Decomposition of Supra M-Continuous Mappings

Decomposition of Supra M-Continuous Mappings

1 M. Sathya Bama

Department of Mathematics, Sree Sevugan Annamalai College,

Devakottai, Sivagangai

2O. Ravi

Department of Mathematics P.M.T.College, Madurai Tamil Nadu, India

Abstract:- In this paper, supra A-set, supra t-set, supra h-set and supra C-set and some new supra topological maps are introduced. Characterizations and properties of such new notions are studied. Also investigate the relationships with other mappings like supra *- continuous.

Keywords and Phrases:- Supra A-set, supra t-set, supra h- set and supra C-set, supra A-continuous map, supra M- continuous map, supra B-continuous map,supra *- continuous map, supraA*-continuous map, supra B*- continuous map, supra topological space.

1. INTRODUCTION

   Njastad [7] initiated the concept of nearly open sets in topological spaces. Following it many research papers were introduced by Tong[12, 13], Przemski [2] and Ganster[3] in the name of Decomposition of Continuity in topological spaces. In 1983,

   Mashhour et al. [6] introduced the supra topological spaces and studied Scontinuous maps and S* – continuous maps. In 2008, Devi et al. [1] introduced and studied a class of sets called supra -open and a class of maps called s- continuous maps between topological spaces, respectively. Ravi et al. [9] introduced and studied a class of sets called supra -open and a class of maps called supra – continuous, respectively. Kamaraj et al. [5] introduced and studied the concepts of supra regular-closed set. It is an effort based on them to bring out a paper in the name of Decomposition of supra M-continuity in supra topological spaces using the new sets like supra A-set, supra t-set, supra h-set and supra C-set and new mappings like supra A-continuous, supra B-continuous map, supra

   *-continuous map, supra A*-continuous map, supra B*- continuous. In this paper, we obtain some important results in supra topological spaces. In most of the occasions, our ideas are illustrated and substantiated by suitable examples.

2. PRELIMINARIES:

   Throughout this paper (X,), (Y, ) and (Z, ) (or simply, X, Y and Z) denote topological spaces on which no separation axioms are assumed unless explicitly stated.

   Definition 2.1 [6, 10] : Let X be a non-empty set. The subfamily P(X) where P(X) is the power set of X is said to be a supra topology on X if X and is closed under arbitrary unions. The pair (X, ) is called a supra topological space.

   The elements of are said to be supra open in (X, ). Complements of supra open sets are called supra closed sets.

   Definition 2.2 [10] : Let A be a subset of (X, ). Then

   1. supra closure of a set A is, denoted by cl(A), defined as cl(A) = { B : B is a supra closed and A B};

   2. supra interior of a set A is, denoted by int(A), defined as int(A) = {G : G is a supra open and A G}.

      Definition 2.3 [6] : Let (X, ) be a topological space and be a supra topology on X. We call is a supra topology associated with if .

      Definition 2.4 : Let (X, ) be a supra topological space. A subset A of X is called

      1. supra semi-open set [10] if A

         cl(int(A));

      2. supra -open set [1, 10] if A

         int(cl(int(A)));

      3. supra regular-open [9] if A= int(cl(A));

      4. supra pre-open set [11] if A

   int(cl(A)).

   The complements of the above mentioned open sets are called their respective closed sets. The family of all supra regular-closed sets of X is denoted by SRC(X).

3. SUPRA C-SETS

   In this section we introduce a new type of set as follows:

   Definition 3.1 : A subset S of X is said to be

   i. supra A-set if S = MN where M is supra open set and N is SRC(X)

   ii. supra t-set if ((S)) = (S)

   1. supra B-set if S = MN where M is supra open and N is a supra t-set

   2. supra C-set if S = MN where M is supra open and N is a supra h-set.

   v. supra h-set if (((S))) = (S)

   Theorem 3.2: Let (X, ) be a supra topological space. If A is a supra t-set of X and BX with AB(A) then B is a supra t-set.

   Proof: We note that ( B) ( A). So we have

   ( B) (( B) (( A)) = ( A)

   (B). Thus (B) = ((B)) and hence B is supra t-set.

   Remark 3.3: i. The union of two supra h-set need not be a supra h-set.

   ii. The union of two supra t-set need not be a supra t-set.

   Example 3.4 : Let X = {a, b, c} with = {X, , {a, b} {a, c} {b, c}}. Here {a} and {b} are both supra t-set and supra h set but their union {a, b} is not both supra t-and supra h-sets.

4. COMPARISONS

   Theorem 4.1 : Any supra open set is a supra A-set.

   Proof : S = XS where X SRC(X) and S is supra open. The proof is completed.

   The converse of the above theorem is not true as can be seen from the following examples.

   Example 4.2 : Let X = {a, b, c, d}with = {X,

   , {a}, {a,d},{b,c,d}}.Here {d} is supra A-set but not supra open.

   Theorem 4.3 :Any supra closed set is a supra t-set but not converse.

   Proof: Since A = (A), (A) = ( (A)). The proof is completed.

   Example 4.4: Consider Example 4.2, {b} is supra t-set but not supra closed.

   Theorem 4.5: A supra regular-open set is a supra t-set but not converse.

   Proof: Since S = ( (S)), () = ( (S)). The proof is completed.

   Example 4.5: Consider Example 4.2, {b} is supra t-set but not supra regular-open.

   Theorem 4.6: A supra regular-open set is supra open but not converse.

   Proof:: Suppose S is supra regular-open set then S =

   ((S)). Then (S) = ((S)). Since S is supra regular-open, we have (S) = S. Thus S is supra open. The proof is completed.

   Example 4.7: Consider the Example 4.2, {a, d} is supra open set but not supra regular open.

   Theorem 4.8: Every supra t-set is supra B-set.

   Proof: Let S be any supra t-set S = XS where X is supra open and S is supra t-set. The proof is completed.

   The converse of the above theorem is not true as can be seen from the following example.

   Example 4.9: Consider Example 4.2, {d} is supra B-set but not supra t-set.

   Theorem 4.10: Any supra open set is a supra B-set.

   Proof: Since S = XS where S is supra open and X is supra regular open, by Theorem 4.5, X is supra t-set. The proof is completed.

   The converse of the above theorem is not true as can be seen the following example.

   Example 4.11: Consider Example 4.2, {c} is supra B-set but not supra open set.

   Theorem 4.12: Any supra closed is a supra B-set. Proof: It follows from Theorem 4.3 and Theorem 4.5 Theorem 4.13: Every supra A-set is a supra B-set.

   Proof: S = XS where X is supra open and S is supra regular-closed. Since S is supra closed, by Theorem 4.3, S is supra t-set. The proof is completed.

   The converse of the above Theorem is not true as can be seen from the following example.

   Example 4.14: Consider Example 4.2, {c} is supra B-set but not supra A-set.

   Theorem 4.15: Any supra t-set is supra h-set but not converse.

   Proof: Let S be supra t-set, then (S)

   = ( (S)), ( (S))= ( ( (S)) implies

   ((int (S)) = ( (S)) = (S). The proof is completed.

   The converse of the above theorem is not true as can seen from the following example.

   Example 4.16: Let X = {a, b, c} with = {X, , {a, b}, {b, c}. Here {b} is supra h-set but not supra t-set.

   Theorem 4.17: Let (X, ) be the supra topological Space.

   1. Any supra A-set is supra C-set.

   2. Any supra open set is supra C-set

   Proof: (a) S = XS where X is supra open and S is supra h-set. The proof is completed.

   (b) S = XS where X is supra h-set and S is supra open set. The proof is completed.

   The converse of the above Theorem is not true as can be seen from the following example.

   Example 4.18: Let X = {a, b, c} with = {X, , {a, b},{b, c}}. Here {b, c} is supra C-set but not supra h-set. Also {a} is supra C-set but not supra open set.

   Theorem 4.19: Every supra B-set is supra C-set.

   Proof: S = XS where X is supra open and S is supra t-set. By Theorem 4.15, S is supra h-set. The proof is completed. The converse of the above theorem is not true as can be seen from following Example.

   Example 4.20: Consider Example 4.18, {b} is supra C-set but not supra B-set.

   Remark 4.21: Supra A-set and supra semi open-sets are independent.

   Consider Example 4.2. Here {d} is supra A-set but not supra semi-open set. Also {a, b, d} is supra semi open set but not supra A-set.

   Remark 4.22: From the above discussions we have the following diagram of implications

   None of the above implications is reversible.

5. DECOMPOSITION OF SUPRA M-CONTINUITY

   Definition 5.1: Let (X,) and (Y, ) be two topological spaces with and . A map f: (X, ) (Y, ) is said to be

   1. supra M-continuous (supra-irresolute [9]) if f- 1(V) is supra open in X for every supra open V of Y

   2. supra -continuous [1] if f-1(V) is supra – open in X for every open V of Y.

We introduce a new class of mappings as follows. Definition 5.2: Let (X, ) and (Y, ) be two topological spaces with and . A map f: (X, )(Y, ) is said to be

1. supra *-continuous if f-1(V) is supra -open in X for every supra open set V of Y

2. supra A-continuous if f-1(V) is supra A set in X for every open set V of Y

3. supra A*-continuous if f-1(V) is supra A set in X for every supra open set V of Y;

4. supra B-continuous if f-1(V) is supra B-set in X for every open set V of Y;

5. supra B*-continuous if f-1(V) is supra B-set in X for every supra open set V of Y;

6. supra C-continuous if f-1(V) is supra C-set in X for every open set V of Y;

7. supra C*-continuous if f-1(V) is supra C-set in X for every supra open set V of Y.

Theorem 5.3: A set S of X is supra regular-open if and only if S is supra pre-open and supra t-set.

Proof: Let S be supra regular-open. By theorem 4.5, S is supra t-set. Also By Theorem 4.6, S is supra open. Thus S is supra pre-open.

Conversely, Let S be supra pre-open and supra t-set. Since

(S) S ( (S)) = (S), S = ( (S)). Hence, S is supra regular open.

Theorem 5.4: A subset S of X is supra open if and only if it is both supra -open and supra A-set.

Proof: Let S be supra open. Then S is supra -open and by Theorem 4.1, S is supra A-set. Conversely, Let S be supra

-open and supra A-set. Since S is supra A-set, S = XS

where X is supra open and S SRC(X). Since S is supra

-open,

XS (((XS))

= (((X)(S)))

= ((X(S))) (as X is supra open)

((X)((S)))

= ((X)S) as S SRC(X)

((X)(S) ———- (1) Now since X ((X), by(1)

S = XS = (XS)X

(((X) (S)) X

X((S))X

= X(S)

=(S)

Therefore S (S) But (S)S. Hence S is supra- open.

Theorem 5.5: A subset S of X is supra open if and only if supra -open and supra B-set.

Proof: Let S be a supra open set. Then S is supra -open. Also, by Theorem 4.10, S is supra B-set.

Conversely let S be supra -open and supra B-set. Since S is supra B-set, S = XS where X is supra open and S is supra t-set.Then S = XS X ((S)) (as S is supra pre-open) = X(S) (as S is supra t-set). We have S X(S) implies S (S). But always (S)

1. Thus S = (S) and S is supra open.

   Theorem 5.6: A subset S is supra open in X if and only if S is supra -open set and supra C-set.

   Proof: Let S be supra open in X. Then S is supra -open set and by Theorem 4.17, S is supra C-set.

   Conversely, let S be a supra -open set and supra C-set. Since S is supra C-set, S = XS where X is supra open and S is supra h-set. Since S is supra -open and S is supra h-set. Since S is supra -open set, S (((S))) =

   ( ( (XS))) ( ( (X)))

   ( ( (S))) = ( (X)) (S) (as X is supra open and S is supra h-set). Now S = XS = X(XS) = XS X( ( (X)) (S)) X (S) (as X ( (X)). S (S). but(S)S. Thus S=(S) and S is supra open.

   Theorem 5.7: Let (X, ) and (Y, ) be two topological space with , . Let f: (X, )(Y, ) be a mapping. Then f is supra M-continuity if and only if

   1. f is supra *-continuous and supra A- continuous.

   2. f is supra*-continuous and supra B- continuous.

   3. f is supra *-continuous and supra C- continuous.

Proof: It is the decompositions of supra M- continuity from Theorem 5.4, 5.5, 5.6.

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