FUZZY WEAKLY CLOSED SETS

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1 Chapter 3 FUZZY WEAKLY CLOSED SETS In this chapter, we introduce another type of fuzzy closed set, called fuzzy weakly closed set in an FTS. Alongwith the study of fundamental results of such closed sets, we define and characterize fuzzy weakly compact space and fuzzy weakly closed space. 3.1 Introduction It is known that the different notions of topology are defined in terms of the open sets, which can also be replaced by either closed sets, or closure or interior, or neighborhood system. Among these again open sets and closed sets are very closely linked and are indispensable in the study of the topological notions. This has led to several generalizations of closed sets ([4], [9], [10], [27], [28], [36], [42], [55] etc.) which in turn have provided several new properties of topological spaces. As discussed in Chapter 1, similar generalizations are also found in fuzzy topological spaces. We refer to Chapter 1, for the definitions of various generalized closed sets in fuzzy topological spaces. Contents of this chapter is available online as a paper entitled On fuzzy weakly-closed sets in arxiv: v1 [math.gm] 31 Jul

2 3.2. FUZZY WEAKLY-CLOSED SETS Fuzzy weakly-closed sets Weakly closed sets in topological space was introduced by Sundaram et al. in [55]. In this section, we introduce fuzzy weakly-closed set in an FTS and study some general properties of such sets. Definition Let (X, τ) be an FTS and A be a fuzzy subset of X. Then A is said to be fuzzy weakly-closed if cl(a) U, whenever A U, U is fuzzy semi-open set. Complement of a fuzzy weakly-closed set is fuzzy weakly-open set. Example Consider the FTS (X, τ), where X = {a, b, c}, τ = {0 X, 1 X, B} and 1 if x = b; B(x) = 0 otherwise. 1 if x = a, c; Then F (x) = 0 otherwise, is fuzzy weakly-closed, whereas 1 if x = a, b; E(x) = 0 otherwise, is not fuzzy weakly-closed. Theorem Union of two fuzzy weakly-closed set is fuzzy weakly-closed. Proof. Let A and B be two fuzzy weakly-closed sets in X. Let U be a fuzzy semi-open set in X such that A B U A U and B U. Then by definition of fuzzy weakly closed set, we have cl(a) U and cl(b) U cl(a B) U A B is fuzzy weakly-closed. Remark Intersection of two fuzzy weakly-closed sets is not necessarily fuzzy weakly-closed.

3 3.2. FUZZY WEAKLY-CLOSED SETS 38 Example Consider the fuzzy sets A, I, N, F, E defined on X = {a, b, c, d} as follows: 1 if x = a; A(x) = 0 otherwise, 1 if x = b, d; I(x) = 0 otherwise, 1 if x = a, b, d; N(x) = 0 otherwise, 1 if x = a, c; F (x) = 0 otherwise, 1 if x = a, b; and E(x) = 0 otherwise Then τ = {0 X, A, I, N, 1 X } is a fuzzy topology on X. Here F and E are fuzzy weakly-closed sets but F E = A is not a fuzzy weakly closed set. Remark Arbitrary union of fuzzy weakly-closed sets is not necessarily fuzzy weakly-closed. Theorem Intersection of two fuzzy weakly-open sets is fuzzy weakly-open. Proof. Let A and B be two fuzzy weakly-open sets A c and B c are fuzzy weaklyclosed sets A c B c is a fuzzy weakly-closed set (AB) c is a fuzzy weakly-closed set A B is a fuzzy weakly-open set. Remark Union of two fuzzy weakly-open sets need not be fuzzy weakly-open. Remark Arbitrary intersection of fuzzy weakly-open sets need not be fuzzy weakly-open.

4 3.2. FUZZY WEAKLY-CLOSED SETS 39 Theorem A fuzzy set A in an FTS (X, τ) is fuzzy weakly open iff U int(a) whenever U A and U is fs-closed. Proof. Let A be fuzzy weakly open set such that U A and U is fs-closed A c U c, where U c is fs-open. Now A c is fuzzy weakly closed, so cl(a c ) U c U (cl(a c )) c = int(a). Conversely, assume that U int(a) whenever U A where U is fs-closed. Let A c V where V is fs-open V c A V c int(a) (by hypothesis) (int(a)) c V cl(a c ) V A c is fuzzy weakly closed A is fuzzy weakly open. Theorem If a fuzzy set A in an FTS (X, τ) is both fuzzy open and fuzzy generalized closed, then it is fuzzy weakly closed. Proof. Let A be an fuzzy open set which is also fg-closed cl(a) A as A is a fuzzy open set containing itself. We know A cl(a) for any fuzzy set in X. Hence A = cl(a) A is fuzzy closed set A is fuzzy weakly-closed set. Remark If A and B are two fuzzy sets such that A B cl(a), then cl(a) = cl(b). Theorem Let A be a fuzzy weakly-closed subset of an FTS (X, τ) and suppose the fuzzy subset B is such that A B cl(a), then B is also fuzzy weakly-closed. Proof. Let A be a fuzzy weakly-closed set and U be an fuzzy semi-open set such that B U A U. Then by definition of fuzzy weakly-closed set cl(a) U. But cl(b) = cl(a) cl(b) U B is fuzzy weakly-closed set. Definition Let A be a fuzzy set in an FTS X. Then fuzzy weakly-closure and weakly-interior of A(denoted by wcl(a) and wint(a) respectively) are defined as follows: wcl(a) = {B B is fuzzy weakly-closed set and A B}. wint(a) = {C C is fuzzy weakly-open set and C A}.

5 3.3. COMPARATIVE STUDY 40 If A is a fuzzy weakly-closed set then wcl(a) = A but the converse is not necessarily true since intersection of any two fuzzy weakly-closed sets may not be fuzzy weakly-closed. Following results are obtained from the definitions: wcl(u c ) = (wint(u)) c and wint(u c ) = (wcl(u)) c. 3.3 Comparative study In this section, fuzzy weakly closed set is compared with different fuzzy closed sets. Below is a diagram describing inter relationship of fuzzy weakly closed sets and other types of fuzzy closed sets. As implications are direct outcomes of the definitions, we are omitting the proofs for the same. However, counter examples are given wherever necessary. F-closed Fg*s-closed Fs-closed Fp-closed Fsp-closed Fα-closed F weakly-closed Fg-closed Fgs-closed Fsg-closed Fgp-closed Fgα-closed Fαg-closed Fpg-closed Fspg-closed Fgsp-closed Figure 3.1: Comparison of fuzzy weakly-closed set with different fuzzy closed sets As shown in the figure the following are some examples that the converses of

6 3.3. COMPARATIVE STUDY 41 some implications are not necessarily true and some of the classes of closed sets are independent of the class of fuzzy weakly closed sets. Example Taking the same FTS as defined in Example 3.2.5, we have N is a fuzzy weakly-closed set but not a fuzzy closed set. Example For the FTS in Example 3.2.5, 1 if x = b, d; I(x) = 0 otherwise is fgs-closed and fsg-closed set but not fuzzy weakly-closed set. Example For the FTS in Example 3.2.5, 1 if x = a, d; G(x) = 0 otherwise is a fg-closed set but not fuzzy weakly-closed set. Example But taking the FTS in Example 3.2.5, A is a fgα-closed set but not fuzzy weakly-closed set. Example Again for the FTS in Example 3.2.5, A is a fspg-closed set but not fuzzy weakly-closed set. Example Let X = {a, b, c} and H be a fuzzy set on it defined by 1 if x = a, b; E(x) = 0 otherwise. Then τ = {0 X, 1 X, E} is a fuzzy topology on X. Here E is a fgsp-closed but not fuzzy weakly-closed.

7 3.4. COMPACTNESS 42 Example For Example 3.2.5, 1 if x = c; C(x) = 0 otherwise is a fgp-closed set but not a fuzzy weakly-closed set. Example Taking the same example as Example 3.2.5, A is a fs-closed, fαclosed and fg s-closed, N is a fsp-closed, C is fp-closed and fgp-closed but are not fuzzy weakly-closed; on the other hand E is fuzzy weakly-closed set but not fs-closed, fα-closed, fp-closed set and fpg-closed set and 1 if x = c, d; J(x) = 0 otherwise closed. is fuzzy weakly-closed but not fsp-closed as well as fg s- 3.4 Compactness A topological space is said to be compact if every open cover of it has a finite subcover, i.e., the property of a topological space being compact depends on open sets. As open set is generalized to weakly open set, the concept of compactness can also be generalized in the similar manner. In this section, we introduce fuzzy weaklycompactness and fuzzy weakly-closed space, which are then characterized using fuzzy filterbases ( Definition 2.5.2). Theorem Let (X, τ) be a fuzzy quasi compact topological space ( Section 2.5). If A is a fuzzy weakly-closed subset of X, then A is also quasi compact. Proof. Let {U λ λ Λ} be a fuzzy open cover of A, i.e., A U λ. But is fs-open set containing A, which is fuzzy weakly-closed cl(a) U λ. U λ have cl(a) is a closed fuzzy set and hence is quasi compact a finite subfamily {U λi i = 1, 2,..., n} such that cl(a) n U λi A n A is fuzzy quasi compact. U λi We

8 3.4. COMPACTNESS 43 Definition An FTS (X, τ) is said to be fuzzy weakly-compact iff for every family {G λ λ Λ} of fuzzy weakly-open sets satisfying G λ = 1 X, there is a finite subfamily {G λi i = 1, 2,..., n} such that n G λi = 1 X. Example Let X = [0, 1] and A n : X [0, 1] be defined by 1 if n is odd; A n (x) = 1 1 if n is even. n Then τ = {0 X, 1 X, A n }, where n N is a fuzzy topology on X. If µ is a weakly open cover of X, then 1 X must be a member of µ, otherwise there does not exist a cover of X. Hence X is fuzzy weakly-compact. Definition A fuzzy set U in an FTS X is said to be fuzzy weakly-compact relative to X iff for every family {G λ λ Λ} of fuzzy weakly-open sets satisfying G λ (x) U(x), x Supp(U), there is a finite subfamily {G λi i = 1, 2,..., n} such that ( n G λi )(x) U(x), for every x Supp(U). Theorem An FTS X is fuzzy weakly-compact iff X does not contain a fuzzy filterbase of fuzzy weakly-closed sets such that the corresponding collection of fuzzy weakly-open sets forms a cover of X. Proof. Equivalently we show, an FTS X is not fuzzy weakly-compact iff X contains atleast one fuzzy filterbase of fuzzy weakly-closed sets such that the corresponding collection of fuzzy weakly-open sets forms a cover of X. Let X be an FTS which is not fuzzy weakly-compact a fuzzy weakly-open cover of X, say {A λ λ Λ} without a finite subcover. i.e., every finite subcollection {A λi i = 1, 2,..., n} is such that n A λi 1 X n A c λ i 0 X {A c λ λ Λ} forms a fuzzy filterbase of fuzzy weakly-closed sets. Conversely, assume X contains atleast one fuzzy filterbase of fuzzy weakly-closed sets in X. If possible, let X be fuzzy weakly-compact. So, every fuzzy weakly-open

9 3.4. COMPACTNESS 44 cover of X, say {A λ λ Λ}, has a finite subcover, {A λi i = 1, 2,..., n}, i.e., n = 1 X n A c λ i = 0 X for every collection of fuzzy weakly-closed sets in X, A λi there is atleast one subcollection with empty intersection. Therefore, a collection of fuzzy weakly-closed sets can not form a fuzzy filterbase. Theorem An FTS X is fuzzy weakly-compact if for every fuzzy filterbase Γ in X, wcl(a) 0 X. Proof. If possible, suppose {A λ λ Λ} is a fuzzy weakly-open cover of X which does not have a finite subcover. Then for every finite subcollection {A λi i = 1, 2,..., n}, there exists x X such that A λi (x) < 1 for each i = 1, 2,..., n. Then A c λi > 0 X n A λic 0 X {A c λ λ Λ} forms a fuzzy filterbase of fuzzy weakly-closed set in X. Since {A λ λ Λ} is fuzzy weakly-open cover of X, ( A λ )(x) = 1 for every x X and hence ( A c λ )(x) = 0 ( wcla c λ )(x) = 0, a contradiction. Hence X is fuzzy weakly-compact. Theorem If an FTS X is fuzzy weakly-compact then for every fuzzy filterbase Γ of fuzzy weakly-closed sets in X, cl(a) 0 X. Proof. Let X be a weakly-compact FTS. If possible, assume that, there exists a filterbase Γ of weakly-closed fuzzy sets in X such that cl(a) = 0 X (cla) c = 1 X {(cla) c A Γ} is a fuzzy weakly-open cover of X, Then by definition of fuzzy weakly-compactness, a finite subcollection {(cla i ) c A Γ, i = 1, 2,..., n} such that n (cla i ) c = 1 X ( n A c i ) = 1 X ( n A i ) = 0 X, a contradiction. Hence cl(a) 0 X. Theorem A fuzzy subset U in an FTS X is fuzzy weakly-compact relative to X if for every fuzzy filterbase Γ of fuzzy weakly-closed sets in X, every finite subcollection of Γ is quasi coincident with U and ( wclg) U 0 X. Proof. Assume U is not fuzzy weakly-compact relative to X, then a fuzzy weaklyopen covering {A λ λ Λ} of U without any finite subcover.

10 3.4. COMPACTNESS 45 This implies ( n A λi )(x) < U(x) for some x Supp(U), for every finite subfamily of {A λi i = 1, 2,..., n}. So, ( n A c λ i )(x) > 0 Γ = {A c λ λ Λ} forms a filterbase of weakly-closed fuzzy sets in X and ( n A c λ i ) q U. Next by hypothesis ( wcl(a c A c λ Γ λ )) U 0 X ( A c A c λ Γ λ ) U 0 X. Then for some x Supp(U), ( A c A c λ Γ λ )(x) > 0 ( A λ )(x) < 1, a contradiction. Hence U is fuzzy weakly-compact relative to X. A c λ Γ Theorem If U is a fuzzy weakly-compact set relative to an FTS X then for every fuzzy filterbase Γ of fuzzy weakly-closed sets in X, every finite subcollection of Γ is quasi coincident with U and ( clg) U 0 X. Proof. Let U be fuzzy weakly-compact relative to X and a filterbase Γ of fuzzy weakly-closed sets in X such that every finite subcollection of Γ is quasi coincident with U and ( clg) U = 0 X for every x Supp(U), ( clg)(x) = 0 ( (clg) c )(x) = 1 {(clg) c G Γ} is a fuzzy weakly open covering of U. But U is fuzzy weakly-compact relative to X, so a finite subfamily {(clg i ) c i = 1, 2,..., n} such that for all x Supp(U), ( n (clg i ) c )(x) U(x) ( n (clg i ))(x) U c (x) ( n clg i ) q U, a contradiction. Hence the result. Definition An FTS X is said to be fuzzy weakly-closed iff for every family {G λ λ Λ} of fuzzy weakly-open sets with G λ (x) = 1, there exists a finite subfamily {G λi i = 1, 2,..., n} such that ( n wclg λi )(x) = 1 for every x X. Definition A fuzzy set U in an FTS X is said to be fuzzy weakly-closed relative to X iff for every family {G λ λ Λ} of fuzzy weakly-open sets with G λ (x) = U(x), for all x Supp(U), there exists a finite subfamily {G λi i = 1, 2,..., n} such that ( n wclg λi )(x) = U(x), for every x Supp(U). Remark Every fuzzy weakly-compact space is fuzzy weakly-closed but not conversely.

11 3.4. COMPACTNESS 46 Theorem An FTS X is fuzzy weakly-closed if for every fuzzy filterbase Γ in X, ( wcla) 0 X. Proof. Let every fuzzy filterbase Γ in X be such that ( wcla) 0 X. If possible, assume {G λ λ Λ} be a fuzzy weakly-open cover of X and let for every finite subfamily {G λi i = 1, 2,..., n}, ( n wclg λi )(x) < 1 for some x X ( n wclg c λ i )(x) > 0, for some x X {(wclg λ ) c λ Λ} forms a fuzzy filterbase in X. We have G c λ (x) = 0 wcl(wclg λ ) c (x) = 0, a contradiction. Hence X is fuzzy weaklyclosed. Theorem If an FTS X is fuzzy weakly-closed then for every fuzzy weaklyopen filterbase Γ in X, ( cl(g)) 0 X. Proof. Let X be a fuzzy weakly-closed space, assume there exists a fuzzy weaklyopen filterbase Γ in X such that cl(g) = 0 X (clg) c = 1 X {(clg λ ) c λ Λ, G λ Γ} is a fuzzy weakly-open covering of X. Then by the definition of fuzzy weakly-closed space, it has a finite subfamily {(clg λi ) c i = 1, 2,..., n} such that n wcl(clg λi ) c = 1 X n (wcl(clg λi ) c ) c = 0 X n G λi = 0 X, a contradiction since all G s are members of a filterbase. Hence cl(g) 0 X. Theorem If a fuzzy subset U in an FTS X is fuzzy weakly-closed relative to X then for every fuzzy weakly-open filterbase {G λ λ Λ} in X with ( cl(g λ ))U 0 X, there exists a finite subfamily {G λi i = 1, 2,..., n} such that ( n G λi ) q U. Proof. Let U be fuzzy weakly-closed relative to X and {G λ λ Λ} be a fuzzy weakly-open filterbase in X, such that for every finite subfamily {G λi i = 1, 2,..., n} of {G λ λ Λ}, we have ( n G λi ) q U but ( cl(g λ ))U = 0 X ( cl(g λ ))(x) = 0 for every x Supp(U) (clg λ ) c (x) = 1 for every x Supp(U) {(clg λ ) c λ Λ} forms a fuzzy weakly-open cover of U and hence there is a finite subfamily {cl(g λi ) c i = 1, 2,.., n} such that n wcl(clg λi ) c U n (wcl(clg λi ) c ) c U c n G λi U c n q U, a contradiction. G λi

12 3.5. CONCLUSION 47 Theorem A fuzzy subset U in an FTS X is fuzzy weakly-closed relative to X if for every fuzzy filterbase {G λ λ Λ} in X with ( wclg λ ) U 0 X, there exists a finite subfamily {G λi i = 1, 2,..., n} such that ( n G λi ) q U. Proof. Suppose U satisfies the hypothesis and if possible, let U be not a fuzzy weaklyclosed set relative to X, then there exists a collection {G λ λ Λ} of fuzzy weaklyopen sets that covers U such that for every finite subfamily {G λi i = 1, 2,..., n} we have ( n wclg λi )(x) U(x) for some x Supp(U) and hence n (wclg λi ) c (x) U c (x) 0 for some x Supp(U) {(wclg λ ) c λ Λ} forms a fuzzy filterbase in X. Now we claim that, n (wclg λi ) c q U for every finite subfamily of {G λ λ Λ}, for otherwise n (wclg λi ) c q U U n wclg λi, a contradiction. n So, wcl(wclg λi ) c U 0X at least one x Supp(U) such that n wcl(wclg λi ) c > 0 X n (wcl(wclg λi ) c ) c < 1 X and hence n < 1 X, a contradiction. G λi 3.5 Conclusion In this chapter, we have introduced fuzzy weakly closed set in an FTS and studied some of its set theoretic properties. We have also examined inter relationship of fuzzy weakly-closed sets and other generalizations of closed sets in FTS. Further, fuzzy compactness and fuzzy closed space are studied through fuzzy weakly closed sets.