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International Journal of Computer Application Available online on http://www.rspublication.com/ijca/ijca_index.htm Issue 4, Volume 1 (February 2014) ISSN: 2250-1797 STRONGLY *- CLOSED SETS IN TOPOLOGICAL SPACES Veronica Vijayan , Associate Professor, Nirmala College for Women , Coimbatore. Akilandeswari. K, PG student, Nirmala College for Women, Coimbatore. ABSTRACT In this paper we introduce a new class of sets namely, strongly *- closed sets which settled in between the class of closed sets and the class of strongly g-closed sets .Applying these sets we introduce three new classes of spaces namely T s * spaces , Keywords: Strongly *T s * spaces and s * T sg spaces . *-closed sets, T s * spaces , *T s * spaces and s * T sg spaces . 1. INTRODUCTION Levine.N[5] introduced the class of g-closed sets, a super class of closed sets in 1970. S.P Arya and Tour [1] defined gs-closed sets in 1990. . Levine.N[4] introduced semi-open sets and pre-open sets respectively. Maki.et.al[6] defined g-closed sets in 1994. M.K.R.S Veerakumar[12] introduced the generalized g*-closed sets in 2000. Pauline Mary Helen, Ponnuthai Selvarani ,Veronica Vijayan , Punitha Tharani [8]introduced strongly g-closed sets and strongly g**-closed sets in 2012. The purpose of this paper is to introduce the concept of strongly *-closed sets, T s * spaces , *T s * spaces and s * T sg spaces and investigate some of their properties. 2.PRELIMINARIES Throughout this paper(X, ) represents a non-empty topological spaces on which no separation axioms are assumed unless otherwise mentioned . For a subset A of a space (X, ), Cl(A) ,and int(A) denote the closure and the interior of A respectively. Definition 2-1: A subset A of a topological space (X, ) is called a 1) - open set [7] if A int(Cl(int(A)) and an - closed set if Cl(int(Cl(A))) A. R S. Publication (rspublication.com), [email protected] Page 12 International Journal of Computer Application Available online on http://www.rspublication.com/ijca/ijca_index.htm Issue 4, Volume 1 (February 2014) ISSN: 2250-1797 2) Semi-open set [4] if A Cl(int(A)) and a semi-closed set if int(Cl(A)) A. 3) g-closed set [5] if Cl(A) U whenever A U and U is open in (X, ). 4) Strongly g-closed set [8] if Cl(int(A)) U whenever A U and U is open in (X, ). 5) * -closed set [13] if Cl(A) U whenever A U and U is - open in (X, ). 6) gs-closed set [1] if Scl(A) U whenever A U and U is open in (X, ). 7) g-closed set [6] if cl(A) U whenever A U and U is open in (X, ). 8) -closed set [11] if Cl(A) U whenever A U and U is semi-open in (X, ). 9) g*-closed set [12] if Cl(A) U whenever A U and U is g-open in (X, ). 10) g*s-closed set [10] if Scl(A) U whenever A U and U is gs-open in (X, ). 11) g**-closed set [9] if Cl(A) U whenever A U and U is g*-open in (X, ). 12) Sg-closed set [2]if Scl(A) U whenever A U and U is semi-open in (X, ). 13) -closed set [14] if Cl(A) U whenever A U and U is sg-open in (X, ). Definition 2-2: A topological space (X, ) is said to be a 1) T * space[13] if every *-closed set in it is closed. 2) T 1/ 2S space[15] if every strongly g -closed set in it is closed. 3) Td space[3] if every gs-closed set in it is g-closed. 3. BASIC PROPERTIES OF STRONGLY *- CLOSED SETS We introduce the following definition. Definition 3.1: A subset A of a topological space (X, ) is said to be a strongly whenever A U and U is *-closed set if Cl(int(A)) U - open in X. Proposition 3.2: Every closed set is a strongly *- closed set. Proof follows from the definition. The following example supports that a strongly *- closed set need not be closed in general. R S. Publication (rspublication.com), [email protected] Page 13 International Journal of Computer Application Available online on http://www.rspublication.com/ijca/ijca_index.htm Issue 4, Volume 1 (February 2014) ISSN: 2250-1797 Example 3.3: Let X={a,b,c} , = { ,X,{a},{a,b}}. Then A= {b} is strongly *- closed but not closed in (X, ). Proposition 3.4: Every strongly *- closed set is strongly g-closed. Proof: Let A be strongly * - closed . Let A U and U be open. Then A U and U is -open and Cl(int (A)) U since A is strongly * -closed. Hence A is strongly g -closed . The converse of the above proposition need not be true in general as seen in the following example. Example 3.5: Let X= {a,b,c} , = { ,X,{b}}.Then A= {a,b} is strongly g - closed but not strongly *- closed in (X, ). Proposition 3.6: Every *- closed set is strongly *- closed . *- closed set. Let A U and U be -open. Then Cl(A) U since A is *-closed and Cl(int(A)) Cl(A ) U. Hence A is strongly *-closed. Proof: Let A be a The following example supports that a strongly * closed need not be *- closed in general. Example 3.7: Let X={a,b,c} , ={ ,X,{b,c}}.Then A={b} is strongly *- closed but not a *- closed set in (X, ). Remark 3.8: Strongly *- closedness is independent from g-closedness , gs-closedness and g – closedness . Example 3.9: Let X={a,b,c} , ={ ,X,{b,c}}.Then A={b} is strongly *- closed but not g-closed in (X, ). Let X={a,b,c}, ={ ,X,{b}}.Then A={a,b} is g-closed but not a strongly Hence strongly *- closed set in (X, ). *- closedness is independent of g-closedness . Example 3.10: Let X={a,b,c} , ={ ,X,{c}}.Then A={a,c} is gs-closed but not a strongly set in (X, ).Let X={a,b,c} , *- closed ={ ,X,{a,c}}. Then A={a} is strongly *- closed but not a gs-closed set in (X, ). Hence strongly *- closedness is independent of gs-closedness. R S. Publication (rspublication.com), [email protected] Page 14 International Journal of Computer Application Available online on http://www.rspublication.com/ijca/ijca_index.htm Issue 4, Volume 1 (February 2014) ISSN: 2250-1797 Example 3.11: Let X={a,b,c} , ={ ,X,{a,c}}.Then A={a} is strongly * closed but not g-closed in (X, ). Let X={a,b,c}, ={ ,X,{b},{b,c}}.Then A={a,b} is g -closed but not a strongly *- closed set in (X, ). Hence strongly *- closedness is independent of g -closedness. Proposition 3.12: Every -closed set is strongly *- closed . Proof follows from the definitions. The reverse implication does not hold as it can be seen from the following example. Example 3.13: Let X= {a,b,c} , = { ,X,{a}}.Then A={b}is strongly *- closed but not a -closed set in (X, ). Remark 3.14: Strongly *- closedness is independent from g*-closedness , g*s- closedness and g**closedness. Example 3.15: Let X={a,b,c} , ={ ,X,{a},{a,c}}.Then A={c} is strongly closed set in (X, ). B={a,b} is g*-closed but not strongly *- closed but not a g*- *- closed in (X, ).Hence strongly *- closedness is independent of g*-closedness. Example 3.16: Let X={a,b,c} , ={ ,X,{a},{b},{a,b}}.Then A= {b} is g*s- closed but not a strongly * -closed set in (X, ). Let X={a,b,c} , ={ ,X,{a,b},{c}}.Then A={b} is strongly *- closed but not a g*s-closed set in (X, ).Hence strongly *- closedness is independent of g*s-closedness. Example 3.17: Let X={a,b,c} , ={ ,X,{a},{a,c}}.Then A={c} is strongly closed set in (X, ). B={a,b} is g**-closed but not strongly *- closed but not a g**- *- closed in (X, ).Hence strongly *- closedness is independent of g**-closedness. Proposition 3.18: Every - closed set is strongly *- closed but not conversely. Example 3.19: Let X={a,b,c} , ={ ,X,{b,c}} . Then A={b} is strongly *- closed but not a - closed set in(X, ). Theorem 3.20: If A is strongly *- closed and A B Cl(int(A)), then B is strongly *- closed . R S. Publication (rspublication.com), [email protected] Page 15 International Journal of Computer Application Available online on http://www.rspublication.com/ijca/ijca_index.htm Issue 4, Volume 1 (February 2014) ISSN: 2250-1797 - open. Then A U where U is - open. Cl(int(A)) U,since A is *-closed. Cl(int(B)) Cl(B) Cl(int(A)) U. B is strongly *-closed. Proof: Let B U where U is strongly Theorem 3.21: If a subset of a topological space (x, ) is both open and strongly *- closed, then it is closed. Proof: Suppose A is both open and strongly *- closed. Cl(int(A)) A and Cl(A) = Cl(int(A)) A . Hence A is closed. Corollary 3.22: If A is both open and strongly *- closed in X, then it is both regular open and regular closed . Corollary 3.23: If a subset A of a topological space (x, ) is both strongly is *- closed and open , then it *-closed. The above results can be represented in the following figure. Strongly g-closed * closed closed g-closed g**-closed strongly *- closed gs-closed - closed g -closed g*s-closed g*-closed * where A B(resp. A - closed B) represents A implies B but not conversely(resp. A and B are independent). R S. Publication (rspublication.com), [email protected] Page 16 International Journal of Computer Application Available online on http://www.rspublication.com/ijca/ijca_index.htm Issue 4, Volume 1 (February 2014) ISSN: 2250-1797 4. APPLICATIONS OF STRONGLY *-CLOSED SETS As applications of strongly sg *-closed sets, new spaces, namely, T s * space , * T s * space and s * T space are introduced. We introduce the following definitions. Definition 4.1: A space (X, ) is called a T s * space if every strongly Definition 4.2: A space (X, ) is called a *-closed set is closed. * T s * space if every strongly *-closed set is *- closed. Definition 4.3: A space (X, ) is called a s * T sg space if every strongly g-closed set is strongly *- closed. Theorem 4.4: Every T s * space is a T * space . Proof: Let (X, ) be a T s * space. Let A be *-closed .Then A is a strongly *-closed set, and since (X, ) is a T s * space, A is a closed set. Hence the space (X, ) is a T * space . The reverse implication does not hold as it can be seen from the following example. Example 4.5: Let X= {a,b,c} , = { ,X,{b}}.Then (X, ) is a T * space. A={a} is strongly *- closed but not a closed set. The space (X, ) is not a T s * space. Theorem 4.6: Every T s * space is a * T s * space . Proof: Let (X, ) be a T s * space. Let A be strongly *-closed. Then A is closed since (X, ) is a T s * space. Since every closed set is a * - closed set, A is *-closed . Hence the space (X, ) is a * T s * space . The converse of the above theorem is not true in general as it can be seen from the following example. Example 4.7: Let X={a,b,c} , ={ ,X,{a},{b,c}}.Then (X, ) is a * T s * space .A={b}is strongly *- closed but not a closed set . The space (X, ) is not a T s * space. R S. Publication (rspublication.com), [email protected] Page 17 International Journal of Computer Application Available online on http://www.rspublication.com/ijca/ijca_index.htm Issue 4, Volume 1 (February 2014) ISSN: 2250-1797 Theorem 4.8: Every T 1/ 2 S space is a s * T sg space. The converse of the above theorem need not be true in general as seen in the following example. Example 4.9: Let X={a,b,c} , ={ ,X,{b,c}}. Then (X, ) is a s * T sg space. A={b} is a strongly g – closed set but not a closed set . The space (X, ) is not a T 1/ 2 S space . Theorem 4.10: A space which is both Proof: Let A be strongly * T s * and T * space is a T s * space . *- closed . Then A is *- closed, since the space is a * T s * space. Now A is closed, since the space is a T * space . The space (X, ) is a T s * space . Theorem 4.11: A space which is both s * T sg and T s * space is a T 1/ 2 S space . Proof follows from the definitions. Remark 4.12: T s * ness , * T s * ness and s * T sg ness are independent from T d ness. Example 4.13: Let X={a,b,c} , ={ ,X,{b,c}}. Then (X, ) is a T d space . A={b} is strongly *closed but not closed. Hence the space (X, ) is not a T s * space but it is a T d space . Let X={a,b,c} , ={ ,X,{a},{b},{a,b}} .Then (X, ) is a T s * space . A={a}is gs-closed but not a gclosed set. Hence the space (X, ) is not a T d space .Hence T s * ness is independent from T d ness. Example 4.14: Let X={a,b,c} , ={ ,X,{b,c}}.Then (X, ) is a T d space .A={b} is strongly *closed but not a *-closed set .Hence (X, ) is not a * T s * space .Let X={a,b,c} , ={ ,X,{a},{b},{a,b}}. Then(X, ) is a * T s * space. A={a} is gs-closed but not a g-closed set. The space (X, ) is not a T d space . Hence * T s * ness is independent from T d ness. Example 4.15: Let X={a,b,c} , ={ ,X,{b}}.Then (X, ) is a T d space .A= {a,b} is strongly g closed but not strongly *- closed. The space (X, ) is not a s * T sg space . R S. Publication (rspublication.com), [email protected] Page 18 International Journal of Computer Application Available online on http://www.rspublication.com/ijca/ijca_index.htm Issue 4, Volume 1 (February 2014) ISSN: 2250-1797 Let X={a,b,c} , ={ ,X,{a},{b},{a,b}}.Then (X, ) is a s * T sg space. A={a} is gs-closed but not a g-closed set. The space (X, ) is not a T d space .Hence s * T sg ness is independent from T d ness. Remark 4.16: s * T sg ness is independent from T * ness. Example 4.17: Let X={a,b,c} , ={ ,X,{b}}.Then (X, ) is a T * space . A= {a,b} is strongly g closed but not strongly *- closed. The space (X, ) is not a s * T sg space . Example 4.18: Let X={a,b,c} , ={ ,X,{b,c}}. Then space (X, ) is a s * T sg space. A= {a,b} is *-closed but not closed. The space (X, ) is not a T * space .Hence s * T sg ness is independent from T * ness. The above results can be represented in the following figure. * T s * T s * Td s * T * where A B(resp. A T sg T 1/ 2 S B ) represents A implies B(resp. A and B are independent). REFERENCES [1] Arya.S.P and Nour.T, Characterizations of s-normal spaces , Indian.J. Pure. Appl. Math., 21(8)(1990),717-719. [2] Bhattacharya.P and Lahiri.B.K, Semi-generalized closed sets in topology , Indian J. Math., 29(3)(1987), 375-382. R S. Publication (rspublication.com), [email protected] Page 19 International Journal of Computer Application Available online on http://www.rspublication.com/ijca/ijca_index.htm Issue 4, Volume 1 (February 2014) ISSN: 2250-1797 [3]Devi.R, Balachandran.K and Maki.H, Semi-generalized closed maps and generalized closed maps, Mem. Fac. Sci. Kochi Univ. Ser.A. Math., 14(1993), 41-54. [4] Levine.N, Semi-open sets and semi-continuity in topological spaces, Amer. Math. Monthly,70(1963), 36-41. [5]Levine.N, Generalized closed sets in topology , Rend. Circ. Math. Palermo, 19(2)(1970),89-96. [6] Maki.H, Devi.R and Balachandran.K, Associated topologies of generalized -closed maps and - generalized closed sets, Mem. Fac. Sci. Kochi Univ. Ser.A, Math., 15(1994), 51-63. [7]Njastad.O, On some classes of nearly open sets, Pacific J.Math., 15(1965), 961-970. [8]Pauline Mary Helen, Veronica Vijayan , Ponnuthai Selvarani , Punitha Tharani, Strongly g-closed sets and Strongly g**-closed sets , IJMA , 3(8)(2012), 3108-3112. [9] Pauline Mary Helen.M, Veronica Vijayan, Ponnuthai Selvarani.S, g**-closed sets in topological spaces,IJMA,3(5)(2012),1-15. [10]Pushpalatha.P and Anitha .K , g*s-closed sets in topological space, Int.J. contemp.Math.Sciences, Vol.6., March 2011, no 19, 917-929. [11]Sundaram.P and Shiek john.M, on -closed sets in topological spaces , Actaciencia Indica 4(2000), 389-392. [12] Veerakumar M.K.R.S , Between closed sets and g-closed sets , Mem. Fac. Sci. Kochi Univ. Ser.A, Math.,17(1996),33-42 [13]Veronica Vijayan, Priya.F, *-closed sets in topological spaces, IJCA, Issue 3, Volume 4 (July – August2013), ISSN:2250-1797,49-60. [14] Veronica Vijayan, Selvapriya.K , A study on *, , * -closed sets in topological spaces , IJCA, Issue 3,Volume 2(April 2013) ISSN:2250-1797, 105-115. [15] Veronica Vijayan, Thenmozhi.B, Strongly g, g*,g** - closed sets in bitopological spaces , IJCA, Issue 3, Volume 3 (May-June 2013) ISSN:2250-1797, 28-35. R S. Publication (rspublication.com), [email protected] Page 20
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