, {a}, {b},
{a, b}} = . Define  :(X, )  (Y, ) by (a) = b, (b) = c and (c) = a. Define  : (Y, )
 (Z, ) by (a) = a, (b) = c and (c) = a. Clearly  and  are pre-semi-continuous maps. 
o  :(X, )  (Z, ) is not pre-semi-continuous maps since {c} is a closed set of (Z, ) but (
o  ) –1({c}) =  –1( –1({c})) = –1({b}) = {a} is not a pre-semi-closed set of (X, ).
We introduce the following definition:
DEFINITION 5.11 - A function f : (X, )  (Y, ) is called pre-semi-irresolute if
–1
f (V) is a pre-semi-closed set of (X, ) for every pre-semi-closed set V of (Y, ).
THEOREM 5.12 - Every pre-semi-irresolute map is pre-semi-continuous but not
conversely.
PROOF: Let f : (X, )  (Y, ) be a pre-irresolute map and V be a closed set of (Y, ). By the
Theorem 3.04, V is a pre-semi-closed set of (Y, ). Since f is a pre-semi-irresolute map, then f
–1
(V) is a pre-semi-closed set of (Y, ). Therefore f is pre-semi-continuous. The function  :(X,
)  (Y, ) in the above example 5.10 is not a pre-semi-irresolute map since {b} is a
pre-semi-closed set of (Y, ) but  –1({b}) = {a} is not a pre-semi-closed set of (X, ).
However  is a pre-semi-continuous map.
Thus the class of pre-semi-continuous maps properly contains the class of
pre-semi-irresolute maps.
THEOREM 5.13 - Let f :(X, )  (Y, ) and g :(Y, )  (Z, ) be any two
functions. Then
(i) g o f : (X, )  (Z, ) is pre-semi-continuous if f is pre-semi-irresolute and g is
pre-semi-continuous.
(ii) g o f : (X, )  (Z, ) is pre-semi-irresolute if both f and g are pre-semi-irresolute.
(iii) g o f : (X, )  (Z, ) is pre-semi-continuous if f is pre-semi-irresolute and g is
pre-semi-continuous.
DEFINITION 5.14 - 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, ).
THEOREM 5.15 - If the domain of a surjective, pre-g-closed and pre--closed map is a
pre-semi-T1/2 space, then so is the range (=codomain).
PROOF: Let f :(X, )  (Y, ) be a surjective, pre-g-closed and pre--closed map, where (X,
) is a pre-semi-T1/2 space. Let y  Y. Since f is a surjection, then y = f(x) for some x  X.
Since (X, ) is a pre-semi-T1/2 space, then by the Theorem 4.04, {x} is either g-closed or
semi-preopen . If {x} is g-closed, then {y} = f({x}) is g-closed as f being a pre-g-closed map. If
{x} is semi-preopen, then {y} = f({x}) is semi-preopen as f being a pre--closed map. Thus {y}
is either a g-closed set or a semi-preopen set of
(Y, ). By the Theorem 4.04 again, (Y, ) is a pre-semi-T1/2 space.
THEOREM 5.16 - Let f : (X, )  (Y, ) be a pre-semi-continuous map. If
(X, ), the domain of f is pre-semi-T1/2 space, then f is -continuous.
PROOF: Let V be a closed set of (Y, ). Then f –1(V) is a pre-semi-closed set of (X, ). Since (X,
) is a pre-semi-T1/2 space, then f -1(V) is a semi-preclosed set of (X, ). Therefore f is
-continuous.
THEOREM 5.17 - Let f : (X, )  (Y, ) be a pre-semi-continuous map. If
(X, ), the domain of f is a pre-semi-Tb space, then f is semi-continuous.
PROOF: Let V be a closed set of (Y, ). Then f –1(V) is a pre-semi-closed set of (X, ). Since (X,
) is a pre-semi-Tb space, then f –1(V) is a semi-closed set of (X, ). Therefore f is
semi-continuous.
THEOREM 5.18 - Let f : (X, )  (Y, ) be a pre-semi-continuous map. If
(X, ), the domain of f is a pre-semi-T3/4 space, then f is pre-continuous.
PROOF: Let V be a closed set of (Y, ). Then f –1(V) is a pre-semi-closed set of (X, ). Since
(X, ) is a pre-semi-T3/4 space, then f –1(V) is a preclosed set of (X, ). Therefore f is
pre-continuous.
THEOREM 5.19 - Let f : (X, )  (Y, ) be a gsp-continuous map. If (X, ), the
domain of f is semi-pre-T1/3 space, then f is pre-semi-continuous.
–1
PROOF: Let V be a closed set of (Y, ). Then f (V) is a gsp-closed set of (X, ). Since (X, ) is
a semi-pre-T1/3 space, then f –1(V) is a pre-semi-closed set of (X, ). Therefore f is
pre-semi-continuous.
THEOREM 5.20 - Let f : (X, )  (Y, ) be a gc-irresolute and a pre--closed map.
Then f(A) is a pre-semi-closed of (Y, ) for every pre-semi-closed set A of (X, ).
PROOF: Let A be a pre-semi-closed set of (X, ) and U be a g-open set of (Y, ) such that f(A)
 U. Since f is gc-irresolute, then f –1(V) is a g-open set of (X, ). Since A 
f –1(U) and A is a pre-semi-closed set of (X, ), then spcl(A)  f –1(U). Then f(spcl(A)))  U.
Since f is pre--closed map, then f(spcl(A))  spcl(f(spcl(A))) = f(spcl(A))  U. Thus spcl(f(A))
 U. Therefore f(A) is a pre-semi-closed 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., 21 (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]. P.Bhattacharya and B.K.Lahiri, Semi-generalized closed sets in topology, Indian J.
Math., 29 (3) (1987), 375-382.
[6]. 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.
[7]. J.Dontchev, On some separation axioms associated with the -topology, Mem. Fac.
Sci. Kochi Univ. Ser.A. Math., 18 (1997), 31-35.
[8]. J.Dontchev, On generalizing semi-preopen sets, Mem. Fac. Sci. Kochi Univ. Ser.A.
Math., 16 (1995), 35-48.
[9]. Y.Gnanambal, On generalized preregular closed sets in topological spaces, Indian J.
Pure. Appl. Math., 28 (3) (1997), 351-360.
[10]. D.S.Jankovic and I.L.Reilly, On semi separation properties, Indian J. Pure. Appl.
Math., 16 (9) (1985), 957-964.
[11]. N.Levine, Generalized closed sets in topology, Rend. Circ. Mat. Palermo,
19 (2) (1970), 89-96.
[12]. N.Levine, Semi-open sets and semi-continuity in topological spaces, Amer. Math.
Monthly, 70 (1963), 36-41.
[13]. R.A. Mahmoud and M.E. Abd El-Monsef , -irrsolute and  -topological invariant,
Proc. Pakistan Acad. Sci., 27 (3) (1990), 285-296.
[14]. 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.
[15]. H.Maki, R.Devi and K.Balachandran, Generalized -closed sets in topology, Bull.
Fukuoka Univ. Ed. Part III, 42 (1993), 13- 21.
[16]. 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.
[17]. A.S.Mashhour, I.A.Hasanein and S.N.El-Deeb, -continuous and -open
mappings, Acta Math. Hung., 41 (3-4) (1983), 213-218.
[18]. O.Njåstad, On some classes of nearly open sets, Pacific J. Math., 15 (1965), 961970.
[19]. P.Sundaram, H.Maki and K.Balachandran, Semi-generalized continuous maps and
semi-T1/2 spaces, Bull. Fukuoka Univ. Ed. Part III, 40 (1991), 33-40.
AUTHOR'S ADDRESS FOR CORRESPONDENCE:
M.K.R.S.VEERA KUMAR
J.K.C. COLLEGE
GUNTUR-522 006
ANDHRA PRADESH
I N D I A