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On a class of sets between μ-closed sets and μg-closed sets
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On a class of sets between μ-closed sets and μg-closed sets Bishwambhar Roy a,∗,1 , Ritu Sen b a b
Department of Mathematics, Women’s Christian College, 6, Greek Church Row, Kolkata 700 026, India Department of Mathematics, S. A. Jaipuria College, 10, Raja Naba Krishna Street, Kolkata 700 005, India Received 28 August 2014; accepted 30 August 2015
Abstract In this paper, a new class of sets called μ* g-closed sets are introduced and investigated with the help of μ-open and μg-open sets. Relationships between this new class and other related classes of sets are established. Some separation axioms has also being studied. Finally, some preservation theorems have been given. © 2015 The Authors. Production and hosting by Elsevier B.V. on behalf of Taibah University. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/). Keywords: μ-Open set; μg-Open set; μ* g-Closed set
1. Introduction In the past few years, different forms of open sets have been studied. Recently, a significant contribution to the theory of generalized open sets, was extended by A. Császár. Especially, the author defined some basic operators on generalized topological spaces. It is observed that a large number of papers is devoted to the study of generalized open like sets of a topological space,
∗
Corresponding author. Tel.: +91 033 9433401976. E-mail addresses: bishwambhar [email protected] (B. Roy), ritu [email protected] (R. Sen). 1 The author acknowledges the financial support from UGC for the Major Research Project having grant No. 41-799/2012(SR) UGC, New Delhi. Peer review under responsibility of Taibah University.
containing the class of open sets and possessing properties more or less similar to those of open sets. We recall some notions defined in [1]. Let X be a nonempty set, expX denotes the power set of X. We call a class μ expX a generalized topology [1], (briefly, GT) if ∅ ∈ μ and union of elements of μ belongs to μ. A set X with a GT μ on it is said to be a generalized topological space (briefly, GTS) and is denoted by (X, μ). For a GTS (X, μ), the elements of μ are called μopen sets and the complement of μ-open sets are called μ-closed sets. For A X, we denote by cμ (A) the intersection of all μ-closed sets containing A, i.e., the smallest μ-closed set containing A; and by iμ (A) the union of all μ-open sets contained in A, i.e., the largest μ-open set contained in A (see [1,2]). It is easy to observe that iμ and cμ are idempotent and monotonic, where γ: expX → expX is said to be idempotent if A X implies γ(γ(A)) = γ(A) and monotonic if A B X implies γ(A) γ(B). It is also well known from [2,3] that if μ is a GT on X, x ∈ X and A X, then x ∈ cμ (A) iff x ∈ M ∈ μ ⇒ M∩ A = / ∅ and cμ (X \ A) = X \ iμ (A).
http://dx.doi.org/10.1016/j.jtusci.2015.08.008 1658-3655 © 2015 The Authors. Production and hosting by Elsevier B.V. on behalf of Taibah University. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).
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Throughout the paper μ, λ will always mean GT’s on the respective spaces. 2. Properties of μ* g-closed sets Definition 2.1. Let (X, μ) be a GTS. Then a subset A of X is called a μ-generalized closed set (in short, μgclosed set) [4] iff cμ (A) U whenever A U where U is μ-open in X. The complement of a μg-closed set is called a μg-open set. Definition 2.2. Let (X, μ) be a GTS. Then a subset A of X is called strongly μ-generalized closed (in short, μ* gclosed) iff cμ (A) U whenever A U and U is μg-open. Remark 2.3. Let (X, μ) be a GTS. Then every μclosed set is μ* g-closed. Also, every μ* g-closed set is μg-closed. Thus we have the following relation: μ-closed set ⇒ μ* g-closed set ⇒ μg-closed set. Example 2.4. (a) Let X = {a, b, c} and μ = {∅ , {a, b}, {a, c}, X}. Then (X, μ) is a GTS. It can be easily verified that {b, c} is a μ* g-closed set but not a μ-closed set. (b) Let X = {a, b, c, d} and μ = {∅ , {a}, {a, b}, {a, c, d}, X}. Then it is easy to check that {b, d} is a μg-closed set but not a μ* g-closed set. Example 2.5. Let X = {a, b, c, d}, μ = {∅ , {a, b, c}, {a, c, d}, {b, c, d}, X}. Then μ is a GT on X. It can be verified that {a} and {b} are two μ* g-closed sets but their union is not μ* g-closed.
Theorem 2.9. Let (X, μ) be a GTS. Then A X is μ* gopen if and only if F iμ (A) whenever F is μg-closed and F A. Proof. Suppose that A is a μ* g-open set and F is a μg-closed set with F A. Then X \ F is μg-open and X \ A X \ F. Thus by μ* g-closedness of X \ A, cμ (X \ A) (X \ F). Hence F iμ (A). Conversely, suppose that F iμ (A) whenever F is μgclosed and F A. If G is a μg-open set in X containing X \ A, then X \ G is a μg-closed set contained in A. Hence by the hypothesis, X \ G iμ (A). Thus cμ (X \ A) G and hence X \ A is μ* g-closed i.e., A is μ* g-open. Theorem 2.10. If A is μ* g-open in a GTS (X, μ), then G = X, whenever G is μg-open and iμ (A) ∪ (X \ A) G. Proof. Let us assume that G is μg-open and iμ (A) ∪ (X \ A) G. Then X \ G cμ (X \ A) ∩ A = cμ (X \ A) \ (X \ A). Since, X \ G is μg-closed and X \ A is μ* g-closed, then by Theorem 2.6, X\ G = ∅ and hence G = X. Theorem 2.11. If A is μ* g-closed in a GTS (X, μ), then cμ (A) \ A is μ* g-open. Proof. Suppose that A is μ* g-closed and F cμ (A) \ A, where F is a μg-closed subset of X. Then by Theorem 2.6, F =∅ and hence F iμ [cμ (A) \ A]. Therefore by Theorem 2.9, cμ (A) \ A is μ* g-open. Theorem 2.12. For each x ∈ X, either {x} is μg-closed or X \ {x} is μ* g-closed.
Theorem 2.6. Let A be a set in a GTS (X, μ). Then cμ (A) \ A does not contain any non-empty μgclosed set.
Proof. If {x} is not μg-closed, then the only μg-open set containing X \ {x} is X (at most if X ∈ μ). Hence cμ (X \ {x}) X is contained in X and therefore X \ {x} is μ* g-closed.
Proof. Let F be a μg-closed subset contained in cμ (A) \ A. Then F cμ (A) and F X \ A. Thus F cμ (A) ∩ (X \ A) = ∅ i.e., F =∅.
Theorem 2.13. If A is a μ* g-closed set of a GTS (X, μ) such that A B cμ (A), then B is also a μ* g-closed set.
Theorem 2.7. Let (X, μ) be a GTS. If A X is μg-open and μ* g-closed then A is μ-closed.
Proof. Let U be a μg-open set of (X, μ) such that B U. Then A U. Since A is μ* g-closed, then cμ (A) U. Now cμ (B) cμ (cμ (A)) = cμ (A) U. Therefore B is also a μ* g-closed set.
μ* g-closed
Proof. Let A be a μg-open and μ* g-closed subset of X. Then A A where A is μg-open in X. Thus by μ* gclosedness of A, cμ (A) A. Hence A is μ-closed. set in
Definition 2.14. Let (X, μ) be a GTS. Then the μgeneralized kernel of A X is denoted by μg -ker(A) and defined as μg -ker(A) = ∩ {G: G is μg-open and A G}.
Proof. Follows from the fact that every μ-open set is a μg-open set.
Theorem 2.15. A subset A of a GTS (X, μ) is μ* gclosed if and only if cμ (A) μg -kerA.
Corollary 2.8. If A be a μ-open and a GTS (X, μ) then A is μ-closed.
μ* g-closed
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3. Strongly μ-T1/2 space
Proof. (i) Suppose that {x} is not μ-closed, for some x ∈ X. Then the only μ-open set containing X \ {x} is X at most, hence X \ {x} is μg-closed. Since X is a strongly μ-T1/2 space, then X \ {x} is μ* g-closed. Therefore, {x} is μ* g-open in X. (ii) Assume that {x} is not μ-closed, for some x ∈ X. Then X is a μ-open set (at most) containing X \ {x}, hence X \ {x} is μg-closed and therefore {x} is μg-open in X.
Definition 3.1. A GTS (X, μ) is said to be:
4. Strongly μ-regular and strongly μ-normal spaces
Proof. Assume that A is a μ* g-closed set in X. Then cμ (A) G whenever A G and G is μg-open in X. This implies that cμ (A) ∩ {G : A G and G is μg-open} = μg -ker(A). For the converse, assume that cμ (A) μg ker(A). This implies that cμ (A) ∩ {G : A G and G is μg-open}. This shows that cμ (A) G for all μg-open sets G containing A. This proves that A is μ* g-closed.
(i) Strongly μ-T1/2 if every μg-closed set in X is μ* gclosed. (ii) μ* -T1/2 if every μ* g-closed set in X is μ-closed. (iii) μ-T1/2 [4] if every μg-closed set in X is μ-closed. Remark 3.2. μ* -T1/2 and strongly μ-T1/2 spaces are independent as may be seen from Example 3.3. Example 3.3. (a) Let X = {a, b, c} and μ = {∅ , {a}, {a, b}, {a, c, d}, X}. Then (X, μ) is a GTS. It can be checked that (X, μ) is μ* -T1/2 but not a strongly μ-T1/2 space. (b) Let X = {a, b, c, d}, μ = {∅ , {a, b}, {a, d}, {a, b, d}, X}. Then (X, μ) is a GTS. It can be checked that (X, μ) is strongly μ-T1/2 but not μ* -T1/2 .
Definition 4.1. A GTS (X, μ) is said to be strongly μregular if for each μg-closed set F and each point x ∈ / F, there exist disjoint μ-open sets U and V such that F U and x ∈ V. Definition 4.2. A GTS (X, μ) is said to be μ-regular [4] if for each μ-closed set F of X and each point x ∈ / F, there exist disjoint μ-open sets U and V such that x ∈ U and F V. As every μ-closed set is μg-closed, every strongly μ-regular space is μ-regular but the converse is false as shown by the next example.
Theorem 3.4. For a GTS (X, μ), the following statements hold:
Example 4.3. Let X = {a, b, c, d} and μ = {∅ , {a, b}, {c, d}, X}. Then (X, μ) is a GTS. It can be checked that X is μ-regular but not strongly μ-regular.
(i) Every μ-T1/2 space is strongly μ-T1/2 . (ii) Every μ-T1/2 space is μ* -T1/2 .
Theorem 4.4. A GTS (X, μ) is strongly μ-regular if and only if (X, μ) is μ-regular and μ-T1/2 .
Proof. Obvious.
Proof. Suppose that (X, μ) is strongly μ-regular. Then clearly (X, μ) is μ-regular. Now let A X be μg-closed. For each x ∈ / A there exists a μ-open set Vx containing / A}, then V is μx such that Vx ∩ A = ∅. If V = ∪ {Vx : x ∈ open and V = X \ A, hence A is μ-closed. The converse is obvious.
The converses of the above theorem need not be true as may be seen from the following examples. Example 3.5. (a) Let X = {a, b, c, d} and μ = {∅ , {a, b}, {b, c}, {a, b, c}, X}. Then (X, μ) is a GTS. It can be verified that (X, μ) is strongly μ-T1/2 but not μ-T1/2 . (b) Let X = {a, b, c, d} and μ = {∅ , {a, b, c}, {b, c, d}, X}. Then (X, μ) is a GTS. It can be checked that (X, μ) is a μ* -T1/2 space but not a μ-T1/2 space. Theorem 3.6. If (X, μ) is a strongly μ-T1/2 space, then the following statements hold: (i) Every singleton in X is μ-closed or μ* g-open in X. (ii) Every singleton in X is μ-closed or μg-open.
Theorem 4.5. For a GTS (X, μ) the following statements are equivalent: (i) (X, μ) is strongly μ-regular. (ii) For each point x ∈ X and for each μg-open set W containing x, there exists a μ-open set U in X such that cμ (U) W. (iii) For each point x ∈ X and for each μg-closed set F not containing x, there exists a μ-open set V in X such that cμ (V)∩ F = ∅.
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(iv) For each x ∈ X and each μg-closed set F with x ∈ /F there exist a μ-open set U and a μ* g-open set V such that x ∈ U, F V and U∩ V = ∅. (v) For each A X and each μg-closed set F with A∩ F = ∅ there exist a μ-open set U and a μ* g-open set V such that A∩ U = / ∅, F V, U∩ V = ∅. Proof. (i) ⇒ (ii): Let W be any μg-open set containing x. Then Wc is μg-closed and x ∈ / Wc . By hypothesis, there exist μ-open sets U and V such that Wc V, x ∈ U and U∩ V = ∅ and so U Vc . Now cμ (U) cμ (Vc ) = Vc and Wc V implies that Vc W. Thus cμ (U) W. (ii) ⇒ (i): Let F be any μg-closed set and x ∈ / F. Then x ∈ Fc and Fc is a μg-open set containing x. By hypothesis, there exists a μ-open set V containing x such that x ∈ V and cμ (V) Fc , which implies that F (cμ (V))c . Then (cμ (V))c is a μ-open set containing F and V∩ (cμ (V))c = ∅. Hence X is strongly μ-regular. (ii) ⇒ (iii): Let x ∈ X and F be a μg-closed set such that x ∈ / F. Then Fc is a μg-open set containing x and by hypothesis, there exists a μ-open set V of X such that cμ (V) Fc and hence cμ (V)∩ F = ∅. (iii) ⇒ (ii): Let x ∈ X and W be a μg-open set contain/ Wc , by hypothesis ing x. Since Wc is μg-closed and x ∈ there exists a μ-open set U containing x such that cμ (U)∩ Wc = ∅. Hence cμ (U) W. (i) ⇒ (iv): This is obvious as every μ-open set is μ* gopen (by Remark 2.3). (iv) ⇒ (v): Let F be a μg-closed set with A∩ F = ∅ for any subset A of X. Then for each a ∈ A, a ∈ / F. So there exist U ∈ μ and a μ* g-open set V such that a ∈ U, F V / ∅. and U∩ V = ∅. Thus A∩ U = (v) ⇒ (i): Let F be a μg-closed set such that x ∈ / F. Then {x}∩ F = ∅. Thus by (v), there exist a μ-open set U and a μ* g-open set W such that x ∈ U, F W and U∩ W = ∅. Put V = iμ (W). Then F V (by Theorem 2.9) and U∩ V = ∅. Definition 4.5. A GTS (X, μ) is said to be strongly μnormal if for every pair of disjoint μg-closed sets A and B, there exist disjoint μ-open sets U, V ( X) such that A U and B V. Definition 4.6. A GTS (X, μ) is said to be μ-normal [4] if for every pair of disjoint μ-closed sets A and B of X, there exist disjoint μ-open sets U and V such that A U and B V. Example 4.7. Let X = {a, b, c, d} and μ = {∅ , {a}, {a, c}, {b, c, d}, X}. Then (X, μ) is a GTS. It can be checked that X is μ-normal but not strongly μ-normal.
Definition 4.8. A GTS (X, μ) is said to be μ-symmetric [4] iff for each x, y ∈ X, x ∈ cμ ({y}) ⇒ y ∈ cμ ({x}). We recall [4] that a GTS (X, μ) is μ-symmetric iff {x} is μg-closed for each x ∈ X. Theorem 4.9. (i) Every μ-normal, μ-symmetric GTS (X, μ) is μ-regular. (ii) Every strongly μ-normal, μ-symmetric GTS (X, μ) is strongly μ-regular and hence μ-regular. / A. Proof. (i) Suppose that A X is μ-closed and x ∈ Then {x} is μg-closed. By μ-normality of X, there exist disjoint μ-open sets U and V such A V and {x} U. Thus X is μ-regular. (ii) Obvious. Example 4.10. (a) Consider Example 4.7. It is easy to observe that X is μ-normal but not μ-regular. (b) Consider X = {a, b, c} and μ = {∅ , {a}, {a, b}}. Then (X, μ) is a GTS. It can be checked that (X, μ) is strongly μ-normal but not strongly μ-regular. Theorem 4.11. Let (X, μ) be a GTS. Then the following statements are equivalent: (i) (X, μ) is strongly μ-normal. (ii) For each μg-closed set F and for each μg-open set U containing F, there exists a μ-open set V containing F such that cμ (V) U. (iii) For each pair of disjoint μg-closed sets A and B in X, there exists a μ-open set U containing A such that cμ (U)∩ B = ∅. (iv) For each pair of disjoint μg-closed sets A and B in X, there exist μ-open sets U and V such that A U, B V and cμ (U)∩ cμ (V) = ∅. (v) For any pair of disjoint μg-closed sets A and B, there exist disjoint μ* g-open sets U and V such that A U and B V. (vi) For every μg-closed set A and μg-open set B containing A, there exists a μ* g-open set U such that A U cμ (U) B. Proof. (i) ⇒ (ii): Let F be a μg-closed set and U be a μg-open set such that F U. Then F∩ Uc = ∅. By assumption, there exist μ-open sets V and W such that F V, Uc W and V∩ W = ∅, which implies that cμ (V)∩ W = ∅. Now cμ (V)∩ Uc cμ (V) ∩ W = ∅ and so cμ (V) U. (ii) ⇒ (iii): Let A and B be two disjoint μg-closed sets in X. Since A∩ B = ∅, A Bc and Bc isμg-open. By
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assumption, there exists a μ-open set U containing A such that cμ (U) Bc and so cμ (U)∩ B = ∅. (iii) ⇒ (iv): Let A and B be two disjoint μg-closed sets in X. Then by assumption, there exists a μ-open set U containing A such that cμ (U)∩ B = ∅. Since cμ (U) is μ-closed, it is μg-closed and so B and cμ (U) are disjoint μg-closed sets in X. Therefore again by assumption, there exists a μ-open set V containing B such that cμ (U)∩ cμ (V) = ∅. (iv) ⇒ (i): Let A and B be any two disjoint μg-closed sets in X. By assumption, there exist μ-open sets U and V such that A U, B V and cμ (U)∩ cμ (V) = ∅, so that we have U∩ V = ∅ and thus X is strongly μ-normal. (i) ⇒ (v): This is obvious as every μ-open set is μ* gopen by Remark 2.3. (v) ⇒ (vi): Let A be a μg-closed set and B be a μgopen set with A B. Then A and Bc are two disjoint μg-closed sets in X. Then by (v), there exist disjoint μ* g-open sets U and V such that A U and Bc V. Thus A U X \ V B. Again, since B is μg-open and X \ V is μ* g-closed, cμ (X \ V) B. Hence A U cμ (U) B. (vi) ⇒ (vii): Let A be a μg-closed subset of X and B be a μ* g-open set with A B. Since B is a μ* g-open set containing A and A is μg-closed, by Theorem 2.9, A iμ (B). Thus by (vi) there exists a μ* g-open set U such that A U cμ (U) iμ (B). (vi) ⇒ (i): Let A and B be two disjoint μg-closed sets in X. Then A X \ B and A is μg-closed and X \ B is μg-open. Then by (vi), there exists a μ* g-open set U such that A U cμ (U) (X \ B). Thus A iμ (U), B X \ cμ (U) and iμ (U)∩ X \ cμ (U) = ∅. 5. Preservation theorems
Definition 5.1. A mapping f : (X, μ) → (Y, λ) is called (μ, λ)-open [5] (resp. (μ, λ)-closed [4]) if f(V) is λ-open (resp. λ-closed) in Y for every μ-open (resp. μ-closed) subset V of X. Definition 5.2. A mapping f : (X, μ) → (Y, λ) is called almost (μ, λ)-irresolute (resp. (μ, λ)-continuous [1]) if f−1 (V) is μg-open (resp. μ-open) in X for every λg-open (resp. λ-open) subset V of Y. Theorem 5.3. If (X, μ) is a strongly μ-regular space and f : (X, μ) → (Y, λ) is bijective, almost (μ, λ)irresolute and (μ, λ)-open, then (Y, λ) is strongly λ-regular. Proof. Let y ∈ Y and F be any λg-closed subset of (Y, λ) with y ∈ / F. Since f is almost (μ, λ)-irresolute, f−1 (F)
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is μg-closed in (X, μ) (as f is bijective). Let f(x) = y, then x = f−1 (y). By hypothesis, there exist disjoint μ-open sets U and V such that x ∈ U and f−1 (F) V. Since f is (μ, λ)open and bijective f(U) and f(V) are λ-open and also we have y ∈ f(U) and F f(V) and f(U)∩ f(V) = f(U ∩ V) = ∅. Hence (Y, λ) is a strongly λ-regular space. Theorem 5.4 ([4]). If f : (X, μ) → (Y, λ) is (μ, λ)continuous, (μ, λ)-closed and A is a μg-closed subset of (X, μ), then f(A) is λg-closed. Theorem 5.5. If f : (X, μ) → (Y, λ) is (μ, λ)-continuous, (μ, λ)-closed injection and (Y, λ) is strongly λ-regular, then (X, μ) is strongly μ-regular. Proof. Let F be any μg-closed subset of (X, μ) and x∈ / F. Since f is (μ, λ)-continuous, (μ, λ)-closed, by Theorem 5.4, f(F) is λg-closed in Y and f(x) ∈ / f(F). Since (Y, λ) is strongly λ-regular, there exist disjoint λ-open sets U and V in (Y, λ) such that f(x) ∈ U and f(F) V. By hypothesis, f−1 (U) and f−1 (V) are μ-open in X, such that x ∈ f−1 (U) and F f−1 (V) and f−1 (U)∩ f−1 (V) = ∅. Therefore (X, μ) is strongly μ-regular. Theorem 5.6. If f : (X, μ) → (Y, λ) is an almost (μ, λ)irresolute, (μ, λ)-open bijection and (X, μ) is strongly μ-normal, then (Y, λ) is strongly λ-normal. Proof. Let A and B be any two disjoint μg-closed sets of (Y, λ). Since the mapping f is almost (μ, λ)-irresolute, f−1 (A) and f−1 (B) are disjoint μg-closed sets of (X, μ). As (X, μ) is strongly μ-normal, there exist disjoint μopen sets U and V such that f−1 (A) U and f−1 (B) V. Since f is bijective and (μ, λ)-open, we have f(U) and f(V) are λ-open sets in (Y, λ) such that A f(U) and B f(V) and f(U)∩ f(V) = ∅. Therefore (Y, λ) is strongly λ-normal. Theorem 5.7. If f : (X, μ) → (Y, λ) is a (μ, λ)continuous, (μ, λ)-closed injection and (Y, λ) is strongly λ-normal, then (X, μ) is strongly μ-normal. Proof. Let A and B be any two disjoint μg-closed sets of (X, μ). Since the mapping f is (μ, λ)-continuous and (μ, λ)-closed, f(A) and f(B) are disjoint λg-closed sets of (Y, λ) (by Theorem 5.4). Since (Y, λ) is strongly λ-normal, there exist disjoint μ-open sets U and V such that f(A) U and f(B) V. Thus A f−1 (U) and B f−1 (V) and f−1 (U)∩ f−1 (V) = ∅. Since f is (μ, λ)irresolute, f−1 (U) and f−1 (V) are μ-open sets in (X, μ) and thus (X, μ) is strongly λ-normal.
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Conclusion: The new class of subsets introduced in this paper can be applied in the field of rough set theory [6] and granular computing [7]. References [1] Á. Császár, Generalized topology, generalized continuity, Acta Math. Hungar. 96 (2002) 351–357. [2] Á. Császár, Generalized open sets in generalized topologies, Acta Math. Hungar. 106 (2005) 53–66.
[3] Á. Császár, δ- and θ-modifications of generalized topologies, Acta Math. Hungar. 120 (3) (2008) 275–279. [4] B. Roy, On a type of generalized open sets, Appl. Gen. Topol. 12 (2) (2011) 163–173. [5] B. Roy, A note on weakly (μ, λ)-closed function, Math. Bohemica 138 (4) (2013) 397–405. [6] A. Wiweger, On topological rough sets, Bull. Pol. Ac. Math. 37 (1989) 89–93. [7] R. Bello, R. Falcon, W. Pedrycz, J. Kacprzyk, Granular computing: at the junction of rough sets and fuzzy sets, in: Studies in Fuzziness and Soft Computing, Springer-Verlag, Berlin, Heidelberg, 2008, p. 224.
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On a class of sets between μ-closed sets and μg-closed sets Bishwambhar Roy a,∗,1 , Ritu Sen b a b
Department of Mathematics, Women’s Christian College, 6, Greek Church Row, Kolkata 700 026, India Department of Mathematics, S. A. Jaipuria College, 10, Raja Naba Krishna Street, Kolkata 700 005, India Received 28 August 2014; accepted 30 August 2015
Abstract In this paper, a new class of sets called μ* g-closed sets are introduced and investigated with the help of μ-open and μg-open sets. Relationships between this new class and other related classes of sets are established. Some separation axioms has also being studied. Finally, some preservation theorems have been given. © 2015 The Authors. Production and hosting by Elsevier B.V. on behalf of Taibah University. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/). Keywords: μ-Open set; μg-Open set; μ* g-Closed set
1. Introduction In the past few years, different forms of open sets have been studied. Recently, a significant contribution to the theory of generalized open sets, was extended by A. Császár. Especially, the author defined some basic operators on generalized topological spaces. It is observed that a large number of papers is devoted to the study of generalized open like sets of a topological space,
∗
Corresponding author. Tel.: +91 033 9433401976. E-mail addresses: bishwambhar [email protected] (B. Roy), ritu [email protected] (R. Sen). 1 The author acknowledges the financial support from UGC for the Major Research Project having grant No. 41-799/2012(SR) UGC, New Delhi. Peer review under responsibility of Taibah University.
containing the class of open sets and possessing properties more or less similar to those of open sets. We recall some notions defined in [1]. Let X be a nonempty set, expX denotes the power set of X. We call a class μ expX a generalized topology [1], (briefly, GT) if ∅ ∈ μ and union of elements of μ belongs to μ. A set X with a GT μ on it is said to be a generalized topological space (briefly, GTS) and is denoted by (X, μ). For a GTS (X, μ), the elements of μ are called μopen sets and the complement of μ-open sets are called μ-closed sets. For A X, we denote by cμ (A) the intersection of all μ-closed sets containing A, i.e., the smallest μ-closed set containing A; and by iμ (A) the union of all μ-open sets contained in A, i.e., the largest μ-open set contained in A (see [1,2]). It is easy to observe that iμ and cμ are idempotent and monotonic, where γ: expX → expX is said to be idempotent if A X implies γ(γ(A)) = γ(A) and monotonic if A B X implies γ(A) γ(B). It is also well known from [2,3] that if μ is a GT on X, x ∈ X and A X, then x ∈ cμ (A) iff x ∈ M ∈ μ ⇒ M∩ A = / ∅ and cμ (X \ A) = X \ iμ (A).
http://dx.doi.org/10.1016/j.jtusci.2015.08.008 1658-3655 © 2015 The Authors. Production and hosting by Elsevier B.V. on behalf of Taibah University. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).
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Throughout the paper μ, λ will always mean GT’s on the respective spaces. 2. Properties of μ* g-closed sets Definition 2.1. Let (X, μ) be a GTS. Then a subset A of X is called a μ-generalized closed set (in short, μgclosed set) [4] iff cμ (A) U whenever A U where U is μ-open in X. The complement of a μg-closed set is called a μg-open set. Definition 2.2. Let (X, μ) be a GTS. Then a subset A of X is called strongly μ-generalized closed (in short, μ* gclosed) iff cμ (A) U whenever A U and U is μg-open. Remark 2.3. Let (X, μ) be a GTS. Then every μclosed set is μ* g-closed. Also, every μ* g-closed set is μg-closed. Thus we have the following relation: μ-closed set ⇒ μ* g-closed set ⇒ μg-closed set. Example 2.4. (a) Let X = {a, b, c} and μ = {∅ , {a, b}, {a, c}, X}. Then (X, μ) is a GTS. It can be easily verified that {b, c} is a μ* g-closed set but not a μ-closed set. (b) Let X = {a, b, c, d} and μ = {∅ , {a}, {a, b}, {a, c, d}, X}. Then it is easy to check that {b, d} is a μg-closed set but not a μ* g-closed set. Example 2.5. Let X = {a, b, c, d}, μ = {∅ , {a, b, c}, {a, c, d}, {b, c, d}, X}. Then μ is a GT on X. It can be verified that {a} and {b} are two μ* g-closed sets but their union is not μ* g-closed.
Theorem 2.9. Let (X, μ) be a GTS. Then A X is μ* gopen if and only if F iμ (A) whenever F is μg-closed and F A. Proof. Suppose that A is a μ* g-open set and F is a μg-closed set with F A. Then X \ F is μg-open and X \ A X \ F. Thus by μ* g-closedness of X \ A, cμ (X \ A) (X \ F). Hence F iμ (A). Conversely, suppose that F iμ (A) whenever F is μgclosed and F A. If G is a μg-open set in X containing X \ A, then X \ G is a μg-closed set contained in A. Hence by the hypothesis, X \ G iμ (A). Thus cμ (X \ A) G and hence X \ A is μ* g-closed i.e., A is μ* g-open. Theorem 2.10. If A is μ* g-open in a GTS (X, μ), then G = X, whenever G is μg-open and iμ (A) ∪ (X \ A) G. Proof. Let us assume that G is μg-open and iμ (A) ∪ (X \ A) G. Then X \ G cμ (X \ A) ∩ A = cμ (X \ A) \ (X \ A). Since, X \ G is μg-closed and X \ A is μ* g-closed, then by Theorem 2.6, X\ G = ∅ and hence G = X. Theorem 2.11. If A is μ* g-closed in a GTS (X, μ), then cμ (A) \ A is μ* g-open. Proof. Suppose that A is μ* g-closed and F cμ (A) \ A, where F is a μg-closed subset of X. Then by Theorem 2.6, F =∅ and hence F iμ [cμ (A) \ A]. Therefore by Theorem 2.9, cμ (A) \ A is μ* g-open. Theorem 2.12. For each x ∈ X, either {x} is μg-closed or X \ {x} is μ* g-closed.
Theorem 2.6. Let A be a set in a GTS (X, μ). Then cμ (A) \ A does not contain any non-empty μgclosed set.
Proof. If {x} is not μg-closed, then the only μg-open set containing X \ {x} is X (at most if X ∈ μ). Hence cμ (X \ {x}) X is contained in X and therefore X \ {x} is μ* g-closed.
Proof. Let F be a μg-closed subset contained in cμ (A) \ A. Then F cμ (A) and F X \ A. Thus F cμ (A) ∩ (X \ A) = ∅ i.e., F =∅.
Theorem 2.13. If A is a μ* g-closed set of a GTS (X, μ) such that A B cμ (A), then B is also a μ* g-closed set.
Theorem 2.7. Let (X, μ) be a GTS. If A X is μg-open and μ* g-closed then A is μ-closed.
Proof. Let U be a μg-open set of (X, μ) such that B U. Then A U. Since A is μ* g-closed, then cμ (A) U. Now cμ (B) cμ (cμ (A)) = cμ (A) U. Therefore B is also a μ* g-closed set.
μ* g-closed
Proof. Let A be a μg-open and μ* g-closed subset of X. Then A A where A is μg-open in X. Thus by μ* gclosedness of A, cμ (A) A. Hence A is μ-closed. set in
Definition 2.14. Let (X, μ) be a GTS. Then the μgeneralized kernel of A X is denoted by μg -ker(A) and defined as μg -ker(A) = ∩ {G: G is μg-open and A G}.
Proof. Follows from the fact that every μ-open set is a μg-open set.
Theorem 2.15. A subset A of a GTS (X, μ) is μ* gclosed if and only if cμ (A) μg -kerA.
Corollary 2.8. If A be a μ-open and a GTS (X, μ) then A is μ-closed.
μ* g-closed
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3. Strongly μ-T1/2 space
Proof. (i) Suppose that {x} is not μ-closed, for some x ∈ X. Then the only μ-open set containing X \ {x} is X at most, hence X \ {x} is μg-closed. Since X is a strongly μ-T1/2 space, then X \ {x} is μ* g-closed. Therefore, {x} is μ* g-open in X. (ii) Assume that {x} is not μ-closed, for some x ∈ X. Then X is a μ-open set (at most) containing X \ {x}, hence X \ {x} is μg-closed and therefore {x} is μg-open in X.
Definition 3.1. A GTS (X, μ) is said to be:
4. Strongly μ-regular and strongly μ-normal spaces
Proof. Assume that A is a μ* g-closed set in X. Then cμ (A) G whenever A G and G is μg-open in X. This implies that cμ (A) ∩ {G : A G and G is μg-open} = μg -ker(A). For the converse, assume that cμ (A) μg ker(A). This implies that cμ (A) ∩ {G : A G and G is μg-open}. This shows that cμ (A) G for all μg-open sets G containing A. This proves that A is μ* g-closed.
(i) Strongly μ-T1/2 if every μg-closed set in X is μ* gclosed. (ii) μ* -T1/2 if every μ* g-closed set in X is μ-closed. (iii) μ-T1/2 [4] if every μg-closed set in X is μ-closed. Remark 3.2. μ* -T1/2 and strongly μ-T1/2 spaces are independent as may be seen from Example 3.3. Example 3.3. (a) Let X = {a, b, c} and μ = {∅ , {a}, {a, b}, {a, c, d}, X}. Then (X, μ) is a GTS. It can be checked that (X, μ) is μ* -T1/2 but not a strongly μ-T1/2 space. (b) Let X = {a, b, c, d}, μ = {∅ , {a, b}, {a, d}, {a, b, d}, X}. Then (X, μ) is a GTS. It can be checked that (X, μ) is strongly μ-T1/2 but not μ* -T1/2 .
Definition 4.1. A GTS (X, μ) is said to be strongly μregular if for each μg-closed set F and each point x ∈ / F, there exist disjoint μ-open sets U and V such that F U and x ∈ V. Definition 4.2. A GTS (X, μ) is said to be μ-regular [4] if for each μ-closed set F of X and each point x ∈ / F, there exist disjoint μ-open sets U and V such that x ∈ U and F V. As every μ-closed set is μg-closed, every strongly μ-regular space is μ-regular but the converse is false as shown by the next example.
Theorem 3.4. For a GTS (X, μ), the following statements hold:
Example 4.3. Let X = {a, b, c, d} and μ = {∅ , {a, b}, {c, d}, X}. Then (X, μ) is a GTS. It can be checked that X is μ-regular but not strongly μ-regular.
(i) Every μ-T1/2 space is strongly μ-T1/2 . (ii) Every μ-T1/2 space is μ* -T1/2 .
Theorem 4.4. A GTS (X, μ) is strongly μ-regular if and only if (X, μ) is μ-regular and μ-T1/2 .
Proof. Obvious.
Proof. Suppose that (X, μ) is strongly μ-regular. Then clearly (X, μ) is μ-regular. Now let A X be μg-closed. For each x ∈ / A there exists a μ-open set Vx containing / A}, then V is μx such that Vx ∩ A = ∅. If V = ∪ {Vx : x ∈ open and V = X \ A, hence A is μ-closed. The converse is obvious.
The converses of the above theorem need not be true as may be seen from the following examples. Example 3.5. (a) Let X = {a, b, c, d} and μ = {∅ , {a, b}, {b, c}, {a, b, c}, X}. Then (X, μ) is a GTS. It can be verified that (X, μ) is strongly μ-T1/2 but not μ-T1/2 . (b) Let X = {a, b, c, d} and μ = {∅ , {a, b, c}, {b, c, d}, X}. Then (X, μ) is a GTS. It can be checked that (X, μ) is a μ* -T1/2 space but not a μ-T1/2 space. Theorem 3.6. If (X, μ) is a strongly μ-T1/2 space, then the following statements hold: (i) Every singleton in X is μ-closed or μ* g-open in X. (ii) Every singleton in X is μ-closed or μg-open.
Theorem 4.5. For a GTS (X, μ) the following statements are equivalent: (i) (X, μ) is strongly μ-regular. (ii) For each point x ∈ X and for each μg-open set W containing x, there exists a μ-open set U in X such that cμ (U) W. (iii) For each point x ∈ X and for each μg-closed set F not containing x, there exists a μ-open set V in X such that cμ (V)∩ F = ∅.
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(iv) For each x ∈ X and each μg-closed set F with x ∈ /F there exist a μ-open set U and a μ* g-open set V such that x ∈ U, F V and U∩ V = ∅. (v) For each A X and each μg-closed set F with A∩ F = ∅ there exist a μ-open set U and a μ* g-open set V such that A∩ U = / ∅, F V, U∩ V = ∅. Proof. (i) ⇒ (ii): Let W be any μg-open set containing x. Then Wc is μg-closed and x ∈ / Wc . By hypothesis, there exist μ-open sets U and V such that Wc V, x ∈ U and U∩ V = ∅ and so U Vc . Now cμ (U) cμ (Vc ) = Vc and Wc V implies that Vc W. Thus cμ (U) W. (ii) ⇒ (i): Let F be any μg-closed set and x ∈ / F. Then x ∈ Fc and Fc is a μg-open set containing x. By hypothesis, there exists a μ-open set V containing x such that x ∈ V and cμ (V) Fc , which implies that F (cμ (V))c . Then (cμ (V))c is a μ-open set containing F and V∩ (cμ (V))c = ∅. Hence X is strongly μ-regular. (ii) ⇒ (iii): Let x ∈ X and F be a μg-closed set such that x ∈ / F. Then Fc is a μg-open set containing x and by hypothesis, there exists a μ-open set V of X such that cμ (V) Fc and hence cμ (V)∩ F = ∅. (iii) ⇒ (ii): Let x ∈ X and W be a μg-open set contain/ Wc , by hypothesis ing x. Since Wc is μg-closed and x ∈ there exists a μ-open set U containing x such that cμ (U)∩ Wc = ∅. Hence cμ (U) W. (i) ⇒ (iv): This is obvious as every μ-open set is μ* gopen (by Remark 2.3). (iv) ⇒ (v): Let F be a μg-closed set with A∩ F = ∅ for any subset A of X. Then for each a ∈ A, a ∈ / F. So there exist U ∈ μ and a μ* g-open set V such that a ∈ U, F V / ∅. and U∩ V = ∅. Thus A∩ U = (v) ⇒ (i): Let F be a μg-closed set such that x ∈ / F. Then {x}∩ F = ∅. Thus by (v), there exist a μ-open set U and a μ* g-open set W such that x ∈ U, F W and U∩ W = ∅. Put V = iμ (W). Then F V (by Theorem 2.9) and U∩ V = ∅. Definition 4.5. A GTS (X, μ) is said to be strongly μnormal if for every pair of disjoint μg-closed sets A and B, there exist disjoint μ-open sets U, V ( X) such that A U and B V. Definition 4.6. A GTS (X, μ) is said to be μ-normal [4] if for every pair of disjoint μ-closed sets A and B of X, there exist disjoint μ-open sets U and V such that A U and B V. Example 4.7. Let X = {a, b, c, d} and μ = {∅ , {a}, {a, c}, {b, c, d}, X}. Then (X, μ) is a GTS. It can be checked that X is μ-normal but not strongly μ-normal.
Definition 4.8. A GTS (X, μ) is said to be μ-symmetric [4] iff for each x, y ∈ X, x ∈ cμ ({y}) ⇒ y ∈ cμ ({x}). We recall [4] that a GTS (X, μ) is μ-symmetric iff {x} is μg-closed for each x ∈ X. Theorem 4.9. (i) Every μ-normal, μ-symmetric GTS (X, μ) is μ-regular. (ii) Every strongly μ-normal, μ-symmetric GTS (X, μ) is strongly μ-regular and hence μ-regular. / A. Proof. (i) Suppose that A X is μ-closed and x ∈ Then {x} is μg-closed. By μ-normality of X, there exist disjoint μ-open sets U and V such A V and {x} U. Thus X is μ-regular. (ii) Obvious. Example 4.10. (a) Consider Example 4.7. It is easy to observe that X is μ-normal but not μ-regular. (b) Consider X = {a, b, c} and μ = {∅ , {a}, {a, b}}. Then (X, μ) is a GTS. It can be checked that (X, μ) is strongly μ-normal but not strongly μ-regular. Theorem 4.11. Let (X, μ) be a GTS. Then the following statements are equivalent: (i) (X, μ) is strongly μ-normal. (ii) For each μg-closed set F and for each μg-open set U containing F, there exists a μ-open set V containing F such that cμ (V) U. (iii) For each pair of disjoint μg-closed sets A and B in X, there exists a μ-open set U containing A such that cμ (U)∩ B = ∅. (iv) For each pair of disjoint μg-closed sets A and B in X, there exist μ-open sets U and V such that A U, B V and cμ (U)∩ cμ (V) = ∅. (v) For any pair of disjoint μg-closed sets A and B, there exist disjoint μ* g-open sets U and V such that A U and B V. (vi) For every μg-closed set A and μg-open set B containing A, there exists a μ* g-open set U such that A U cμ (U) B. Proof. (i) ⇒ (ii): Let F be a μg-closed set and U be a μg-open set such that F U. Then F∩ Uc = ∅. By assumption, there exist μ-open sets V and W such that F V, Uc W and V∩ W = ∅, which implies that cμ (V)∩ W = ∅. Now cμ (V)∩ Uc cμ (V) ∩ W = ∅ and so cμ (V) U. (ii) ⇒ (iii): Let A and B be two disjoint μg-closed sets in X. Since A∩ B = ∅, A Bc and Bc isμg-open. By
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assumption, there exists a μ-open set U containing A such that cμ (U) Bc and so cμ (U)∩ B = ∅. (iii) ⇒ (iv): Let A and B be two disjoint μg-closed sets in X. Then by assumption, there exists a μ-open set U containing A such that cμ (U)∩ B = ∅. Since cμ (U) is μ-closed, it is μg-closed and so B and cμ (U) are disjoint μg-closed sets in X. Therefore again by assumption, there exists a μ-open set V containing B such that cμ (U)∩ cμ (V) = ∅. (iv) ⇒ (i): Let A and B be any two disjoint μg-closed sets in X. By assumption, there exist μ-open sets U and V such that A U, B V and cμ (U)∩ cμ (V) = ∅, so that we have U∩ V = ∅ and thus X is strongly μ-normal. (i) ⇒ (v): This is obvious as every μ-open set is μ* gopen by Remark 2.3. (v) ⇒ (vi): Let A be a μg-closed set and B be a μgopen set with A B. Then A and Bc are two disjoint μg-closed sets in X. Then by (v), there exist disjoint μ* g-open sets U and V such that A U and Bc V. Thus A U X \ V B. Again, since B is μg-open and X \ V is μ* g-closed, cμ (X \ V) B. Hence A U cμ (U) B. (vi) ⇒ (vii): Let A be a μg-closed subset of X and B be a μ* g-open set with A B. Since B is a μ* g-open set containing A and A is μg-closed, by Theorem 2.9, A iμ (B). Thus by (vi) there exists a μ* g-open set U such that A U cμ (U) iμ (B). (vi) ⇒ (i): Let A and B be two disjoint μg-closed sets in X. Then A X \ B and A is μg-closed and X \ B is μg-open. Then by (vi), there exists a μ* g-open set U such that A U cμ (U) (X \ B). Thus A iμ (U), B X \ cμ (U) and iμ (U)∩ X \ cμ (U) = ∅. 5. Preservation theorems
Definition 5.1. A mapping f : (X, μ) → (Y, λ) is called (μ, λ)-open [5] (resp. (μ, λ)-closed [4]) if f(V) is λ-open (resp. λ-closed) in Y for every μ-open (resp. μ-closed) subset V of X. Definition 5.2. A mapping f : (X, μ) → (Y, λ) is called almost (μ, λ)-irresolute (resp. (μ, λ)-continuous [1]) if f−1 (V) is μg-open (resp. μ-open) in X for every λg-open (resp. λ-open) subset V of Y. Theorem 5.3. If (X, μ) is a strongly μ-regular space and f : (X, μ) → (Y, λ) is bijective, almost (μ, λ)irresolute and (μ, λ)-open, then (Y, λ) is strongly λ-regular. Proof. Let y ∈ Y and F be any λg-closed subset of (Y, λ) with y ∈ / F. Since f is almost (μ, λ)-irresolute, f−1 (F)
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is μg-closed in (X, μ) (as f is bijective). Let f(x) = y, then x = f−1 (y). By hypothesis, there exist disjoint μ-open sets U and V such that x ∈ U and f−1 (F) V. Since f is (μ, λ)open and bijective f(U) and f(V) are λ-open and also we have y ∈ f(U) and F f(V) and f(U)∩ f(V) = f(U ∩ V) = ∅. Hence (Y, λ) is a strongly λ-regular space. Theorem 5.4 ([4]). If f : (X, μ) → (Y, λ) is (μ, λ)continuous, (μ, λ)-closed and A is a μg-closed subset of (X, μ), then f(A) is λg-closed. Theorem 5.5. If f : (X, μ) → (Y, λ) is (μ, λ)-continuous, (μ, λ)-closed injection and (Y, λ) is strongly λ-regular, then (X, μ) is strongly μ-regular. Proof. Let F be any μg-closed subset of (X, μ) and x∈ / F. Since f is (μ, λ)-continuous, (μ, λ)-closed, by Theorem 5.4, f(F) is λg-closed in Y and f(x) ∈ / f(F). Since (Y, λ) is strongly λ-regular, there exist disjoint λ-open sets U and V in (Y, λ) such that f(x) ∈ U and f(F) V. By hypothesis, f−1 (U) and f−1 (V) are μ-open in X, such that x ∈ f−1 (U) and F f−1 (V) and f−1 (U)∩ f−1 (V) = ∅. Therefore (X, μ) is strongly μ-regular. Theorem 5.6. If f : (X, μ) → (Y, λ) is an almost (μ, λ)irresolute, (μ, λ)-open bijection and (X, μ) is strongly μ-normal, then (Y, λ) is strongly λ-normal. Proof. Let A and B be any two disjoint μg-closed sets of (Y, λ). Since the mapping f is almost (μ, λ)-irresolute, f−1 (A) and f−1 (B) are disjoint μg-closed sets of (X, μ). As (X, μ) is strongly μ-normal, there exist disjoint μopen sets U and V such that f−1 (A) U and f−1 (B) V. Since f is bijective and (μ, λ)-open, we have f(U) and f(V) are λ-open sets in (Y, λ) such that A f(U) and B f(V) and f(U)∩ f(V) = ∅. Therefore (Y, λ) is strongly λ-normal. Theorem 5.7. If f : (X, μ) → (Y, λ) is a (μ, λ)continuous, (μ, λ)-closed injection and (Y, λ) is strongly λ-normal, then (X, μ) is strongly μ-normal. Proof. Let A and B be any two disjoint μg-closed sets of (X, μ). Since the mapping f is (μ, λ)-continuous and (μ, λ)-closed, f(A) and f(B) are disjoint λg-closed sets of (Y, λ) (by Theorem 5.4). Since (Y, λ) is strongly λ-normal, there exist disjoint μ-open sets U and V such that f(A) U and f(B) V. Thus A f−1 (U) and B f−1 (V) and f−1 (U)∩ f−1 (V) = ∅. Since f is (μ, λ)irresolute, f−1 (U) and f−1 (V) are μ-open sets in (X, μ) and thus (X, μ) is strongly λ-normal.
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Conclusion: The new class of subsets introduced in this paper can be applied in the field of rough set theory [6] and granular computing [7]. References [1] Á. Császár, Generalized topology, generalized continuity, Acta Math. Hungar. 96 (2002) 351–357. [2] Á. Császár, Generalized open sets in generalized topologies, Acta Math. Hungar. 106 (2005) 53–66.
[3] Á. Császár, δ- and θ-modifications of generalized topologies, Acta Math. Hungar. 120 (3) (2008) 275–279. [4] B. Roy, On a type of generalized open sets, Appl. Gen. Topol. 12 (2) (2011) 163–173. [5] B. Roy, A note on weakly (μ, λ)-closed function, Math. Bohemica 138 (4) (2013) 397–405. [6] A. Wiweger, On topological rough sets, Bull. Pol. Ac. Math. 37 (1989) 89–93. [7] R. Bello, R. Falcon, W. Pedrycz, J. Kacprzyk, Granular computing: at the junction of rough sets and fuzzy sets, in: Studies in Fuzziness and Soft Computing, Springer-Verlag, Berlin, Heidelberg, 2008, p. 224.
Please cite this article in press as: B. Roy, R. Sen. On a class of sets between μ-closed sets and μg-closed sets, J. Taibah Univ. Sci. (2015), http://dx.doi.org/10.1016/j.jtusci.2015.08.008