Regular Weakly Star Closed Sets in Generalized Topological Spaces 1

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1 Applied Mathematical Sciences, Vol. 9, 2015, no. 79, HIKARI Ltd, Regular Weakly Star Closed Sets in Generalized Topological Spaces 1 Amelia Torregosa Laniba College of Technology and Allied Sciences Bohol Island State University-Main Campus CPG North Avenue, 6300 Tagbilaran City, Bohol, Philippines Helen Moso Rara Department of Mathematics and Statistics Mindanao State University-Iligan Institute of Technology Tibanga Highway, 9200 Iligan City, Philippines Copyright c 2015 Amelia Torregosa Laniba and Helen Moso Rara. This article is distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Abstract The main purpose of this paper is to introduce the concept of a new class of sets called µ-regular weakly star closed sets in generalized topological spaces. Basic properties of µ-regular weakly star closed sets are discussed and relationships among other types of closed sets in generalized topological spaces are investigated. Keywords: µ-rw -closed sets, µ-rw -open sets 1 Introduction The notion of regular open sets in topological spaces was introduced and investigated by Stone [8] in In 1978, Cameron [2] introduced and investigated the concept of a regular semiopen sets. Using the latter study, Benchalli 1 This research is partially funded by the Commission on Higher Education, Philippines under Faculty Development Program Phase II.

2 3918 Amelia T. Laniba and Helen M. Rara and Wali [1] came up with another class of set called regular weakly closed sets in topological spaces in Later in 2002, Á. Császár [3] introduced the concept of generalized topological spaces and the aforementioned closed sets are studied in this spaces. These concepts paved the way for the notion of another class of closed sets. In this paper, the concept of µ-regular weakly star closed sets in generalized topological spaces is introduced and investigated. 2 Preliminaries Here, we recall some preliminary concepts needed in the study. Let X be a nonempty set and denote by P(X) the set of all subsets of X. A subset µ of P(X) is said to be a generalized topology (briefly GT) on X [3] if µ and the arbitrary union of elements of µ belongs to µ. A GT µ is said to be a strong generalized topology (briefly SGT) if X µ. If µ is a GT on X, then (X, µ) is said to be a generalized topological space (briefly GT-space) and the elements of µ are called µ-open sets, the complement of µ-open sets are called µ-closed sets and the subsets of P(X) that are both µ-closed and µ-open are called µ-clopen sets. If A X, then the µ-closure of A, denoted by c µ (A), is the intersection of all µ-closed sets containing A and the µ-interior of A, denoted by i µ (A), is the union of all µ-open sets contained in A. Definition 2.1 [5, 9] A subset A of a GT-space (X, µ) is said to be µ-regular open if A = i µ (c µ (A)), µ-regular closed if A = c µ (i µ (A)), µ-regular semiopen if there exists a µ-regular open set U such that U A c µ (U) and µ-regular semiclosed if there exists a µ-regular closed set U such that i µ (U) A U. Definition 2.2 Let (X, µ) be a GT-space and A X. Then the µ-regular interior of A, denoted by ri µ (A), is the union of all µ-regular open sets contained in A. Moreover, the µ-regular closure of A, denoted by rc µ (A), is the intersection of all µ-regular closed sets containing A. The µ-preclosure of A (briefly pc µ (A)) is defined analogously as in µ-regular closure of A. Definition 2.3 [5] Let (X, µ) be a GT-space. The intersection of all µ-regular semiopen subsets of X containing A is called the µ-regular semikernel of A and is denoted by µ-rsker(a). Theorem 2.4 Let (X, µ) be a GT-space and A, B, U X. Then the following are true: (i) If U is µ-regular open set in X, then X \ c µ (U) is µ-regular open; (ii) X is µ-regular semiopen; (ii) If A is µ-regular open, then A is µ-regular semiopen; (iv) If A is µ-regular closed, then rc µ (A) = A; (v) c µ (A) rc µ (A);

3 Regular weakly star closed sets 3919 (vi) c µ (i µ (A)) c µ (A); (vii) pc µ (A) c µ (A); (viii) rc µ (rc µ (A)) = rc µ (A); (ix) i µ (A) = X \ (c µ (X \ A)) [3]; (x) If A is µ-regular semiopen, then X \ A is µ-regular semiopen; (xi) X \ (rc µ (X \ A)) = ri µ (A); (xii) If A B, then ri µ (A) ri µ (B) and rc µ (A) rc µ (B); and (xiii) ri µ (ri µ (A)) = ri µ (A). Definition 2.5 [4, 5, 6, 7, 9] A subset A of a GT-space (X, µ) is said to be: (i) µ-regular weakly closed (briefly µ-rw-closed) if c µ (A) U whenever A U and U is µ-regular semiopen; (ii) µ-regular generalized closed (briefly µ-rg-closed) if c µ (A) U whenever A U and U is µ-regular open; (iii) µ-regular generalized b-closed (briefly µ-rgb-closed) if bc µ (A) U whenever A U and U is µ-regular open; (iv) µ-generalized preregular closed (briefly µ-gpr-closed) if pc µ (A) U whenever A U and U is µ-regular open; (v) µ-regular generalized weakly closed (briefly µ-rgw-closed) if c µ (i µn (A)) U whenever A U and U is µ-regular semiopen; and (vi) µ-regular weakly generalized closed (briefly µ-rwg-closed) if c µ (i µn (A)) U whenever A U and U is µ-regular open. 3 µ-rw -closed and µ-rw -open Sets In this section, we introduce µ-regular weakly star closed sets in GT-spaces. All sets considered in this section are subsets of GT-space (X, µ) unless otherwise stated. Definition 3.1 A subset A of a GT-space (X, µ) is said to be µ-regular weakly star closed (briefly µ-rw -closed) if rc µ (A) U whenever A U and U is µ- regular semiopen. The complement of µ-rw -closed set with respect to X is said to be µ-rw -open set. If A is both µ-rw -closed and µ-rw -open, then A is called µ-rw -clopen. Example 3.2 Let X = {a, b, c, d} with the GT µ = {, X, {a}, {c, d}, {a, b, d}, {a, c, d}}. Then the µ-rw -closed sets are, X, {b}, {a, b}, {a, c}, {a, d}, {b, c}, {b, d}, {a, b, c}, {a, b, d}, {a, c, d} and {b, c, d}. Hence, the µ- rw -open sets are X,, {a, c, d}, {c, d}, {b, d}, {b, c}, {a, d}, {a, c}, {d}, {c}, {b} and {a}. Furthermore, the µ-rw -clopen sets are X,, {b}, {a, c}, {a, d}, {b, c}, {b, d} and {a, c, d}.

4 3920 Amelia T. Laniba and Helen M. Rara Remark 3.3 Let (X, µ) be a GT-space and A X. Then the following are true. (i) If rc µ (A) = A, then A is µ-rw -closed. (ii) If there is no µ-regular semiopen proper subset of X that contains A, then A is µ-rw -closed. (iii) X is µ-rw -closed. (iv) is µ-rw -open. Remark 3.4 For any GT-space (X, µ), the following are true. (i) The intersection of two µ-rw -closed sets in X need not be µ-rw -closed. (ii) The union of two µ-rw -open sets in X need not be µ-rw -open, implying that the collection of µ-rw -open sets does not form a generalized topology on X. (iii) The concept of µ-closed sets is independent with the concept of of µ-rw - closed sets. To see these, consider Example 3.2. Then, observe that (i) {a, b} and {a, c} are µ-rw -closed sets but {a, b} {a, c} = {a} is not a µ-rw -closed set in X. (ii) {a} and {b} are µ-rw -open sets but {a} {b} = {a, b} is not a µ-rw - open set in X. (iii) {c} is a µ-closed set but not µ-rw -closed and {a, c} is a µ-rw -closed set but not µ-closed. Theorem 3.5 Every µ-regular closed set is µ-rw -closed. Proof : Let A be µ-regular closed and A U where U is µ-regular semiopen. By Theorem 2.4 (iv), rc µ (A) = A U. Hence, A is µ-rw -closed. Remark 3.6 The converse of Theorem 3.5 need not be true in general. To see this, consider the set X = {a, b, c, d} with the GT µ = {, X, {b}, {d}, {b, d}, {a, b, c}, {a, b, d}, {a, c, d}}. Then, observe that {b} is a µ-rw - closed set but not µ-regular closed. Theorem 3.7 Every µ-rw -closed set is µ-rw-closed.

5 Regular weakly star closed sets 3921 Proof : Suppose that A is µ-rw -closed and A U where U is µ-regular semiopen. Thus, rc µ (A) U. By Theorem 2.4 (v), c µ (A) rc µ (A) U. Therefore, A is µ-rw-closed. Remark 3.8 The converse of Theorem 3.7 need not be true in general. To see this, consider the set X = {a, b, c, d} with the GT µ = {, X, {b}, {d}, {b, d}, {a, b, c}, {a, b, d}, {a, c, d}}. Then, observe that {c} is a µ-rwclosed set but not µ-rw -closed. Lemma 3.9 Let (X, µ) be a BGT-space and A X. Then the following are true. (i) Every µ-rw-closed set is µ-rg-closed. (ii) Every µ-rg-closed set is µ-gpr-closed. (iii) [9] Every µ-gpr-closed set is µ-rgb-closed. (iv) Every µ-rw-closed set is µ-rgw-closed. (v) Every µ-rgw-closed set is µ-rwg-closed. Proof : (i) Suppose that A is µ-rw-closed and A U where U is µ-regular open. Since every µ-regular open is µ-regular semiopen, U is µ-regular semiopen. Thus, c µ (A) U. Therefore, A is µ-rg-closed. (ii) Suppose that A is µ-rg-closed and A U where U is µ-regular open. Thus, c µ (A) U. By Theorem 2.4 (vii), pc µ (A) c µ (A) U. Therefore, A is µ-gpr-closed. (iv) Suppose that A is µ-rw-closed and A U where U is µ-regular semiopen. Thus, c µ (A) U. By Theorem 2.4 (vi), c µ (i µ (A)) c µ (A) U. Therefore, A is µ-rgw-closed. (v) Suppose that A is µ-rgw-closed and A U where U is µ-regular open. Since every µ-regular open is µ-regular semiopen, U is µ-regular semiopen. Thus, c µ (i µ (A)) U. Therefore, A is µ-rwg-closed. Corollary 3.10 Every µ-rw -closed set is µ-rg-closed, µ-gpr-closed, µ-rgbclosed, µ-rgw-closed and µ-rwg-closed. Remark 3.11 The converses of Corollary 3.10 need not be true in general.

6 3922 Amelia T. Laniba and Helen M. Rara To see these, consider the set X = {a, b, c, d} with the GT µ = {, X, {b}, {d}, {b, d}, {a, b, c}, {a, b, d}, {a, c, d}}. Then, observe that {a} is µ- rg-closed, µ-gpr-closed, µ-rgb-closed, µ-rgw-closed and µ-rwg-closed but not µ-rw -closed. Remark 3.12 The following diagram shows the relationships among different types of weakly closed sets discussed in Theorem 3.5, Remark 3.6, Theorem 3.7, Remark 3.8, Lemma 3.9, Corollary 3.10 and Remark µ-rgb-closed µ-gpr closed µ-rg-closed µ-regular closed µ-rw -closed µ-rwg-closed µ-rgw-closed µ-rw-closed A B means A implies B Theorem 3.13 If A is both µ-rw -closed and µ-regular semiopen, then A is µ-closed. Proof : Suppose that A is both µ-rw -closed and µ-regular semiopen. Then rc µ (A) A. By Theorem 2.4 (v), c µ (A) rc µ (A) A. Thus, c µ (A) A. Note that A c µ (A). Therefore, c µ (A) = A and so A is µ-closed. Corollary 3.14 If A is both µ-rw -open and µ-regular semiclosed, then A is µ-open. Lemma 3.15 Every µ-clopen set is µ-regular closed. Proof : Suppose that A is µ-clopen. Then c µ (A) = A and i µ (A) = A. It follows that c µ (i µ (A)) = c µ (A) = A implying that A is µ-regular closed. Remark 3.16 The converse of Lemma 3.15 need not be true in general. To see this, consider the set X = {a, b, c, d} with the GT µ = {, X, {a}, {c, d}, {a, b, d}, {a, c, d}} in Example 3.2. Take A = {a}. Then observe that {a} is a µ-regular closed set but not µ-clopen. Lemma 3.17 Let (X, µ) be a GT-space and A X. Then A is µ-regular semiopen if and only if X \ A is µ-regular semiopen.

7 Regular weakly star closed sets 3923 Proof : In Theorem 2.4(x), if A is µ-regular semiopen, then X \ A is µ-regular semiopen. Hence, it remains to show that the converse holds. Suppose that X \ A is µ-regular semiopen. Then U X \ A c µ (U) for some µ-regular open set U. Hence, X \ (c µ (U)) A X \ U. Since U is µ-regular open and by Theorem 2.4(xi), U = i µ (c µ (U)) = X \ (c µ (X \ (c µ (U)))). Then, X \ U = c µ (X \ (c µ (U))). By Theorem 2.4(i), X \ (c µ (U)) is µ-regular open and X \ (c µ (U)) A X \ U = c µ (X \ (c µ (U))). Thus, A is µ-regular semiopen. Corollary 3.18 Let (X, µ) be a GT-space and A X. Then A is µ-regular semiopen if and only if A is µ-regular semiclosed. Proof : Suppose that A is µ-regular semiopen. By Lemma 3.17, X \ A is µ- regular semiopen. Hence, X \ (X \ (A)) = A is µ-regular semiclosed. The converse is proved similarly. Theorem 3.19 If A is both µ-rw -clopen and µ-regular semiopen, then A is µ-regular closed. Proof : Suppose that A is both µ-rw -clopen and µ-regular semiopen. By Theorem 3.13, A is µ-closed and by Corollary 3.14 and Corollary 3.18, A is µ-open. Hence, A is µ-clopen and by Lemma 3.15, A is µ-regular closed. Lemma 3.20 If A is both µ-rw-clopen and µ-regular semiopen, then A is µ-regular closed. Proof : Suppose that A is both µ-rw-clopen and µ-regular semiopen. Since A is both µ-rw-closed and µ-regular semiopen, c µ (A) A. Note that A c µ (A). Hence, c µ (A) = A implying that A is µ-closed. By Corollary 3.18, A is also both µ-rw-open and µ-regular semiclosed, A i µ (A). Note that i µ (A) A. Hence, i µ (A) = A implying that A is µ-open. Thus, A is µ-clopen. By Lemma 3.15, A is µ-regular closed. Corollary 3.21 If A is both µ-rw-clopen and µ-regular semiopen, then A is µ-rw -closed. Proof : By Lemma 3.20 and by Theorem 3.5, the corollary follows. Theorem 3.22 Let (X, µ) be an GT-space and A X. Then A is µ-rw -closed if and only if rc µ (A) µ-rsker(a).

8 3924 Amelia T. Laniba and Helen M. Rara Proof : Suppose that A is µ-rw -closed. By Definition 3.1, rc µ (A) U whenever A U and U is µ-regular semiopen. Let x rc µ (A) and suppose that x / µ-rsker(a). Then there exists a µ-regular semiopen set U containing A such that x / U. By assumption, it follows that x / rc µ (A), a contradiction. Therefore, x µ-rsker(a) and so rc µ (A) µ-rsker(a). Conversely, let rc µ (A) µ-rsker(a). Now, for any µ-regular semiopen set U containing A, µ-rsker(a) U. That is, rc µ (A) µ-rsker(a) U. Therefore, A is µ-rw -closed in X. Theorem 3.23 The empty set is µ-rw -closed if and only if µ is an SGT. Proof : Suppose that is a µ-rw -closed set. Hence, X is µ-rw -open set. By Theorem 2.4 (ii), X is µ-regular semiopen. Hence, by Corollary 3.14 and Corollary 3.18, X is µ-open. Therefore, µ n is an SGT. Conversely, let µ be an SGT. Then is µ-open and µ-closed. Hence, i µ ( ) = and c µ ( ) =. Now, c µ (i µ ( )) = c µ ( ) =. Therefore, is µ- regular closed. Hence, U and rc µ ( ) = U for all µ-regular semiopen set U. Therefore, is µ-rw -closed. Corollary 3.24 The set X is µ-rw -open if and only if µ is an SGT. Theorem 3.25 If A is a µ-rw -closed set such that A B rc µ (A), then B is µ-rw -closed. Proof : Let A B rc µ (A). Suppose that A is µ-rw -closed. Let B U where U is µ-regular semiopen. Then A U and rc µ (A) U. Since B rc µ (A), rc µ (B) rc µ (rc µ (A)) rc µ (A) U. Therefore, B is µ-rw -closed. Theorem 3.26 For an element x X, the set X \ {x} is µ-rw -closed or µ-regular semiopen. Proof : Suppose X \ {x} is not µ-regular semiopen. Then X is the only µ- regular semiopen set containing X \ {x}. Hence, rc µ (X \ {x}) X. Thus, X \ {x} is µ-rw -closed. Lemma 3.27 Let (X, µ) be a GT-space and A X. Then X and are both µ-regular semiopen and µ-regular semiclosed. Proof : By Theorem 2.4(ii), X is µ-regular semiopen and so is µ-regular semiclosed. By Corollary 3.18, X is µ-regular semiclosed and is µ-regular semiopen. Theorem 3.28 Let (X, µ) be an GT-space and A X. If A is µ-rw -closed, then [rc µ (A)] \ A does not contain a nonempty µ-regular semiclosed set.

9 Regular weakly star closed sets 3925 Proof : Suppose that A is µ-rw -closed. Let F be a µ-regular semiclosed set such that F [rc µ (A)] \ A. We claim that F =. Since F [rc µ (A)] \ A, we have F [rc µ (A)] [X \ A]. Hence, F X \ A and F rc µ (A). It follows that A X \ F where X \ F is µ-regular semiopen. Since A is µ-rw -closed, [rc µ (A)] X \ F. Thus, F X \ rc µ (A) rc µ (A) =. Therefore, F =. Theorem 3.29 A subset A of a GT-space (X, µ) is µ-rw -open if and only if F ri µ (A) whenever F A and F is µ-regular semiclosed. Proof : Let A be any µ-rw -open set in X. Then X \A is µ-rw -closed. Hence, X \A U and rc µ (X \A) U for some µ-regular semiopen set U in X. Thus, X \ U A and X \ U X \ (rc µ (X \ A)) where X \ U is µ-regular semiclosed. Take F = X \ U. By Theorem 2.4 (xi), F X \ A and F ri µ (X \ A) where F is µ-regular semiclosed. Conversely, suppose that F ri µ (A) whenever F A and F is µ-regular semiclosed. Let X \ A U where U is µ-regular semiopen. Then X \ U A and X \ U is µ-regular semiclosed. By assumption, X \ U ri µ (A). By Theorem 2.4 (xi), we have X \ U A and X \ U X \ (rc µ (X \ A)) where X \ U is µ-regular semiclosed. Hence, X \ A U and rc µ (X \ A) U where U is µ-regular semiopen. Thus, X \ A is µ-rw -closed and it follows that A is a µ-rw -open set in X. Theorem 3.30 Let (X, µ) be a GT-space and A, B X where ri µ (A) B A. If A is µ-rw -open, then B is µ-rw -open. Proof : Let ri µ (A) B A. Suppose that A is µ-rw -open. Let F B where F is µ-regular semiclosed. Then F A and F ri µ (A). Since ri µ (A) B, ri µ (ri µ (A)) ri µ (B). Consequently, ri µ (A) ri µ (B) and it follows that F ri µ (B). Therefore, B is µ-rw -open. Lemma 3.31 Let (X, µ) be a GT-space and A X. Then X and are both µ-regular semiopen and µ-regular semiclosed. Proof : Follows immediately from Theorem 2.4 (ii) and (xii). Theorem 3.32 Let A be a subset of a GT-space (X, µ). If A is µ-rw -closed, then rc µ (A) \ A is µ-rw -open. Proof : Suppose that A is a µ-rw -closed set. Let X \ (rc µ (A) \ A) U where U is µ-regular semiopen. Then X \ U rc µ (A) \ A and X \ U is µ-regular semiclosed. By Theorem 3.28, rc µ (A) \ A does not contain a nonempty µ- regular semiclosed set. Consequently, X \ U = and thus U = X. Hence, rc µ (X \ (rc µ (A) \ A)) U implying that X \ (rc µ (A) \ A) is µ-rw -closed set. Therefore, rc µ (A) \ A is µ-rw -open.

10 3926 Amelia T. Laniba and Helen M. Rara Theorem 3.33 Let A be a subset of a GT-space (X, µ). If A is µ-rw -open, then U = X whenever U is µ-regular semiopen and ri µ (A) [X \ A] U. Proof : Suppose that A is µ-rw -open, U is µ-regular semiopen and ri µ (A) [X \ A] U. Then X \ U X \ {ri µ (A) [X \ A]} = X \ ri µ (A) A = rc µ (X \ A)\[X \A]. Since X \A is µ-rw -closed, by Theorem 3.28, rc µ (X \A)\(X \A) does not contain µ-regular semiclosed set. Since X \ U is µ-regular semiclosed and X \ U rc µ (X \ A) \ [X \ A], it follows that X \ U =. Therefore, U = X. 4 µ-rw -interior and µ-rw -closure of a Set Definition 4.1 Let (X, µ) be a GT-space and A X. The µ-rw -interior of A, denoted by rw i µ (A), is the union of all µ-rw -open sets contained in A. Remark 4.2 Let (X, µ) be a GT-space and A X. Then (i) rw i µ (A) = {G X : G is µ-rw -open and G A} (ii) x rw i µ (A) if and only if there exists a µ-rw -open set U with x U A (iii) rw i µ (A) A Theorem 4.3 Let (X, µ) be a GT-space and A X. Then (i) If A is a µ-regular open set, then A is µ-rw -open. (ii) If A B, then rw i µ (A) rw i µ (B). (iii) If A is µ-rw -open, then A = rw i µ (A). (iv) rw i µ (A) = rw i µ (rw i µ (A)). (v) ri µ (A) rw i µ (A). Proof : (i) Suppose that A is a µ-regular open set. Then X \ A is µ-regular closed. By Theorem 3.5, X \ A is µ-rw -closed. Therefore, A is µ-rw -open. (ii) Suppose that A B and let x rw i µ (A). Then there exists µ-rw - open set O A such that x O. Since A B, there exists µ-rw -open set O B such that x O. Therefore, x rw i µ (B). (iii) Suppose that A is µ-rw -open. Then A {G X : G is µ-rw -open and G A} = rw i µ (A). By Remark 4.2 (iii), rw i µ (A) = A.

11 Regular weakly star closed sets 3927 (iv) Let x rw i µ (A) and let G be µ-rw -open such that G rw i µ (A). Since G rw i µ (A) A and x rw i µ (A), we have x G for all G X. Hence, x rw i µ (rw i µ (A)). Therefore, rw i µ (A) rw i µ (rw i µ (A)). To show the other inclusion, let x rw i µ (rw i µ (A)) and let G be µ- rw -open such that G A. By (ii) and (iii), G = rw i µ (G) rw i µ (A) and x rw i µ (rw i µ (A)) implying that x G for all G X. Hence, x rw i µ (A). Therefore, rw i µ (rw i µ (A)) rw i µ (A). Consequently, rw i µ (A) = rw i µ (rw i µ (A)). (v) Let x ri µ (A). Then there exists a µ-regular open set O with x O such that O A. Thus, by (i), there exists a µ-rw -open set O with x O such that O A. Hence, x rw i µ (A). Definition 4.4 Let (X, µ) be a GT-space and A X. The µ-rw -closure of A, denoted by rw c µ (A), is the intersection of all µ-rw -closed sets containing A. Remark 4.5 Let (X, µ) be a GT-space and A X. Then (i) rw c µ (A) = {F X : F is µ-rw -closed and F A} (ii) x rw c µ (A) if and only if there exists a µ-rw -closed set F with x F A (iii) A rw c µ (A) Theorem 4.6 Let (X, µ) be a GT-space and A, B X. Then (i) x rw c µ (A) if and only if for every µ-rw -open set O with x O, O A (ii) If A B, then rw c µ (A) rw c µ (B). (iii) If A is µ-rw -closed, then A = rw c µ (A). (iv) rw c µ (A) = rw c µ (rw c µ (A)) (v) rw c µ (A) rc µ (A) Proof : (i) Let x rw c µ (A) and let O be a µ-rw -open set with x O. Suppose that O A =. Then A X \ O and X \ O is µ-rw -closed. Since x rw c µ (A), x X \ O by Definition 4.4. Thus, x / O which is a contradiction to the assumption. Hence, O A.

12 3928 Amelia T. Laniba and Helen M. Rara Conversely, suppose that for every µ-rw -open set O with x O, O A. Suppose further that x / rw c µ (A). Then there exists a µ-rw -closed set F with A F and x / F. Thus x X \ F and X \ F is µ-rw - open. But A F implies that A X \ F = which is a contradiction. Therefore, x rw c µ (A). (ii) Suppose that A B and x / rw c µ (B). Then there exists µ-rw - closed set F B such that x / F. Since A B, x / rw c µ (A). rw c µ (A) rw c µ (B). (iii) Suppose that A is µ-rw -closed. Then rw c µ (A) = {F X : F is µ-rw -closed and F A} A. By Remark 4.5 (ii), A = rw c µ (A). (iv) Let x rw c µ (A) and let F be any µ-rw -closed set such that rw c µ (A) F. Since A rw c µ (A) F and x rw c µ (A), we have x F. Hence, x rw c µ (rw c µ (A)). Therefore, rw c µ (A) rw c µ (rw c µ (A)). To show the other inclusion, let x rw c µ (rw c µ (A)) and let F be any µ-rw -closed set such that A F. By (ii) and (iii), rw c µ (A) rw c µ (F ) = F and x rw c µ (rw c µ (A)) implying that x F. Hence, x rw c µ (A). Therefore, rw c µ (rw c µ (A)) rw c µ (A). Consequently, rw c µ (A) = rw c µ (rw c µ (A)). (v) Suppose that x / rc µ (A). Then there exists a µ-regular closed set F A such that x / F. By Theorem 2.4, F is µ-rw -closed. Hence, x rw c µ (A). Therefore, rw c µ (A) rc µ (A). Theorem 4.7 Let (X, µ) be a GT-space and A X. Then X\rw c µ (X\A) = rw i µ (A). Proof : Let x X \ rw c µ (X \ A). Hence, x / rw c µ (X \ A). Thus, there exists a µ-rw -closed set F X \ A such that x / F. That is, there exists a µ-rw -open set X \ F A such that x X \ F. Therefore, x rw i µ (A). Conversely, let x rw i µ (A). Hence, there exists a µ-rw -open set U A such that x U. That is, there exists a µ-rw -closed set X \ U X \ A such that x / X \ U. Hence, x / rw c µ (X \ A). Thus, x X \ rw c µ (X \ A). Theorem 4.8 Let (X, µ) be a GT-space and A X. Then X\rw i µ (X\A) = rw c µ (A). Proof : Let x X \ rw i µ (X \ A). Hence, x / rw i µ (X \ A). Thus, x / U for all µ-regular open set U X \ A. That is, x X \ U for all µ-regular closed set X \ U A. Therefore, x rw c µ (A). Conversely, let x rw c µ (A). Hence, x F for all µ-regular closed set F A. That is, x / X \ F for all µ-regular open set X \ F X \ A. Hence, x / rw i µ (X \ A). Therefore, x X \ rw i µ (X \ A).

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