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Some Types of µ-semi-compactness Via hereditary classes Abdo Qahis Department of Mathematics, Faculty of Science and Arts Nagran university, Saudi Arabia E-mail: [email protected] Abstract The aim of this paper is to introduce and study the notions of µ-semicompactness and strong µ-semi-compactness in generalized topological spaces with respect to a hereditary class, called µH-semi-compactness and S-µH-semicompactness, respectively. Also several of their properties are presented. Finally some effects of various kinds of functions on them are studied.. Keywords: Generalized topology, hereditary class, µ-semi-covering, µHsemi-compact, S-µH-semi-compact. 2000 Mathematics Subject Classification. 54A05, 54A08, 54D10. 1 Preliminaries and Introduction This work is developed around the concept of µ-semi-compactness with respect a hereditary class which was introduced by Jamal M. Mustafa in [8]. In this research, we use the notions of generalized topology and hereditary class introduced by Császár in [1] and [2], respectively, in order to define and characterize the µH-semi-compactness and S-µH-compactness spaces. Also some properties of these spaces are obtained and the behavior of these spaces under certain kinds of functions also is investigated. The strategy of using generalized topologies and hereditary classes to extend classical topological concepts have been used by many authors such as [2], [9], [11], [13]. Let X be a non-empty set and 2X denote the power set of X. We call a class µ ⊆ 2X a generalized topology [1] (briefly, GT) if ϕ ∈ µ and arbitrary union of elements of µ belongs to µ. A set X with a GT is called a generalized topological space (briefly, GTS) and is denoted by (X, µ). For a GTS (X, µ), the elements of µ are called µ-open 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 54 Abdo Qahis . 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], [3]). A subset A of X is said to be µ-semi-compact if for every µ-semi-open cover U of A there exists a finite subfamily U ′ that also covers A. X is said to be µ-semi-compact if X is µ-semi-compact as a subset[8]. A nonempty family H of subsets of X is called a hereditary class [2] if A ∈ H and B ⊂ A imply that B ∈ H. Given a generalized topological space (X, µ) with a hereditary class H, for a subset A of X, the generalized local function of A with respect to H and µ [2] is defined as follows: A∗ = {x ∈ X : U ∩ A ∈ / H for all U ∈ µx }, where µx = {U : x ∈ U and U ∈ µ}. And for a subset A of X, is defined: c∗µ (A) = A ∪ A∗ . The family µ∗ = {A ⊂ X : X \ A = c∗µ (X \ A)} is a GT on X. The elements of µ∗ are called µ∗ -open and the complement of a µ∗ -open set is called µ∗ -closed set. It is clear that a subset A is µ∗ -closed if and only if A∗ ⊂ A. If the hereditary class H satisfies the additional condition: if A, B ∈ H implies A ∪ B ∈ H, then H is called an ideal on X [10]. We call (X, µ, H) a hereditary generalized topological space and briefly we denote it by HGTS. Definition 1.1 [3] Let (X, µ) generalized topological spaces. Then, A ⊆ X is said to be µ-semi-open set if and only if A ⊂ cµ iµ (A) i.e., if there exist a µ-open set V ∈ µ, such that V ⊆ A ⊆ cµ (V ). The complement of a µ-semi-open set is said to be µ-semi-closed. The collection of all µ-semi-open set ( resp. µ-semi-closed set ) sets in X is denoted by SOµ (X)(resp : SCµ (X)). The µ-semi-closure of A in (X, µ) is the intersection of all µ-semi-closed set containing A and is denoted by scµ (A). Definition 1.2 [11] Let (X, µ, H) hereditary generalized topological spaces and A ⊂ X, Aµ (H, µ) = {x ∈ X : U ∩ A ̸∈ H f or every U ∈ SOµ (X, x)} is called the generalized semi-local function of A with respect to H and µ, where SOµ (X, x) = {U ∈ SOµ (X) : x ∈ U }. We simply write Aµ instead of Aµ (H, µ) is this case there is no ambiguity. It is given in [14], µ∗g (H) is a generalized topology on X, generated by the subbases {U − H : U ∈ SOµ (X) and H ∈ H} or equivalently µ∗g (H) = {U ⊂ X : ∗g c∗g µ (X − U ) = X − U }. The closure operator cµ for a generalized topology ∗g µ∗g (H) is defined as follows: for A ⊆ X, c∗g µ (A) = A ∪ Aµ . iµ denotes the ∗g ∗ ∗g interior of the set µ (H). It is know that µ ⊆ µ (H) ⊆ µ (H). Definition 1.3 [14] A subset of A of an hereditary generalized topological spaces (X, µ, H) is said to be: 1. µ-semi-∗-perfect if A = Aµ ; 2. µ-semi dense in-itself if A ⊆ Aµ ; 55 Some Types of µ-semi-compactness Via hereditary Classes 3. µ-semi-∗-closed if Aµ ⊆ A. Definition 1.4 Let (X, µ) be a GTS. Then a subset A of X is said to be µ-dense [6] if cµ (A) = X. The space (X, µ) is said to be µ-submaximal [7] if every µ-dense subset is µ-open in X. Definition 1.5 [4] A GTS (X, µ) is said to be µ-extremally disconnected if the µ-closure of every µ-open set is µ-open Definition 1.6 [1] Let (X, µ) and (Y, ν) be two GTS’s, then a function f : (X, µ) → (Y, ν) is said to be (µ, ν)-continuous if U ∈ ν implies f −1 (U ) ∈ µ. Definition 1.7 [8] Let (X, µ) and (Y, ν) )be GTS’s. A function f : (X, µ) → (Y, ν) is said to be µ-pre-semi-open if it maps µ-semi-open subsets of X onto µ-semi-open subsets of Y . 2 µH-Semi-Compactness Spaces Definition 2.1 A subset A of a HGTS (X, µ, H) is said to be µH-semicompact if for every µ-semi-open cover {Vα }α∈Λ of A there exists a finite subset ∪ Λ0 of Λ such that A \ Vα ∈ H. The HGTS (X, µ, H) is said to be µHα∈Λ0 semi-compact if X is µH-semi-compact as a subset. Remark 2.2 1. It is clear that (X, µ) is µ-semi-compact if and only if (X, µ, {ϕ}) is µ{ϕ}-semi-compact. 2. If (X, µ) is a µ-semi-compact then (X, µ, H) is µH-semi-compact. The converse is not true as shown by the following example. Example 2.3 Let X = [0, 1), µ = {ϕ} ∪ {[0, a) : 0 < a ≤ 1} and H = {A : A ⊆ [0, 1)}, then: 1. (X, µ, H) is µH-semi-compact because for any µ-semi-open cover {Vα }α∈Λ ∪ we have X \ Vα ∈ H; α∈Λ 2. (X, µ) is not µ-semi-compact. In fact if we take Vn = [0, 1 − n1 ) for all ∪ integer number n ≥ 2, then X = [0, 1 − n1 ) but if k ∈ Z + where Z + n≥2 is the set of positive integers, and n1 , n2 , ..., nk are integer numbers with ni ≥ 2, for all i ∈ {1, 2, ..., k}, then X ̸= k [ ∪ i=1 0, 1 − 1 ni ) . It is clear that the family SOµ (X) of all µ-semi-open subsets of X forms a generalized topology on X which is finer than µ. 56 Abdo Qahis . Proposition 2.4 Let (X, µ, H) be a HGTS and B be a base for the generalized topology SOµ (X), then (X, µ, H) is µH-semi-compact if and only if for ∪ all family {Vα }α∈Λ of µ-semi-open sets in B, if X = Vα then there exists Λ0 ⊆ Λ, finite, with X \ ∪ α∈Λ0 α∈Λ Vα ∈ H. Proof. (=⇒ ) Let {Vα }α∈Λ be a family of non-empty µ-semi-open sub∪ sets of X such that X = Vα . Since B be a base for SOµ (X), then α∈Λ SOµ (X) = {∪B : B ⊆ B}. So, for each α ∈ Λ there exists a family ∪ ∪ {Bαβ : β ∈ Λα } ⊆ B such that Vα = Bαβ . Given that X = Vα and X = such that ∪ ∪ α∈Λ β∈Λα Bβα is µH-semi-compact there exist Bα1 β1 , Bα2 β2 , ..., Bαk βk α∈Λ β∈Λα k ∪ X \ Bαi βi i=1 ∈ H. But X \ k ∪ i=1 Vk ⊆ X \ k ∪ i=1 Bαi βi and so X \ k ∪ i=1 Vk ∈ H. (⇐=) It is obvious. Given a generalized topological space (X, µ), we denote by Hf the hereditary class of finite subsets of X. The following proposition is obvious and thus the proof are omitted. Proposition 2.5 For a HGTS (X, µ, H) the following are equivalent: 1. (X, µ) is µ-semi-compact; 2. (X, µ, Hf ) is µHf -semi-compact; 3. (X, µ, {ϕ}) is µ{ϕ}-semi-compact. Theorem 2.6 A HGTS (X, µ, H) is µH-semi-compact if and only if for ∩ Gα = ϕ, there any family {Gα }α∈Λ of µ-semi-closed subsets of X such that exists a finite subset Λ0 of Λ with ∩ α∈Λ0 ∩ α∈Λ α∈Λ Gα ∈ H. Proof. Let {Gα }α∈Λ be a family of µ-semi-closed subsets of X such that Gα = ϕ. Then {X \ Gα }α∈Λ is a µ-semi-open cover of X. By hypothesis, there exists a finite subset Λ0 ⊂ Λ such that X \ implies that ∩ α∈Λ0 Gα ∈ H. ∪ α∈Λ0 (X\Gα ) ∈ H. This Conversely, Let {Vα }α∈Λ be any µ-semi-pen cover of X. Then {X \ Vα }α∈Λ is a ∩ family of µ-semi-closed subsets of X such that X \ Vα = ϕ. By hypothesis, there exists a finite subset Λ0 ⊂ Λ such that we obtain that, X \ ∪ α∈Λ0 α∈Λ ∩ α∈Λ0 X\Vα ∈ H. Consequently, Vα ∈ H and therefore (X, µ, H) is µH-semi-compact. It is clear that the intersection of any two hereditary classes on a non-empty set X is a heredity class. 57 Some Types of µ-semi-compactness Via hereditary Classes Corollary 2.7 If (X, µ, H1 ∩H2 ) is µ(H1 ∩H2 )-semi-compact, then (X, µ, H1 ) is µH1 -semi-compact and (X, µ, H2 ) is µH2 -semi-compact. Definition 2.8 [5] A HGTS (X, µ, H) is said to be µH-compact if for every µ-open cover {Vα }α∈Λ of X there exists a finite subset Λ0 of Λ such that X \ ∪ Vα ∈ H. α∈Λ0 Proposition 2.9 Every µH-semi-compact space is µH-compact. Proof. The proof is obvious since every µ-open set is µ-semi-open. Proposition 2.10 If (X, µ) is µ-submaximal and extremally disconnected, then µH-compact and µH-semi-compact are equivalent. Proof. The proof comes immediately form the fact: In the extremally disconnected µ-submaximal space µ = SOµ (X). Next we study the behavior of some types of subspaces of a µH-semicompact space relative to X. Theorem 2.11 If Ai , i = 1, 2 are µH-semi-compact subsets of a HGTS (X, µ, H), and H is an ideal then A1 ∪ A2 is µH-semi-compact. Proof. Let {Vα }α∈Λ be a family of µ-semi-open subsets of X such that ∪ Vα . Then {Vα }α∈Λ is a µ-semi-open cover of Ai for i = 1, 2. A1 ∪ A2 ⊆ α∈Λ Since A1 and A2 are µH-semi-compact, then there exists finite subsets Λ0 and ∪ ∪ Λ1 of Λ with A1 \ Vα ∈ H and A2 \ Vα ∈ H. This implies that A1 \ ∪ α∈Λ0 ∪Λ1 that ( A1 \ α∈Λ0 Vα ∈ H and A2 \ ∪ α∈Λ0 ∪Λ1 ) ( Vα ∪ A2 \ α∈Λ1 ∪ α∈Λ0 ∪Λ1 Vα ∈ H and since H is an ideal we have ) ∪ α∈Λ0 ∪Λ1 Vα ∈ H. Thus A1 ∪ A2 \ So A1 ∪ A2 is µH-semi-compact. ∪ α∈Λ0 ∪Λ1 Vα ∈ H. The following example shows that the previous theorem does not hold when H is just a hereditary class, not an ideal. Example 2.12 Let X = (0, 1), µ the usual topology, H = {A ⊂ X : A ⊂ (0, 21 ) or A ⊂ ( 12 , 1)} and if A = (0, 12 ) and B = ( 12 , 1), then: 1. It is clear that A = (0, 12 ) and B = ( 12 , 1) are µH-semi-compact subsets. 2. A ∪ B is not µH-semi-compact if {( n1 , 1 − n1 ) : n ∈ Z + } is a cover of µ-semi-open subsets of X where Z + is the set of positive integers, but if we choose a finite set n1 , ..., nk and we take N = max{n1 , ..., nk }, follows that A ∪ B \ k ( ∪ n=1 1 , ni 1− 1 ni ) =A∪B\ ( 1 , N 1− 1 N ) ∈ / H. 58 Abdo Qahis . A GTS (X, µ) is said to be µ-semi-Hausdroff [12] for each pair of distinct points x and y in X, there exist µ-semi-open sets Ux and Vy containing x and y, respectively, such that U ∩ V = ϕ. / Aµ if and only if (V \ H) ∩ A = ϕ, where V ∈ SOµ (X, x) Lemma 2.13 x ∈ and H ∈ H. Proof. Let x ∈ / Aµ . Then there exist Ux ∈ SOµ (X, x) such that A ∩ V = H ∈ H. It follows that (V \ H) ∩ A = ϕ. Conversely, suppose that (V \ H) ∩ A = ϕ for some V ∈ SOµ (X, x) and H ∈ H. Then V ∩ (X \ H) ∩ A = (A ∩ V ) ∩ (X \ H) = ϕ. This implies that (A ∩ V ) ⊂ H ∈ H. Hence, x ∈ / Aµ . Theorem 2.14 Let (X, µ, H) be a HGTS such that (X, µ) is a µ-semiHausdorff. If A is µH-semi-compact subset of X, then A is µ-semi-∗-closed. Proof. Suppose that A be a µH-semi-compact subset of a µ-semi-Hausdroff HGTS (X, µ, H). Let x ∈ / A then x ∈ X \ A. For each y ∈ A, there exist two µ-semi-open sets Uy and Vy containing x and y, respectively, such that Uy ∩ Vy = ϕ. Note that x ∈ / scµ (Vy ). Then {Vy }y∈A is a µ-semi-open cover of A relative to X. Since A is a µH- semi-compact, there exists a finite sub∪ set Λ0 of A such that A \ Vy ∈ H. Now x ∈ / scµ (Vy ) for each y ∈ A ∪ implies x ∈ / H = A \ scµ ( y∈Λ0 ∪ y∈Λ0 Λ0 scµ (Vy ) = scµ ( Vy ) ⊂ A \ ∪ y∈Λ0 ∪ y∈Λ0 Vy ). Let U = X \ scµ ( ∪ Vy ) and let y∈Λ0 Vy = H1 , where H1 ∈ H. Since U ∈ SOµ (X, x) and H ∈ H, U \ H ∈ µ∗g (H) containing x and (U \ H) ∩ A = ∅. By Lemma 2.13, x ∈ / Aµ . Hence Aµ ⊂ A, so A is µ-semi-∗-closed. Let (X, µ, H) be a HGTS and let A ⊆ X, A ̸= ϕ. We denote by HA the collection {H ∩ A : H ∈ H} and by (A, µA ) the subspace of (X, µ) on A. It is clear that the collection µA is a generalized topology on A and the collection HA is a hereditary class of subsets in A. Then we have the following theorem. Theorem 2.15 Let (X, µ, H) be a µH-semi-compact HGTS and A be a µsemi-closed subset of X then (A, µA , HA ) is semi-compact with respect to the hereditary class HA . Proof. Let {Vα }α∈Λ be a µA -semi-open cover of A where Vα = Uα ∩ A such that Uα ∈ SOµ (X). Then {Uα }α∈Λ ∪ (X \ A) is a µ-semi-covering of X and hence there exists a finite subset Λ0 of Λ such that X \ [∪{Uα : α ∈ Λ0 } ∪ (X \ A)] ∈ H. There exists H ∈ H with X \ [∪{Uλ : λ ∈ Λ0 } ∪ (X \ A)] = H. Since A∩H = A∩(X \[∪{Uα : α ∈ Λ0 }∪(X \A)]) = A∩(X \∪{Uα : α ∈ Λ0 })∩A = A ∩ (X \ ∪{Uα : α ∈ Λ0 }) = A \ ∪{Uα : α ∈ Λ0 } = A \ (A ∩ [∪{Uα : α ∈ Λ0 }]) = 59 Some Types of µ-semi-compactness Via hereditary Classes A \ ∪{Uα ∩ A : α ∈ Λ0 }. Then, we have A \ ∪{Uα ∩ A : α ∈ Λ0 } = A ∩ H ∈ HA . This shows that A is a µHA - semi-compact. The well known result that a µ-semi-closed subspace of a µ-semi-compact space is µ-semi-compact is a special case by taking H = {ϕ}. Definition 2.16 Let (X, µ) be a GTS. Then a subset A of X is called a µsemi-generalized closed if scµ (A) ⊆ U whenever A ⊆ U where U is µ-semi-open in X. Theorem 2.17 If (X, µ, H) is µH-semi-compact and A ⊂ X is µ-semi generalized-closed then A is µH-semi-compact. Proof. Let {Vα }α∈Λ be any cover of A by µ-semi-open sets in X. Since A is µ-semi generalized-closed, scµ (A) ⊂ ∪Vα . Then {Vα }α∈Λ ∪ (X \ scµ (A)) is a µ-semi-open cover of X and hence there exists a finite subset Λ0 of Λ such that [ ] X\ ∪ α∈Λ0 Vα ∪ (X \ scµ (A) ∈ H. Now, we have X \ [∪{Vα : α ∈ Λ0 } ∪ (X \ scµ (A))] = [X \ ∪{Vα : α ∈ Λ0 }] ∩ scµ (A) ⊃ [X \ ∪{Vα : α ∈ Λ0 }] ∩ A = A \ [∪{Vα : α ∈ Λ0 }]. This implies that A \ [∪{Vα : α ∈ Λ0 }] ∈ H, and so A is µH-semi-compact relative to X. Now we study the behavior of µH -semi-compactness under certain types of functions. Theorem 2.18 Let (X, µ) and (Y, ν) be GTSs and let f : (X, SOµ (X)) → (Y, SOν (Y )) be a (SOµ (X), SOν (Y ))-continuous function. If (X, µ, H) is µHsemi-compact, and if G = {B ⊆ Y : f −1 (B) ∈ H} then: 1. G is a hereditary class on Y . 2. (Y, ν, G) is µH-semi-compact. Proof. (1) Suppose that A ⊆ B ⊆ Y and B ∈ G . Since f −1 (A) ⊆ f (B) ∈ H, then f −1 (A) ∈ H, and so A ∈ G. ∪ ∪ −1 (2) Let {Vα }α∈Λ ⊂ SOν (Y ) such that Y = Vα . Since X = f (Vα ) = −1 ( f −1 ∪ ) Vα α∈Λ Λ0 of Λ with f α∈Λ α∈Λ and (X, µ, H) is µH-semi-compact, there exists a finite subset ( −1 Y \ ∪ α∈Λ0 ) Vα = X \ ∪ α∈Λ0 f −1 (Vα ) ∈ H. Thus Y \ ∪ α∈Λ0 Vα ∈ G. The following lemma is very useful in studying the preservation of µHcompact by certain classes of functions 60 Abdo Qahis . Lemma 2.19 [5] Let f : (X, µ) → (Y, ν) be a function. If H is a hereditary class on X, then f (H) = {f (H) : H ∈ H} is a hereditary class on Y . Theorem 2.20 Let (X, µ) and (Y, ν) be GTSs and let f : (X, SOµ (X)) → (Y, SOν (Y )) be a surjection (SOµ (X), SOν (Y )-continuous function. If (X, µ, H) is µH-semi-compact, then (Y, ν, f (H)) is νf (H)-semi-compact. Proof. Let {Vα }α∈Λ ⊂ SOν (Y ) such that Y = ∪ α∈Λ Vα . Then {f −1 (Vα )}α∈Λ is a µ-semi-open ( cover of X ) and hence there exists a finite subset Λ0 of Λ ∪ ∪ ∪ such that f −1 Y \ Vα = X \ f −1 (Vα ) ∈ H. Thus Y \ Vα = f (f −1 (Y \ ∪ α∈Λ0 α∈Λ0 Vα )) ∈ f (H). α∈Λ0 α∈Λ0 Corollary 2.21 Let (X, µ) and (Y, ν) be GTSs and let f : (X, SOµ (X)) → (Y, SOν (Y )) be a bijective µ-semi-open function. If (Y, ν, G) is νG-semi-compact, then (X, µ, f −1 (G)) is µf −1 (G)-semi-compact Proof. Since f : (X, SOµ (X)) → (Y, SOν (Y )) is a bijective µ-semi-open function, f −1 : (Y, SOν (Y )) → (X, SOµ (X)) is a surjection (SOµ (X), SOν (Y ))continuous function. Since (Y, ν, G) is νG-semi-compact, by Theorem 2.20 we obtain that (X, µ, f −1 (G)) is µf −1 (G)-semi-compact. 3 S-µH-Semi-Compactness Spaces Definition 3.1 If (X, µ, H) is a HGTS and A ⊆ X, A is said to be strong µH-semi-compact (briefly S-µH-semi-compact) relative to X if for every fam∪ ily {Vα }α∈Λ of µ-semi-open subsets of X with A \ Vα ∈ H there exists a finite subset Λ0 of Λ, such that A ⊆ ∪ α∈Λ α∈Λ0 Vα . The HGTS (X, µ, H) is said to be S-µH-semi-compact if X is S-µH-semi-compact. Clearly, the following diagram follows immediately from the definitions and facts. ↗ S − µH − semi − compact ↘ µH − semi − compact ↑ µ − semi − compact Remark 3.2 We note that if (X, µ, H) is a HGTS and (X, µ∗ , H) is a SµH- semi-compact, then (X, µ, H) is S-µH-semi-compact, and that (X, µ, H) is S-µH-semi-compact if and only if for any family {Gα }α∈Λ of µ-semi-closed sub∩ ∩ sets of X, if Gα ∈ H then there exists Λ0 ⊆ Λ, finite, such that Gα = ϕ. α∈Λ α∈Λ0 61 Some Types of µ-semi-compactness Via hereditary Classes Remark 3.3 1. It is clear that the GT (X, µ) is µ-semi-compact if and only if (X, µ, {ϕ}) is S-µ{ϕ}-semi-compact. 2. If (X, µ, H) is a S-µH-semi-compact then (X, µ, H) is µH-semi-compact, and (X, µ) is µ-semi-compact. Definition 3.4 A subset A of a HGTS (X, µ, H) is said to be SOµ Hg -closed if for every U ∈ SOµ with A \ U ∈ H then scµ (A) ⊆ U . Theorem 3.5 Every SOµ Hg -closed subset of a S-µH-semi-compact space is S-µH-semi-compact. Proof. Let A be any SOµ Hg -closed subset of (X, µ, H) and {Vα }α∈Λ be ∪ a family of µ-semi-open subsets of X such that A \ Vα ∈ H. Since A is SOµ Hg -closed, scµ (A) ⊆ ∪ α∈Λ α∈Λ Vα . Then (X \ scµ (A)) ∪ ( [ semi-open cover of X and so X \ X \ cµ (A) ∪ ( ] ∪ α∈Λ ∪ Vα ) is a µ- α∈Λ Vα ) = ϕ ∈ H. Given that X is S-µH-semi-compact there exists s finite subset Λ0 of Λ such that ∪ ∪ X = (X \ scµ (A) ∪ Vα ). Then A = A ∩ [(X \ scµ (A)) ∪ Vα ] = A∩ ∪ α∈Λ0 Vα ⊆ ∪ α∈Λ0 α∈Λ0 Vα . α∈Λ0 Theorem 3.6 If A and B are S-µH-semi-compact subsets of a HGTS (X, µ, H), then A ∪ B is S-µH-semi-compact. Proof. Let {Vα }α∈Λ be a family of µ-semi-open subsets of X such that ∪ ∪ ∪ ∪ Vα ⊆ Vα and B \ Vα ⊆ A ∪ B \ Vα ∈ H. Since, A \ A∪B\ A∪B \ α∈Λ ∪ α∈Λ Vα then A \ ∪ α∈Λ subsets Λ0 and Λ1 of Λ such that A ⊆ that A ⊆ ∪ α∈Λ0 ∪Λ1 ∪ α∈Λ Vα ∈ H and B \ Vα and B ⊆ A ∪ B is S-µH-semi-compact. ∪ α∈Λ0 ∪Λ1 ∪ α∈Λ α∈Λ0 α∈Λ α∈Λ Vα ∈ H and so there exist finite Vα and B ⊆ ∪ Vα . This implies α∈Λ1 Vα and so A ∪ B ⊆ ∪ α∈Λ0 ∪Λ1 Vα . Hence Theorem 3.7 Let (X, µ, H) be a HGTS, A ⊆ B ⊆ X and B ⊆ scµ (A). If A is µ-semi-generalized closed and S-µH-semi-compact, then B is S-µH-semicompact. Proof. Let {Vα }α∈Λ be a family of µ-semi-open subsets of X such that ∪ ∪ B\ Vα ∈ H. Since, A \ Vα ∈ H and A is S-µH-semi-compact, there α∈Λ α∈Λ exists a finite subset Λ0 of Λ such that A ⊆ generalized closed, scµ (A) ⊆ ∪ α∈Λ0 ∪ α∈Λ0 Vα . Since A is µ-semi- Vα and this implies B ⊆ ∪ Vα . α∈Λ0 Now we study the behavior of S-µH-semi-compactness under certain types of functions. 62 Abdo Qahis . Theorem 3.8 Let (X, µ) and (Y, ν) be GTSs and let f : (X, SOµ (X)) → (Y, SOν (Y )) be a (SOµ (X), SOν (Y ))-continuous surjective function. If (X, µ, H) is S- µH-semi-compact and G = {B ⊆ Y : f −1 (B) ∈ H} then (Y, ν, G) is SµH-semi-compact. Proof. Let {Vα }α∈Λ be a family of µ-semi-open subsets of Y such that ( ) Y\ ∪ α∈Λ Vα ∈ G. Since X \ ∪ α∈Λ f −1 (Vα ) = f −1 Y \ ∪ α∈Λ Vα ∈ H and (X, µ, H) is S-µH-semi-compact, there exists a finite subset Λ0 of Λ, such that X = ∪ ∪ f −1 (Vα ). Given that f is surjective we have Y = Vα . α∈Λ0 α∈Λ0 Theorem 3.9 Let (X, µ) and (Y, ν) be GTSs and let f : (X, SOµ (X)) → (Y, SOν (Y )) be a bijective (SOµ (X), SOν (Y ))-continuous function. If (X, µ, H) is S-µH-semi-compact then (Y, ν, f (H)) is S-νf (H)-semi-compact. Proof. Let {Vα }α∈Λ be a family of µ-semi-open subsets of Y such that ∪ ∪ Vα = f (H). Then Vα ∈ f (H). There exists H ∈ H with Y \ Y \ α∈Λ H = f −1 (f (H)) = X \ ∪ α∈Λ α∈Λ f −1 (Vα ) ∈ H. Given that (X, µ, H) is S-µH-semi- compact, there exists a finite subset Λ0 of Λ such that X = f is surjective, Y = ∪ ∪ f −1 (Vα ). Since α∈Λ0 Vα . α∈Λ0 Corollary 3.10 If f : (X, µ) → (Y, ν) is a bijective and µ-pre-semi-open function and (Y, ν, G) is S-νG-semi-compact, then (X, µ, f −1 (G)) is S-µf −1 (G)semi-compact. Proof. Let {Vα }α∈Λ be a family of µ-semi-open subsets of X such that ∪ ∪ Vα = f −1 (G). Then Vα ∈ f −1 (G). There exists G ∈ G with X \ X\ Y \ α∈Λ ∪ α∈Λ α∈Λ f (Vα ) = f (f −1 (G)), and given that (Y, ν, G) is S-νG-semi- compact then there exists a finite subset Λ0 of Λ with Y = that X = ∪ ∪ f (Vα ). This implies α∈Λ0 Vα . α∈Λ0 ACKNOWLEDGEMENTS. The author would like to thank the referees for useful comments and suggestions. References [1] Császár Á., Generalized topology, generalized continuity, Acta Math. Hungar., 96 (2002), 351- 357. Some Types of µ-semi-compactness Via hereditary Classes 63 [2] Császár Á., Modification of generalized topologies via hereditary classes, Acta Math. Hungar., 115(1-2) (2007), 29 - 36. [3] Császár Á. , Generalized open sets in generalized topologies, Acta Math. Hungar., 106(2005), 53-66. [4] Császár Á.,Extremally disconneted genealized topologies, Annal Univ. Sci. Budapest., 47 (2004), 91-96. 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