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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, ). 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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