Decomposition of Locally Closed Sets in Topological Spaces

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1 Advances in Fuzzy Mathematics (AFM). ISSN Volume 12, Number 1 (2017), pp Research India Publications Decomposition of Locally Closed Sets in Topological Spaces P. G. Patil, S. S. Benchalli and Pallavi S. Mirajakar Department of Mathematics, Karnatak University, Dharwad , Karnataka, India. psmiraj Abstract The purpose of this paper is to introduce g ωα-lc sets, g ωα -lc sets and g ωα -lc sets and different notions of generalizations of continuous functions in topological spaces and study their properties. AMS subject classification: 54A05, 54C05, 54C08. Keywords: g ωα-closed set, g ωα-lc sets, g ωα -lc sets, g ωα -lc sets and g ωαlc continuous functions. 1. Introduction The notion of locally closed sets in the literature was first introduced and studied by Kuratowaski and Sierpienski [4]. Bourbaki [2] defined, a subset A of a topological space X is locally closed if it is the intersection of an open set and a closed set. Further, Stone [8] has used the term FG for a locally closed subset. Using the concept of a locally closed set, in 1989 Ganster and Reilly [3] continued their work and introduced the concept of LC-continuous and LC-irresolute maps to find a decomposition of continuous functions. In 1996, Balachandran et al. [1] introduced and investigated the concepts of generalized locally closed sets and obtained different notions of continuity called GLCcontinuity and GLC-irresolue maps. Various authors contributed to the development of generalizations of locally closed sets and locally closed continuous functions in topological spaces. In this paper, we introduce new classes of sets called g ωα-lc sets, g ωα -lc sets and g ωα -lc sets by using the notion of g ωα-closed and g ωα-open sets. We study some of their properties and the relationship among these classes and the other existing classes of sets. Finally, we also introduce and study different classes of weaker forms of continuity and irresoluteness and some of their properties in topological spaces.

2 102 P. G. Patil, et al. 2. Preliminaries Throughout this paper (X, τ), (Y, µ) and (Z, σ ) (or simply X, Y and Z) always mean topological spaces on which no separation axioms are assumed unless explicitly stated. For a subset A of a space X the closure and interior of A with respect to τ are denoted by cl(a) and int(a) respectively. Definition 2.1. [5] Let A be a subset of X. Then A is said to be a generalized star ωα-closed (briefly g ωα-closed) if cl(a) U whenever A UandUisωα-open in X. Definition 2.2. A subset A of a space X is called a (i) locally closed [3] if A = G F where G is open and F is closed in X. (ii) generalized locally closed (briefly glc-closed) [1] if A = G F where G is g-open and F is g-closed in X. Theorem 2.3. [6] In a topological space X, if every g ωα-closed set is closed then X is said to be T g ωα-space. Definition 2.4. [7] A function f : X Y is called a (i) g ωα-continuous if f 1 (V) is g ωα-closed in X for every closed set V in Y. (ii) g ωα-irresolute if f 1 (V) is g ωα-closed in X for every g ωα-closed set V in Y. Proposition 2.5. [5] Following statements are true for a topological space X: (i) A is g ωα-closed in X if and only if g ωα-cl(a) = A. (ii) g ωα-cl(a) is g ωα-closed in X. (iii) x g ωα cl(a) if and only if A U = φ for every U G ωαo(x, x). 3. g ωα-locally Closed Sets in Topological Spaces Definition 3.1. Let A X. Then A is said to be a (i) g ωα-locally closed (briefly g ωα-lc) if A = G F where G is g ωα-open and F is g ωα-closed in X. (ii) g ωα -locally closed (briefly g ωα -lc) if A = G F where G is g ωα-open and F is closed in X. (iii) g ωα -locally closed (briefly g ωα -lc) if A = G F where G is open and F is g ωα-closed in X.

3 Decomposition of Locally Closed Sets in Topological Spaces 103 Remark 3.2. The family of all g ωα-lc sets (resp. g ωα -lc sets, g ωα -lc sets) of (X, τ) will be denoted by G ωα-lc(x, τ)(resp. G ωα -LC(X, τ), G ωα -LC(X, τ)). Remark 3.3. It is obvious that every g ωα-closed (resp. g ωα-open) set is g ωα-locally closed. Remark 3.4. Every locally closed set is g ωα-locally closed but not conversely. Example 3.5. X ={a,b,c} with τ ={X, φ, {a}, {b, c}}. A subset {a,b} of the space Xisg ωα-locally closed but not locally closed. Remark 3.6. (i) Every locally closed set is g ωα -lc and g ωα -lc set. (ii) Every g ωα-lc set is g ωα -lc set and g ωα -lc set. However, converses of the above remark need not be true in general as seen from the following example. Example 3.7. Let X ={a,b,c}, τ ={X, φ, {a,b}}. A subset {a,c} of X is g ωαlocally closed but not g ωα -locally closed and locally closed. The set {c} is g ωαlocally closed but not g ωα -locally closed. Remark 3.8. The class of g ωα -locally closed and g ωα -locally closed sets are independent of each other as seen from the following example. Example 3.9. Let X ={a,b,c} and τ ={X, φ, {a}, {a,c}}. In this topological space X, the set A ={b} is g ωα -locally closed but not g ωα -locally closed. Example Let X ={a,b,c} and τ ={X, φ, {a,b}}. In this topological space X, the set A ={a} is g ωα -locally closed but not g ωα -locally closed. We have the following characterizations: Proposition The following properties holds for a T g ωα-space: (i) G ωα LC(X, τ) = LC(X, τ). (ii) G ωα LC(X, τ) Gωα LC(X, τ). Proof. (i) Follows from the fact that every closed set is g ωα-closed by [5] and from the definition of T g ωα-space. (ii) In any space (X, τ), LC(X, τ) Gωα LC(X, τ) and from (i), we have G ωα LC(X, τ) Gωα LC(X, τ). Theorem The following properties are equivalent for any subset A of a space X:

4 104 P. G. Patil, et al. (i) A G ωα-lc(x, τ) (ii) A = G g ωα cl(a) for some g ωα-open set G (iii) g ωα cl(a)\ Aisg ωα-closed (iv) A (X \ g ωα cl(a)) is g ωα-open. Proof. (i) (ii): Let A G ωα-lc(x, τ). Then there exist g ωα-open set G and g ωαclosed set F such that A = G F. Since A G and A g ωα cl(a) then A G g ωα cl(a). Conversely, from Proposition 2.1(ii), we have g ωα cl(a) F and hence G g ωα cl(a) G F = A. (ii) (i): From hypothesis and Proposition 2.1(ii), g ωα cl(a) is g ωα-closed and hence A = G g ωα cl(a) G ωα-lc(x, τ). (ii) (iii): Suppose (ii) holds then g ωα cl(a)\ A=g ωα cl(a) (X \ G) which is g ωα-closed. Hence g ωα cl(a)\ Aisg ωα-closed. (iii) (ii): Let G = X \ (g ωα cl(a) \ A). Then from (iii), G is g ωα-open in X and hence A = G g ωα cl(a) holds. (iii) (iv): Let F = g ωα cl(a) \ A. Then X \ F = A (X \ g ωα cl(a)) holds and X \ F is g ωα-open. Hence A (X \ g ωα cl(a)) is g ωα-open. (iv) (iii): Let G = A (X \ g ωα cl(a)). Since X \ G is g ωα-closed and X \ G = g ωα cl(a) \ A holds. Hence g ωα cl(a) \ A is g ωα-closed. Theorem Let A be any subset of a space (X, τ). Then the following properties are equivalent: (i) A G ωα -LC(X, τ) (ii) A = U cl(a) for some g ωα-open set G (iii) cl(a) \ A is g ωα-closed (iv) A (X \ cl(a)) is g ωα-open. Proposition Let A be a subset of (X, τ). If A G ωα -LC(X, τ) if and only if A = U g ωα cl(a) for some open set U. Proof. Necessity: Let A G ωα -LC(X, τ). Then there exist open set G and g ωαclosed set F such that A = G F. Then from Proposition 2.1 (ii), A F implies

5 Decomposition of Locally Closed Sets in Topological Spaces 105 g ωα-cl(a) F. Now A = A g ωα-cl(a) = G F g ωα-cl(a) = G g ωα-cl(a). Sufficiency: Let A = G g ωα-cl(a) for some open set G. Then from Proposition 2.1 (ii), g ωα-cl(a) is g ωα-closed and hence A = G g ωα-cl(a) G ωα -LC(X, τ). Theorem Let A be a subset of (X, τ). If A G ωα -LC(X, τ) then (i) g ωα-cl(a)\a is g ωα-closed. (ii) A (X \ g ωα-cl(a)) is g ωα-open. Proof. (i) Let A G ωα -LC(X, τ) then A = G F where G is open and F is g ωαclosed. Since A G and A g ωα-cl(a) so A G g ωα-cl(a). Conversely, from Proposition 2.1 (ii), we have g ωα-cl(a) F and hence G g ωαcl(a) G F = A. Therefore A = G g ωα-cl(a). Then it follows from the assumption that g ωα-cl(a) \ A=g ωα-cl(a) (X \ G) is g ωα-closed in X. (ii) From (i), g ωα-cl(a) \ Aisg ωα-closed in X and let F = g ωα-cl(a)\a. Since X \ F = A (X \ g ωα-cl(a)) holds so X \ F is g ωα-open. Therefore A (X \ g ωαcl(a)) is g ωα-open. Definition Let A and B be any two subsets of a space X. Then A and B are said to be separated if A cl(b) = φ and B cl(a) = φ. Theorem Let A, B G ωα -LC(X, τ). Suppose that the collection of all g ωαopen sets of (X, τ) are closed under finite unions. If A and B are separated in (X, τ) then A B g ωα -LC(X, τ). Proof. Since A, B g ωα -LC(X, τ). Then from Theorem 3.2 (ii), there exist g ωαopen sets G and F in X such that A = G cl(a) and B = F cl(b). Put U = G (X cl(b)) and V = F (X cl(a)). Then U and V are g ωα-open subsets of X implies that A = U cl(a), B = U cl(b), U cl(b)) = φ and U cl(a) = φ. Therefore A B = (U V) (cl(a B)), that is A B g ωα -LC(X, τ). Remark From the following example we can observe that assumption A and B are separated cannot be removed. Example Let X ={a,b,c} with τ ={X, φ, {a,b}}. The set {b, c} g ωα -lc and {c} g ωα -lc but {b, c} / g ωα -lc. Remark Union of two g ωα-lc (resp. g ωα -lc, g ωα -lc) sets need not be g ωα-lc (resp. g ωα -lc, g ωα -lc) sets as seen from the following example. Example Let X ={a,b,c}, τ ={X, φ, {a}, {a,b}}.then {a} and {c} are g ωα-lc sets but {a} {c} ={a,c} is not g ωα-lc set in X.

6 106 P. G. Patil, et al. Example Let X ={a,b,c}, τ ={X, φ, {a}, {a,b}, {a,c}}.then {b} and {c} are g ωα -lc and g ωα -lc sets but {b, c} is not g ωα-lc and g ωα -lc in X. Theorem Let A, B X. Suppose that the collection of g ωα-closed set of X is closed under finite intersection then the following properties holds: (i) if A g ωα-lc(x, τ)andbisg ωα-open and g ωα-closed then A B g ωα- LC(X, τ). (ii) if A, B g ωα -LC(X, τ) then A B g ωα -LC(X, τ). Proof. (i) Let A g ωα-lc(x, τ). Then there exist g ωα-open set G and g ωα-closed set F such that A = G F,soA B = (G F) B. IfBisg ωα-open then A B = (G B) F g ωα-lc(x, τ). IfBisg ωα-closed then A B = G (F B) g ωα-lc(x, τ). Hence A B is g ωα-closed. Thus A B g ωα-lc(x, τ). (ii) Let A, B g ωα -LC(X, τ). Then from Theorem 3.2 (ii), there exist g ωαopen sets P and Q such that A = P cl(a)) and B = Q cl(b)). Then A B = (P cl(a)) (Q cl(b)) = (P Q) (cl(a) cl(b)) g ωα -LC(X, τ). Proposition Let A and B be any two subsets of a space X such that A B. Suppose the collection of g ωα-open sets of X are closed under finite intersection. If B is g ωα-open in X and A g ωα LC(B, τ/b) then A g ωα LC(X, τ). Proof. Let A g ωα LC(B, τ/b) then there exists g ωα-open set G in (X, τ/b) such that A = G cl(a) B where cl(a) B = B cl(a). Since G and B are g ωα-open then G B be is also g ωα-open [5]. This implies that A = (G B) cl(a) g ωα LC(X, τ). Definition Let X be a topological space. Then X is said to be g ωα-submaximal if every desne subset is g ωα-open. Remark Every submaximal space is g ωα-submaximal. However the converse need not be true as seen from the following example. Example X ={a,b,c} and τ ={X, φ, {a}, {b, c}}. Let A ={a,b} then A is dense in X such that A is g ωα-open but not open in X. Theorem A topological space X is g ωα-submaximal if and only if P(X) = g ωα LC(X, τ). Proof. LetXbeag ωα-submaximal and A P(X). Let U = A (X \ cl(a)). Then cl(u) = X and hence U is dense in X. Since X is g ωα submaximal, so U is g ωαopen in X. Then from Theorem 3.2, A g ωα LC(X, τ). This implies that P(X) = g ωα LC(X, τ).

7 Decomposition of Locally Closed Sets in Topological Spaces 107 Conversely, let A be dense in X. Then A (X \ cl(a)) = A φ = A. Since A g ωα LC(X, τ), A = A (X \ cl(a)) is g ωα-open from Theorem 3.2 (iv). Hence X is g ωα-submaximal. Proposition Let {Z i : i τ} be a cover of X, where τ is finite set and A be a subset of X. Suppose {Z i : i τ} is g ωα-closed in X and the collection of g ωα-closed sets is closed under finite unions. If A Z i g ωα LC(Z i,τ/z i ) for each i τ then A g ωα LC(X, τ). Proof. Let i τ. Since A Z i g ωα LC(Z i,τ/z i ) there exist an open set U i of (X, τ) and g ωα-closed set F i of (Z i,τ/z i ) such that A Z i = (U i Z i ) F i = U i (Z i F i ). Then A = {A Z i : i τ} = {U i : i τ} ( {Z i F i : i τ}). Hence A g ωα LC(X, τ). 4. g ωα-lc Continuous and g ωα-lc Irresolute Functions in Topological Spaces Definition 4.1. Let f : (X, τ) (Y, σ ) be a function. Then f is called a (a) g ωα LC (resp. g ωα LC and g ωα LC) continuous if for each V (Y, σ ), f 1 (V ) g ωα LC(X, τ) (resp. g ωα LC(X, τ) and g ωα LC(X, τ)). (b) g ωα LC (resp. g ωα LC and g ωα LC) irresolute if for each V g ωα LC(Y, σ ), f 1 (V ) g ωα LC(X, τ) (resp. g ωα LC(X, τ) and g ωα LC(X, τ)). Theorem 4.2. The following properties holds for a function f : (X, τ) (Y, σ ): (a) if f is LC-continuous then f is g ωα LC continuous, g ωα LC continuous and g ωα LC continuous. (b) if f is g ωα LC continuous or g ωα LC continuous then f is g ωα LC continuous. (c) if f is g ωα LC (resp. g ωα LC and g ωα LC) irresolte then it is g ωα LC (resp. g ωα LC and g ωα LC) continuous. Proof. (a) It follows from the Remark 3.4. (b) Since every g ωα -lc and g ωα -lc set is g ωα-lc set and hence the proof follows. (c) Since every open set is g ωα-open, g ωα -open and g ωα -open sets. However the converse of the above statements need not be true as seen from the following examples.

8 108 P. G. Patil, et al. Example 4.3. Let X = Y ={a,b,c}, τ ={X, φ, {a}, {a,c}} and σ ={Y, φ, {a,b}}. Then the identity function f : X Y is g ωα lc continuous but not lc-continuous, since for the set A ={a,b} in Y, f 1 ({a,b}) ={a,c} is not locally closed in X. Let X = Y ={a,b,c}, τ ={X, φ, {a}, {a,b}} and σ ={Y, φ, {a,b}}. Then the identity function f is g ωα lc-continuous but not g ωα lc-irresolute, g ωα lccontinuous and g ωα lc-continuous. Consider the set {a,c} in Y, f 1 ({a,c}) = {a,c} is not g ωα-locally closed in X. Proposition 4.4. Let f : (X, τ) (Y, σ ) be g ωα-irresolute injective map. Then (a) if B g ωα LC(Y, σ ) then f 1 (B) g ωα LC(X, τ). (b) if X is T g ωα-space and B g ωα LC(Y, σ ) then f 1 (B) LC(X, τ). Proof. (a) Let B g ωα LC(X, τ). Then there exist g ωα-open set G and g ωα-closed set F such that B = G F, f 1 (B) = f 1 (G) f 1 (F ). Since f is g ωαirresolute, f 1 (G) and f 1 (F ) are g ωα-open and g ωα-closed sets in X respectively. Hence f 1 (B) g ωα LC(X, τ). (b) Let B g ωα LC(Y, σ ). There exist g ωα-open set G and g ωα-closed set F such that B = G F, f 1 (B) = f 1 (G) f 1 (F ). Since f is g ωαirresolute map, f 1 (G) and f 1 (F ) are g ωα-open and g ωα-closed sets in (X, τ) respectively. From hypothesis f 1 (G) and f 1 (F ) are open and closed sets in X. Hence f 1 (B) LC(X, τ). Theorem 4.5. Any map defined on a door space is g ωα LC continuous (resp. g ωα LC irresolute). Proof. Let f : (X, τ) (Y, σ ) be a function where (X, τ) is a door space. Let A (Y, σ ) (resp. A g ωα LC(Y, σ )) then f 1 (A) is either open or closed. Since every open or closed set is g ωα-open or g ωα-closed [5] respectively and hence f 1 (A) g ωα LC(X, τ). Hence f is g ωα LC continuous (resp. g ωα LC irresolute). Proposition 4.6. g ωα LC continuous and contra-continuous maps defined on a T g ωα-space is g ωα LC irresolute. Proof. Let f : (X, τ) (Y, σ ) be g ωα LC-continuous and contra-continuous maps and (Y, σ) be T g ωα-space. Let G g ωα LC(Y, σ ) then G = U F where U is g ωα-open and F is g ωα-closed then U is open and F is closed in (Y, σ ). Consider f 1 (G) = f 1 (U) f 1 (F ), where f 1 (U) is g ωα-locally closed and f 1 (F ) is open. Therefore f 1 (G) is g ωα-locally closed in (X, τ) by Theorem 3.5.

9 Decomposition of Locally Closed Sets in Topological Spaces 109 Proposition 4.7. A topological space (X, τ) is g ωα-submaximal if and only if every function having (X, τ) as a domain is g ωα LC-continuous. Proof. Let f : (X, τ) (Y, σ ) be a function. Then from Theorem 3.6, P(X) = g ωα LC(X, τ). Let U be an open set in (Y, σ ) then f 1 (U) P(X) = g ωα LC(X, τ), sofisg ωα LC continuous. Conversely, let us consider the Sierpinski space Y ={0, 1} with σ ={φ,y,{0}}. Let V be a subset of X and define a function f : (X, τ) (Y, σ ) asf(x)=0ifx V and f(x) =1ifx/ V. Then it follows from the assumption that f 1 {(0)} =V g ωα LC(X, τ). Therefore, we have P(X) = g ωα LC(X, τ) and so X is g ωα-submaximal. Now we will recall the definition of combination of two functions. Let X = A B and f : A Y and h : B Y be any two functions. We say that f and h are compatible if f A B = h A B, we define a function f h : X Y as follows: (f h)(x) = f(x)for every x A (f h)(x) = h(x) for every x B. Then the function f h : X Y is called the combination of f and h. Proposition 4.8. Let X = A B where A and B are g ωα-closed sets of X and f : (A, τ A) (Y, σ ) and h : (B, τ B) (Y, σ ) be compatible functions. If f and h are g ωα LC continuous (resp. g ωα LC irresolute ) then f h : (X, τ) (Y, σ ) is g ωα LC continuous (resp. g ωα LC irresolute). Proof. Let V (Y, σ ) (resp. g ωα LC(Y, σ )) then (f h) 1 (V ) A = f 1 (V ) and (f h) 1 (V ) B = h 1 (V ) holds. By assumption, we have (f h) 1 (V ) A g ωα LC(A, τ/a) and (f h) 1 (V ) B g ωα LC(B, τ/b). Then it follows that (f h) 1 (V ) g ωα LC(X, τ) and hence f h is g ωα LC continuous (resp. g ωα LC irresolute). Now we have the theorems concerning to composition of maps: Theorem 4.9. Let f : (X, τ) (Y, σ ) and g : (Y, σ ) (Z, η) are any two functions. Then (a) if f and g are g ωα LC irresolute (resp. g ωα LC irresolute and g ωα LC irresolute) then g f is g ωα LC irresolute (resp. g ωα LC irresolute and g ωα LC irresolute). (b) if f is g ωα LC irresolute and g is g ωα LC continuous then g f is g ωα LC continuous. Proof. (a) Let V g ωα LC(Z) (resp. g ωα LC(Z) and g ωα LC(Z)) then g 1 (V ) g ωα LC(Y ) (resp. g ωα LC(Y ) and g ωα LC(Y )) and since f

10 110 P. G. Patil, et al. is g ωα LC irresolute (resp. g ωα LC irresolute and g ωα LC irresolute), (f 1 ) 1 (V )) = (g f) 1 (V ) g ωα LC(X) (resp. g ωα LC(X) and g ωα LC(X)). Therefore (g f)is g ωα LC irresolute (resp. g ωα LC irresolute and g ωα LC irresolute). (b) Let V Z then g 1 (V ) g ωα LC(Y ) and f 1 (g 1 (V )) = (g f) 1 (V ) g ωα LC(X) as f is g ωα LC irresolute. Therefore (g f) 1 (V ) g ωα LC(X). Hence g f is g ωα LC continuous. Acknowledgement The first and second authors are grateful to the University Grants Commission, New Delhi, India for financial support under UGC SAP DRS-III: F-510/3/DRS-III/2016(SAP- I) dated 29th Feb 2016 to the Department of Mathematics, Karnatak University, Dharwad, India. Also the third author is thankful to Karnatak University, Dharwad, India for financial support under No.KU/Sch/UGC-UPE/ /893 dated 24th November, References [1] K. Balachandran, P. Sundaram and H. Maki, Generalized Locally Closed Sets and GLC-Continuous Functions, Indian Jl. Pure. Appl. Math. 27 (1997), No. 5, [2] N. Bourbaki, General Topology, Part I, Addison Wesley, Reading Mass, (1966). [3] M. Ganster, I. L. Reilly and M. K. Vamanamurthy, Locally Closed Sets and LC- Continuous Functions, Int. Jl. Math. Soc. 12, (1989), [4] C. Kuratowski, Sierpinski W. Sar les Differences de deux ensemble fermes, Tobuku Math. Jl. 20, (1921), [5] P. G. Patil, S. S. Benchalli and Pallavi S. Mirajakar, Generalized Star ωα-closed Sets in Topological Spaces, Jl. of New Results in Science, Vol 9, (2015), [6] P. G. Patil, S. S. Benchalli and Pallavi S. Mirajakar, Generalized Star ωα-spaces in Topological Spaces, Int. Jl. of Scientific and Innovative Mathematical Research, Vol. 3, Special Issue 1, (2015), [7] P. G. Patil, S. S. Benchalli and Pallavi S. Mirajakar, Generalization of Some New Continuous Functions in Topological Spaces, Scientia Magna, Vol. 11, No. 2, (2016), [8] A. H. Stone, Absolutely FG Spaces, Proc. Amer. Math. Soc. 80 (1980),