ON µ-compact SETS IN µ-spaces

Transcription

1 Questions and Answers in General Topology 31 (2013), pp ON µ-compact SETS IN µ-spaces MOHAMMAD S. SARSAK (Communicated by Yasunao Hattori) Abstract. The primary purpose of this paper is to introduce and study µ- compact sets in µ-spaces. Several properties and mapping properties of such spaces are studied. 1. Introduction and Preliminaries A generalized topology (breifly GT) [2] µ on a nonempty set X is a collection of subsets of X such that µ and µ is closed under arbitrary unions. Elements of µ will be called µ-open sets, and a subset A of (X, µ) will be called µ-closed if X\A is µ-open. Clearly, a subset A of (X, µ) is µ-open if and only if for each x A, there exists U x µ such that x U x A, or equivalently, A is the union of µ-open sets. The pair (X, µ) will be called generalized topological space (breifly GTS). A space X or (X, µ) will always mean a GTS. A space (X, µ) is called a µ-space [17] if X µ. (X, µ) is called a quasi-topological space [4] if µ is closed under finite intersections. Clearly, every topological space is a quasi-topological space, every quasi-topological space is a GTS, and a space (X, µ) is a topological space if and only if (X, µ) is both µ-space and quasi-topological space. If A is a subset of a space (X, µ), then the µ-closure of A [3], c µ (A), is the intersection of all µ-closed sets containing A and the µ-interior of A [3], i µ (A), is the union of all µ-open sets contained in A. It was pointed out in [3] that each of the operators c µ and i µ are monotonic [5], i.e. if A B X, then c µ (A) c µ (B) and i µ (A) i µ (B), idempotent [5], i.e. if A X, then c µ (c µ (A)) = c µ (A) and i µ (i µ (A)) = i µ (A), c µ is enlarging [5], i.e. if A X, then c µ (A) A, i µ is restricting [5], i.e. if A X, then i µ (A) A, A is µ-open if and only if A = i µ (A), and c µ (A) = X\i µ (X\A) Mathematics Subject Classification. Primary 54A05, 54A10, 54D20. Key words and phrases. µ-open; µ-closed; Generalized topology; µ-space; µ-compact set; µ- Lindelöf set; µ-compact space; µ-lindelöf space. c 2013 Symposium of General Topology. 49

2 50 Mohammad S. Sarsak Clearly, A is µ-closed if and only if A = c µ (A), c µ (A) is the smallest µ-closed set containing A and i µ (A) is the largest µ-open set contained in A, and x c µ (A) if and only if any µ-open set containing x intersects A. A function f : (X, µ) (Y, κ) is called (µ, κ)-continuous [19] if the inverse image of each κ-open set is µ-open, and called (µ, κ)-closed [19] if the image of every µ-closed set is κ-closed. If (X, τ) is a topological space and A X, then A and Int A will stand respectively for the closure of A in X and the interior of A in X. A subset A of a topological space (X, τ) is called semi-open [14] if A Int A, and called semi-closed if X\A is semi-open. A is called preopen [15] if A Int A. It is known that the arbitrary union of semi-open (resp. preopen) sets is semi-open (resp. preopen). A subset A of a topological space (X, τ) is called semi-regular [7] if it is both semi-open and semi-closed. A point x X is called a semi-θ-adherent point [7] of A if A U for every semi-regular set U containing x. The set of all semi-θadherent points of A is usually denoted by scl θ (A) and is called the semi-θ-closure of A [7]. A is called semi-θ-closed [7] if A = scl θ (A), and called semi-θ-open [11] if X\A is semi-θ-closed. Clearly, A is semi-θ-open if and only if for every x A, there exists a semi-regular set U such that x U A, thus, A is semi-θ-open if and only if A is the union of semi-regular sets. It is also clear that the arbitrary union of semi-θ-open sets is semi-θ-open. The families of semi-open (resp. semi-θ-open, preopen) subsets of a topological space (X, τ) will be denoted by SO (X) (resp. SθO (X), P O (X)). Clearly, if µ = SO (X) (resp. SθO (X), P O (X)), then (X, µ) is a µ-space. A subset A of a topological space (X, τ) is called γ-open [1] if A γ (A), where γ : exp X exp X is a mapping such that γ ( ) =, γ (X) = X, A B γ (A) γ (B), and G γ (A) γ (G A) whenever A X and G is open in X. In [1], γ-compact topological spaces where introduced and studied, where a topological space (X, τ) is called γ-compact if any cover of X by γ-open subsets of X has a finite subcover. For the concepts and terminology not defined here, the reader is referred to [10]. 2. µ-compact Sets This section is mainly devoted to introduce and study µ-compact sets and µ- Lindelöf sets in µ-spaces. Definition 2.1. A subset A of a µ-space (X, µ) is called µ-compact (resp. µ- Lindelöf) if any cover of A by µ-open subsets of X has a finite (resp. countable) subcover of A.

3 On µ-compact sets in µ-spaces 51 If (X, τ) is a topological space, A X and µ = SO (X), then A is µ-compact (resp. µ-lindelöf) = (A is semi-compact [6], resp. semi-lindelöf [18]) relative to (X, τ). If (X, τ) is a topological space, A X and µ = SθO (X), then A is µ-compact (resp. µ-lindelöf) = (A is s-closed [8], resp. rs-lindelöf [8]) relative to (X, τ). If (X, τ) is a topological space, A X and µ = P O (X), then A is µ-compact (resp. µ-lindelöf) = (A is strongly compact [16], resp. strongly Lindelöf [13]) relative to (X, τ). Definition 2.2. A µ-space (X, µ) is called µ-compact (resp. µ-lindelöf) if any cover of X by µ-open sets has a finite (resp. countable) subcover. If (X, τ) is a topological space and µ = SO (X), then (X, µ) is µ-compact (resp. µ-lindelöf) = ((X, τ) is semi-compact [9], resp. semi-lindelöf [12]). If (X, τ) is a topological space and µ = SθO (X), then (X, µ) is µ-compact (resp. µ-lindelöf) = ((X, τ) is s-closed [7], resp. rs-lindelöf [8]). If (X, τ) is a topological space and µ = P O (X), then (X, µ) is µ-compact (resp. µ-lindelöf) = ((X, τ) is strongly compact [16], resp. strongly Lindelöf [16]). The proofs of the following two propositions are straightforward, and thus omitted. Proposition 2.3. (i) A subset A of a µ-space (X, µ) is µ-compact (resp. µ- Lindelöf) if and only if for every family F = {F α : α Λ} of µ-closed sets having the property that for every finite (resp. countable) subfamily F i of F, ( F i ) A, then ( F) A. (ii) A µ-space (X, µ) is µ-compact (resp. µ-lindelöf) if and only if for every family F = {F α : α Λ} of µ-closed sets having the property that for every finite (resp. countable) subfamily F i of F, ( F i ), then ( F). Proposition 2.4. The finite (resp. countable) union of µ-compact (resp. µ- Lindelöf) sets in a µ-space (X, µ) is µ-compact (resp. µ-lindelöf). Definition 2.5. A filter base F on a µ-space (X, µ) is said to µ-converge to a point x X if for each µ-open subset U of X such that x U, there exists F F such that F U. F is said to µ-accumulate at x X if U F for every F F and for every µ-open subset U of X such that x U. Example 2.6. Let X = {a, b, c}, µ = {, {a, b}, {a, c}, X} and F = {{b, c}}. Then F µ-accumulates at a, b, and c. F does not µ-converge to a or b or c. Example 2.7. Let X = {a, b, c}, µ = {, {a, b}, {a, c}, X} and F = {{a}}. Then F µ-converges to a, b, and c. The proof of the following proposition is straightforward and thus omitted.

4 52 Mohammad S. Sarsak Proposition 2.8. Let F be a filter base on a µ-space (X, µ) and x X. Then (i) If F µ-converges to x, then F µ-accumulates at x. (ii) If F is a maximal filter base, then F µ-converges to x if and only if F µ- accumulates at x. Proposition 2.9. For a µ-space (X, µ), the following are equivalent: (i) X is µ-compact, (ii) Every maximal filter base on X µ-converges to some point of X, (iii) Every filter base on X µ-accumulates at some point of X. Proof. (i) (ii): Let F be a maximal filter base on X such that F does not µ- converge to any point of X. Since F is maximal, it follows from Proposition 2.8 (ii) that F does not µ-accumulate at any point of X. Thus, for each x X, there exists F x F and a µ-open subset U x of X such that x U x and U x F x =, but X is µ-compact, so there exist x 1, x 2,..., x n X such that X = n i=1 U x i. Since F is a filter base on X, there exists F F such that F n i=1 F x i, but U xi F xi = for each i {1, 2,..., n}, so U xi F = for each i {1, 2,..., n}, i.e. ( n i=1 U x i ) F = X F = F =, a contradiction. (ii) (iii): Let F be a filter base on X. Then F is contained in a maximal filter base H on X. By (ii), H µ-converges to some point x of X, thus by Proposition 2.8 (i), H µ-accumulates at x, but F H, so F µ-accumulates at x. (iii) (i): Suppose that X is not µ-compact. Then by Proposition 2.3 (ii), there exists a cover U = {U α : α Λ} of X by µ-open sets such that for any finite subset Λ 0 of Λ, {X\U α : α Λ 0 } =. For each finite subset Λ 0 of Λ, let F Λ0 = {X\Uα : α Λ 0 }. Then F = {F Λ0 : Λ 0 is a finite subset of Λ} is a filter base on X. Thus by (iii), F µ-accumulates at some point x of X. Since U is a cover of X, there exists α 0 Λ such that x U α0, but F µ-accumulates at x and U α0 is µ-open, so U α0 F for every F F. Let F = X\U α0. Then F F and thus U α0 (X\U α0 ), a contradiction. The proof of the following proposition is similar to that of Proposition 2.9, and thus omitted. Proposition For a subset A of a µ-space (X, µ), the following are equivalent: (i) A is µ-compact, (ii) Every maximal filter base on X, each of whose members meets A, µ-converges to some point of A, (iii) Every filter base on X, each of whose members meets A, µ-accumulates at some point of A. Definition Let A be a nonempty subset of a space (X, µ). The generalized subspace topology on A is the collection {U A : U µ}, and will be denoted by µ A. The generalized subspace A is the generalized topological space (A, µ A ).

5 On µ-compact sets in µ-spaces 53 Remark Let A be a nonempty subset of a space (X, µ). Then it is easy to see the following: (i) If (X, µ) is a µ-space, then (A, µ A ) is a µ A -space. (ii) A subset B of A is µ A -closed if and only if B = F A for some µ-closed set F. Proposition Let B be a nonempty subset of a µ-space (X, µ) and A B. Then A is µ-compact (resp. µ-lindelöf) if and only if A is µ B -compact (resp. µ B -Lindelöf). Proof. Necessity. The case when A is µ-compact will be shown, the other case is similar. Observe first by Remark 2.12 (i) that since (X, µ) is a µ-space, (B, µ B ) is a µ B -space. Suppose that A = {A α : α Λ} is a cover of A by µ B -open sets. Then A α = S α B for each α Λ, where S α is µ-open for each α Λ. Thus S = {S α : α Λ} is a cover of A by µ-open sets, but A is µ-compact, so there exist α 1, α 2,...α n Λ such that A n, and thus A n i=1 (S α i B) = n i=1 A α i. Hence, A is µ B -compact. Sufficiency. The case when A is µ B -compact will be shown, the other case is similar. Suppose that S = {S α : α Λ} is a cover of A by µ-open sets. Then A = {S α B : α Λ} is a µ B -open cover of A, but A is µ B -compact, so there exist α 1, α 2,...α n Λ such that A n i=1 (S α i B) n. Hence, A is µ-compact. Corollary Let A be a nonempty subset of a µ-space (X, µ). Then A is µ-compact (resp. µ-lindelöf) if and only if A is µ A -compact (resp. µ A -Lindelöf). Proposition Let A be a µ-compact (resp. µ-lindelöf) subset of a µ-space (X, µ) and B be a µ-closed subset of X. Then A B is µ-compact (resp. µ- Lindelöf). In particular, a µ-closed subset A of a µ-compact (resp. µ-lindelöf) µ-space (X, µ) is µ-compact (resp. µ-lindelöf). Proof. The case when A is µ-compact will be shown, the other case is similar. Suppose that S = {S α : α Λ} is a cover of A B by µ-open sets. Then A = {S α : α Λ} {X\B} is a cover of A by µ-open sets, but A is µ-compact, so there exist α 1, α 2,..., α n Λ such that A ( n ) (X\B). Thus A B n i=1 (S α i B) n. Hence, A B is µ-compact. Definition Let (X α, µ α ) be a generalized topological space for each α Λ, where {X α : α Λ} is a disjoint family of sets. The collection µ of subsets of X α is defined as follows: { µ = U } X α : U X α µ α, α Λ The following proposition can be easily verified.

6 54 Mohammad S. Sarsak Proposition Let (X α, µ α ) be a generalized topological space for each α Λ, where {X α : α Λ} is a disjoint family of sets, and let µ be as in Definition Then µ is a generalized topology on X α. The generalized topological space ( X α, µ) will be called the generalized topological sum of X α, α Λ, and will be denoted by X α. The following corollary is an immediate consequence of Proposition Corollary Let ( X α, µ) be the generalized topological sum of (X α, µ α ), α Λ. Then a subset A of X α is µ-closed if and only if A X α is µ α -closed for each α. Remark Let (X α, µ α ) be a generalized topological space for each α Λ, and let ( X α, µ) be the generalized topological sum of (X α, µ α ), α Λ. Then it is easy to see the following: (i) µ α µ. (ii) µ Xα = µ α for each α Λ. Remark Let (X α, µ α ) be a µ α -space for each α Λ, and let ( X α, µ) be the generalized topological sum of (X α, µ α ), α Λ. Then it is easy to see the following: (i) ( X α, µ) is a µ-space. (ii) X α is µ-open (µ-closed) for each α Λ. The proof of the following proposition is straightforward and thus omitted. Proposition Let (X α, µ α ) be a µ α -space for each α Λ, and let ( X α, µ) be the generalized topological sum of (X α, µ α ), α Λ. Then X α is µ-compact (resp. µ-lindelöf) if and only if X α is µ α -compact (resp. µ α -Lindelöf) for each α Λ and Λ is finite (resp. countable). 3. Mapping Properties This section is mainly devoted to study several mapping properties of µ-compact sets and µ-lindelöf sets in µ-spaces. Proposition 3.1. Let f : (X, µ) (Y, κ) be a (µ, κ)-continuous function, where (X, µ) is a µ-space and (Y, κ) is a κ-space. Then (i) If A is µ-compact, then f(a) is κ-compact. (ii) If A is µ-lindelöf, then f(a) is κ-lindelöf. Proof. Part (ii) will be shown, the proof of (i) is similar. Suppose that S = {S α : α Λ} is a cover of f(a) by κ-open sets. Then A = {f 1 (S α ) : α Λ} is a cover of A, but f is (µ, κ)-continuous, so f 1 (S α ) is µ-open for each α Λ. Since A is µ-lindelöf, there exist α 1, α 2, α 3,... Λ such that A i=1 f 1 (S αi ). Thus f (A) i=1 f (f 1 (S αi )). Hence, f (A) is κ-lindelöf.

7 On µ-compact sets in µ-spaces 55 Corollary 3.2. Let f : (X, µ) (Y, κ) be a (µ, κ)-continuous surjection, where (X, µ) is a µ-space and (Y, κ) is a κ-space. Then (i) If X is µ-compact, then Y is κ-compact. (ii) If X is µ-lindelöf, then Y is κ-lindelöf. Proposition 3.3. Let f : (X, µ) (Y, κ) be a (µ, κ)-closed surjection, where (X, µ) is a µ-space and (Y, κ) is a κ-space. If for each y Y, f 1 (y) is µ-compact (resp. µ-lindelöf), then f 1 (A) is µ-compact (resp. µ-lindelöf) whenever A is κ-compact (resp. κ-lindelöf). Proof. The case when A is κ-compact will be shown, the other case is similar. Suppose that S = {S α : α Λ} is a cover of f 1 (A) by µ-open sets. Then it follows by assumption that for each y A, there exists a finite subcollection S y of S such that f 1 (y) S y. Let V y = S y. Then V y is µ-open. Let H y = Y \f (X\V y ). Then H y is κ-open as f is (µ, κ)-closed, also y H y for each y A as f 1 (y) V y. Thus, H = {H y : y A} is a cover of A by κ-open sets, but A is κ-compact, so there exist y 1, y 2,..., y n A such that A n i=1 H y i. Thus, f 1 (A) n i=1 f 1 (H yi ) n i=1 V y i. Since S y i is a finite subcollection of S for each i {1, 2,..., n}, it follows that n i=1 Sy i is a finite subcollection of S. Hence, f 1 (A) is µ-compact. The proof of the following proposition is straightforward and thus omitted. Proposition 3.4. Let (X, µ) and (Y, κ) be generalized topological spaces, and let U = {U V : U µ, V κ}. Then U generates a generalized topology σ on X Y, called the generalized product topology on X Y, that is, σ = {all possible unions of members of U} σ will sometimes be denoted by µ κ. Remark 3.5. Let (X, µ) and (Y, κ) be generalized topological spaces, σ be the generalized product topology on X Y, A X, B Y and K X Y. Then it is easy to see the following: (i) K is σ-open if and only if for each (x, y) K, there exist U x µ and V y κ such that (x, y) U x V y K. (ii) c σ (A B) = c µ (A) c κ (B). (iii) i σ (A B) = i µ (A) i κ (B). (iv) σ A B = µ A κ B. Remark 3.6. Let (X, µ) be a µ-space, (Y, κ) be a κ-space, and σ be the generalized product topology on X Y. Then it is clear that (X Y, σ) is a σ-space. The proof of the following lemma is straightforward and thus omitted.

8 56 Mohammad S. Sarsak Lemma 3.7. Let (X, µ) be a µ-space, (Y, κ) be a κ-space, and σ be the generalized product topology on X Y. Then the projection function P X : (X Y, σ) (X, µ) (resp. P Y : (X Y, σ) (Y, κ)) is (σ, µ)-continuous (resp. (σ, κ)-continuous). Corollary 3.8. Let (X, µ) be a µ-space, (Y, κ) be a κ-space, and σ be the generalized product topology on X Y. If X Y is σ-compact (resp. σ-lindelöf), then (X, µ) is µ-compact (resp. µ-lindelöf) and (Y, κ) is κ-compact (resp. κ-lindelöf). Proof. Observe first by Remark 3.6 that since (X, µ) is a µ-space and (Y, κ) is a κ-space, then (X Y, σ) is a σ-space. The result follows from Corollary 3.2 and Lemma 3.7. References [1] Á. Császár, γ-compact spaces, Acta Math. Hungar. 87 (1-2) (2000), [2] Á. Császár, Generalized topology, generalized continuity, Acta Math. Hungar. 96 (2002), [3] Á. Császár, Generalized open sets in generalized topologies, Acta Math. Hungar. 106 (1-2) (2005), [4] Á. Császár, Further remarks on the formula for γ-interior, Acta Math. Hungar. 113 (2006), [5] Á. Császár, Remarks on quasi topologies, Acta Math. Hungar. 119 (1-2) (2008), [6] M. C. Cueva and J. Dontchev, On spaces with hereditarily compact α-topologies, Acta Math. Hungar. 82 (1-2) (1999), [7] Di Maio and T. Noiri, On s-closed spaces, Indian J. Pure Appl. Math. 18 (3) (1987), [8] J. Dontchev and M. Ganster, On covering spaces with semi-regular sets, Ricerche di Matematica 45, fasc. 1 o, (1996), [9] C. Dorsett, Semi-compactness, semi-separation axioms and product spaces, Bull. Malaysian Math. Soc. (2) 4 (1981), [10] R. Engelking. General Topology. Second edition. Sigma Series in Pure Mathematics, 6. Heldermann Verlag, Berlin, [11] S. Ganguly and C. K. Basu, Further characterizations of s-closed spaces, Indian J. Pure Appl. Math. 23 (1992), [12] M. Ganster, On covering properties and generalized open sets in topological spaces, Math. Chronicle 19 (1990), [13] H.Z. Hdeib and M.S. Sarsak, On strongly Lindelöf spaces, Questions Answers Gen. Topology 18 (2) (2000), [14] N. Levine, Semi-open sets and semi-continuity in topological spaces, Amer. Math. Monthly 70 (1963), [15] A. S. Mashhour, M. E. Abd El-Monsef and S. N. El-Deeb, On precontinuous and weak precontinuous mappings, Proc. Math. Phys. Soc. Egypt 53 (1982), [16] A.S. Mashhour, M.E. Abd El-Monsef, I.A. Hasanein, and T. Noiri, Strongly compact spaces, Delta J. Sci. 8 (1) (1984), [17] T. Noiri, Unified characterizations for modifications of R 0 and R 1 topological spaces, Rend. Circ. Mat. Palermo (2) 55 (2006), [18] M. S. Sarsak, On semicompact sets and associated properties, International Journal of Mathematics and Mathematical Sciences 2009 (2009), 1 8.

9 On µ-compact sets in µ-spaces 57 [19] M. S. Sarsak, Weak separation axioms in generalized topological spaces, Acta Math. Hungar. 131 (1-2) (2011), Department of Mathematics, Faculty of Science, The Hashemite University, P.O. Box , Zarqa 13115, Jordan address: Received June 19, 2012 and revised October 9, 2012