 Open Access
 Total Downloads : 16
 Authors : M. Sathya Bama, O. Ravi
 Paper ID : IJERTCONV3IS33004
 Volume & Issue : RACMS – 2015 (Volume 3 – Issue 33)
 Published (First Online): 24042018
 ISSN (Online) : 22780181
 Publisher Name : IJERT
 License: This work is licensed under a Creative Commons Attribution 4.0 International License
Decomposition of Supra MContinuous 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 Aset, supra tset, supra hset and supra Cset 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 Aset, supra tset, supra h set and supra Cset, supra Acontinuous map, supra M continuous map, supra Bcontinuous map,supra * continuous map, supraA*continuous map, supra B* continuous map, supra topological space.

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 regularclosed set. It is an effort based on them to bring out a paper in the name of Decomposition of supra Mcontinuity in supra topological spaces using the new sets like supra Aset, supra tset, supra hset and supra Cset and new mappings like supra Acontinuous, supra Bcontinuous 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.

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 nonempty 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

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

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

supra semiopen set [10] if A
cl(int(A));

supra open set [1, 10] if A
int(cl(int(A)));

supra regularopen [9] if A= int(cl(A));

supra preopen 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 regularclosed sets of X is denoted by SRC(X).


SUPRA CSETS
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 Aset if S = MN where M is supra open set and N is SRC(X)
ii. supra tset if ((S)) = (S)

supra Bset if S = MN where M is supra open and N is a supra tset

supra Cset if S = MN where M is supra open and N is a supra hset.
v. supra hset if (((S))) = (S)
Theorem 3.2: Let (X, ) be a supra topological space. If A is a supra tset of X and BX with AB(A) then B is a supra tset.
Proof: We note that ( B) ( A). So we have
( B) (( B) (( A)) = ( A)
(B). Thus (B) = ((B)) and hence B is supra tset.
Remark 3.3: i. The union of two supra hset need not be a supra hset.
ii. The union of two supra tset need not be a supra tset.
Example 3.4 : Let X = {a, b, c} with = {X, , {a, b} {a, c} {b, c}}. Here {a} and {b} are both supra tset and supra h set but their union {a, b} is not both supra tand supra hsets.


COMPARISONS
Theorem 4.1 : Any supra open set is a supra Aset.
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 Aset but not supra open.
Theorem 4.3 :Any supra closed set is a supra tset but not converse.
Proof: Since A = (A), (A) = ( (A)). The proof is completed.
Example 4.4: Consider Example 4.2, {b} is supra tset but not supra closed.
Theorem 4.5: A supra regularopen set is a supra tset but not converse.
Proof: Since S = ( (S)), () = ( (S)). The proof is completed.
Example 4.5: Consider Example 4.2, {b} is supra tset but not supra regularopen.
Theorem 4.6: A supra regularopen set is supra open but not converse.
Proof:: Suppose S is supra regularopen set then S =
((S)). Then (S) = ((S)). Since S is supra regularopen, 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 tset is supra Bset.
Proof: Let S be any supra tset S = XS where X is supra open and S is supra tset. 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 Bset but not supra tset.
Theorem 4.10: Any supra open set is a supra Bset.
Proof: Since S = XS where S is supra open and X is supra regular open, by Theorem 4.5, X is supra tset. 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 Bset but not supra open set.
Theorem 4.12: Any supra closed is a supra Bset. Proof: It follows from Theorem 4.3 and Theorem 4.5 Theorem 4.13: Every supra Aset is a supra Bset.
Proof: S = XS where X is supra open and S is supra regularclosed. Since S is supra closed, by Theorem 4.3, S is supra tset. 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 Bset but not supra Aset.
Theorem 4.15: Any supra tset is supra hset but not converse.
Proof: Let S be supra tset, 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 hset but not supra tset.
Theorem 4.17: Let (X, ) be the supra topological Space.

Any supra Aset is supra Cset.

Any supra open set is supra Cset
Proof: (a) S = XS where X is supra open and S is supra hset. The proof is completed.
(b) S = XS where X is supra hset 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 Cset but not supra hset. Also {a} is supra Cset but not supra open set.
Theorem 4.19: Every supra Bset is supra Cset.
Proof: S = XS where X is supra open and S is supra tset. By Theorem 4.15, S is supra hset. 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 Cset but not supra Bset.
Remark 4.21: Supra Aset and supra semi opensets are independent.
Consider Example 4.2. Here {d} is supra Aset but not supra semiopen set. Also {a, b, d} is supra semi open set but not supra Aset.
Remark 4.22: From the above discussions we have the following diagram of implications
None of the above implications is reversible.


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

supra Mcontinuous (suprairresolute [9]) if f 1(V) is supra open in X for every supra open V of Y

supra continuous [1] if f1(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

supra *continuous if f1(V) is supra open in X for every supra open set V of Y

supra Acontinuous if f1(V) is supra A set in X for every open set V of Y

supra A*continuous if f1(V) is supra A set in X for every supra open set V of Y;

supra Bcontinuous if f1(V) is supra Bset in X for every open set V of Y;

supra B*continuous if f1(V) is supra Bset in X for every supra open set V of Y;

supra Ccontinuous if f1(V) is supra Cset in X for every open set V of Y;

supra C*continuous if f1(V) is supra Cset in X for every supra open set V of Y.
Theorem 5.3: A set S of X is supra regularopen if and only if S is supra preopen and supra tset.
Proof: Let S be supra regularopen. By theorem 4.5, S is supra tset. Also By Theorem 4.6, S is supra open. Thus S is supra preopen.
Conversely, Let S be supra preopen and supra tset. 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 Aset.
Proof: Let S be supra open. Then S is supra open and by Theorem 4.1, S is supra Aset. Conversely, Let S be supra
open and supra Aset. Since S is supra Aset, 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 Bset.
Proof: Let S be a supra open set. Then S is supra open. Also, by Theorem 4.10, S is supra Bset.
Conversely let S be supra open and supra Bset. Since S is supra Bset, S = XS where X is supra open and S is supra tset.Then S = XS X ((S)) (as S is supra preopen) = X(S) (as S is supra tset). We have S X(S) implies S (S). But always (S)

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 Cset.
Proof: Let S be supra open in X. Then S is supra open set and by Theorem 4.17, S is supra Cset.
Conversely, let S be a supra open set and supra Cset. Since S is supra Cset, S = XS where X is supra open and S is supra hset. Since S is supra open and S is supra hset. Since S is supra open set, S (((S))) =
( ( (XS))) ( ( (X)))
( ( (S))) = ( (X)) (S) (as X is supra open and S is supra hset). 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 Mcontinuity if and only if

f is supra *continuous and supra A continuous.

f is supra*continuous and supra B continuous.

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.
REFERENCES

R. Devi, S. Sampathkumar and M. Caldas, On supra open sets and scontinuous maps, General Mathematics, 16(2) (2008), 77 84.

J. Dontchev and M. Przemski,On the various decomposition of continuous and some weakly continuous function, Acta Math. Hungar.71(1996)(12), 109120.

M. Gansterand I. L. Reilly, A Decomposition of continuity, Acta Math. Hungar., 56(34)(1990), 299301.

E. Hatir, T. Noiri and S. Yuksel, A Decomposition of continuity, Acta Math. Hungar., 94(12)(1996), 145150.

M. Kamaraj, G. Ramkumar, O. Ravi and M. L. Thivagar, Mildly supra normal spaces and some maps, International Journal of Advances in Pure and Applied Mathematics, 1(4)(2011), 6885.

A. S. Mashhour, A. A. Allam, F. S Mahmoud and F. H. Khedr, On supra topological spaces, Indian J. Pure and Appl. Math., 14(4) (1983), 502510.

O. Njastad, On some classes of nearly open sets, Pacific J. Math., 15(1965), 961970.

O. Ravi, M. L. Thivagar and ErdalEkici, On (1, 2)* sets and Decompositions Bitopological (1, 2)* continuous mappings, Kochi J. Math., 3(2008), 181189.

O. Ravi, G. Ramkumar and M. Kamaraj, On supra open sets and supra continuity on topological spaces, Proceed. National Seminar held at Sivakasi, India, (2011), 2231.

O. R. Sayed and T. Noiri, On supra bopen sets and supra b continuity on topological spaces, European J.Pure and Applied Math.,(3) (2) (2010), 295302.

O. R. Sayed, Supra preopen sets and supra precontinuity on topological spaces, Vasile Alecsandri, University of Bacau, Faculty of Sciences, Scientific Studies and Research Series Mathematics and Informatics., 20(2)(2010), 7988.

J. Tong, A decomposition of continuity, Acta Math. Hungar., 48(12)(1986), 1115.

J. Tong, On Decomposition of continuity in topological spaces, Acta Math. Hungar., 54(12)(1989), 5155.