More on μ-semi-Lindelöf sets in μ-spaces


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                  </jats:inline-formula>-<jats:italic>spaces</jats:italic>, Questions Answers Gen. Topology <jats:bold>31</jats:bold> (2013), no. 1, 49–57] introduced and studied the class of <jats:inline-formula>
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                  </jats:inline-formula>-spaces. Mustafa [<jats:inline-formula>
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                  </jats:inline-formula>-<jats:italic>semi compactness and</jats:italic>
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                  </jats:inline-formula>-<jats:italic>semi Lindelöfness in generalized topological spaces</jats:italic>, Int. J. Pure Appl. Math. <jats:bold>78</jats:bold> (2012), no. 4, 535–541] introduced and studied the class of <jats:inline-formula>
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                           <m:mi>μ</m:mi>
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                  </jats:inline-formula>-semi-Lindelöf sets in generalized topological spaces (GTSs); the primary purpose of this paper is to investigate more properties and mapping properties of <jats:inline-formula>
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                           <m:mi>μ</m:mi>
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                  </jats:inline-formula>-semi-Lindelöf sets in <jats:inline-formula>
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                  </jats:inline-formula>-spaces. The class of <jats:inline-formula>
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                  </jats:inline-formula>-spaces is a proper subclass of the class of <jats:inline-formula>
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                  </jats:inline-formula>-spaces. It is shown that the property of being <jats:inline-formula>
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                           <m:mi>μ</m:mi>
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                  </jats:inline-formula>-semi-Lindelöf is not monotonic, that is, if <jats:inline-formula>
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                  </jats:inline-formula>-semi-Lindelöf, then <jats:inline-formula>
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                           <m:mi>μ</m:mi>
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                  </jats:inline-formula>-semi-Lindelöf. We also introduce and study a new type of generalized open sets in GTSs, called <jats:inline-formula>
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                  </jats:inline-formula>-semi-open sets, and investigate them to obtain new properties and characterizations of <jats:inline-formula>
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                           <m:mi>μ</m:mi>
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                  </jats:inline-formula>-semi-Lindelöf sets in <jats:inline-formula>
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                  </jats:inline-formula>-spaces.</jats:p>


Introduction and preliminaries
A topological space X is said to be Lindelöf [1] if every open cover of X has a countable subcover. Lindelof spaces play a vital role in the theory of general topology as a natural generalization of compact spaces. Since then, related concepts had been of special interests to several mathematicians. For instance, nearly Lindelöf spaces [2], strongly Lindelöf spaces [3], almost Lindelöf spaces [4], semi-Lindelöf spaces [5], and rc-Lindelöf spaces [6]. For more related studies, one can see [7][8][9][10][11][12].
The study of generalized topological spaces (GTSs) was first initiated by Csàszàr [13], which in role, motivated a lot of authors to generalize the topological notions including covering properties to the generalized topological surroundings. For instance, Sarsak [14,15], Mustafa [16], Arar [17,18], Abuage et al. [19], and Roy [20] studied several analogous notions via GTSs. The primary purpose of this paper is to continue the study of μ-semi-Lindelöf sets in μ-spaces introduced by Mustafa in [16]. In Section 2, we study the relationship between μ-semi-Lindelöf sets and μ-Lindelöf sets as we also obtain new characterizations of μ-semi-Lindelöf sets; in Section 3, we mainly deduce that if ( ) X μ , is a μ-space and ⊂ ⊂ A B X, where A is μ B -semi-Lindelöf, then A need not be μ-semi-Lindelöf, as we also give corrections to some results in [16]; in Section 4, we introduce and study a new type of generalized open sets in generalized topological spaces, called ω μ -semi-open sets, and investigate them in Sections 5 and 6 to obtain more characterizations of μ-semi-Lindelöf spaces and to obtain several properties of μ-semi-Lindelöf sets related to sums, products, images, and preimages.
A generalized topology (GT) [13] 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 μ x such that ∈ ⊂ x U A x , or equivalently, A is the union of μ-open sets. The pair ( ) X μ , will be called GTS. A space X or ( ) X μ , will always mean a GTS. A space ( ) X μ , is called a μ-space [21] if ∈ X μ. ( ) X μ , is called a quasi-topological space [22] 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 [23], ( ) c A μ , is the intersection of all μ-closed sets containing A and the μ-interior of A [23], ( ) i A μ , is the union of all μ-open sets contained in A. It was pointed out in [23] that each of the operators c μ and i μ are monotonic [24], i.e., if ⊂ ⊂ A B X, then ( ) , is a topological space and ⊂ A X, then A and A Int will stand, respectively, for the closure of A in X and the interior of A in X. A , is a μ-space. For the concepts and terminology not defined here, the reader is referred to [29]. In concluding this section, we recall the following definitions and facts for their importance in the material of our paper.
, be a space. Then each of the families σ, π, and α is a GT.
, is a μ-space, then it is easy to see that is called μ-Lindelöf if any cover of A by μ-open subsets of X has a countable subcover.
, is called μ-Lindelöf if any cover of X by μ-open sets has a countable subcover.

μ-semi-Lindelöf sets
This section is mainly devoted to investigate more properties of μ-semi-Lindelöf sets in μ-spaces.
, is called semi-compact relative to X [32] (resp. semi-Lindelöf in X [33]) if any cover of A by semi-open subsets of X has a finite (resp. countable) subcover. We will usually use the term "in X" to mean "relative to X".
In particular, is semi Lindelof , is semiLindelöf .
, is μ-Lindelöf. However, the converse need not be true even for topological spaces as the following example tells.
Example 2.6. Let X be an uncountable set and consider the topology , and thus, x is a cover of X by τ-semi-open sets, but has no countable subcover, so, ( ) X τ , is not semi-Lindelöf. However, ( ) X τ , is Lindelöf (even compact).
The proofs of the following three propositions are either straightforward or from [35] and thus omitted.
, is μ-semi-Lindelöf if and only if for every family closed sets having the property that for every countable subfamily i of , ( Definition 2.9. [35] A filter base on a nonempty set X is called a strong filter base on X if whenever i is a countable subcollection of , there exists ∈ F such that ⊂ ⋂ F i .

Definition 2.10. [35]
A strong filter base on a nonempty set X is called a maximal strong filter base on X if whenever is a strong filter base on X with ⊂ , then = .
Proposition 2.11. [35] Every strong filter base on a nonempty set X is contained in a maximal strong filter base on X.
is a maximal filter base (maximal strong filter base), then μ σ -converges to x if and only if μ σ -accumulates at x.
, , the following are equivalent: (ii) Every maximal strong filter base on X, each of whose members meets A, μ σ -converges to some point of A; (iii) Every strong filter base on X, each of whose members meets A, μ σ -accumulates at some point of A.
Let be a maximal strong filter base on X, each of whose members meets A, such that does not μ σ -converge to any point of A. Since is maximal, it follows from Proposition 2.13(ii) that does not μ σ -accumulate at any point of A. Thus, for each ∈ iii : Let be a strong filter base on X, each of whose members meets A.
: is a strong filter base on X. Thus by Proposition 2.11, is contained in a maximal strong filter base on X, each of whose members meets A. By (ii), μ σ -converges to some point x of A, thus by Proposition 2.13(i), is a strong filter base on X, each of whose members meets A. Thus by (iii), μ σ -accumulates at some point , , the following are equivalent: (ii) Every maximal strong filter base on X μ σ -converges to some point of X; (iii) Every strong filter base on X μ σ -accumulates at some point of X.

Subspaces
This section is mainly devoted to discuss some statements in [16] and [33] concerning subspaces.
In the proof of Proposition 3.5 [16, Theorem 2.7], the author used without proof the following: For a matter of convenience and importance, we will prove this fact, to proceed, we introduce the following two lemmas. Proof.
(i) Let A be μ B -semi-open. Then it follows from Lemmas 3.6 and 3.7 that there exists a μ-open set U such that , and X S \ is μ-semi-closed. (ii) This is clear from Definition 3.1 and Remark 3.
. Thus by Lemma 3.6, A is μ B -semi-open. □ Remark 3.9. From Proposition 3.8(i), we observe that the condition "preopen" is not essential for the necessity of Proposition 3.3.
Corollary 3.11. Let A be a subset of a topological space X. If A is semi-compact (resp. semi-Lindelöf) in X, then A is a semi-compact (resp. semi-Lindelöf) subspace.
Remark 3.12. From Corollary 3.11, we observe that the condition "preopen" of Proposition 3.4 is not essential. If A is a semi-compact (resp. semi-Lindelöf) subspace, then A is semi-compact (resp. semi-Lindelöf) in X.
The following example shows that the condition "preopen" in Proposition 3.13 is essential.
Example 3.14. Consider the space of Example 2.6, that is, the space ( ) X τ , , where X is an uncountable set, and x is a cover of A by τ-semi-open sets and has no countable subcover of A. Hence, A is not semi-compact (semi-Lindelöf) in X. Now, the subspace topology We point out here to an error in the proof of the sufficiency of [ We observe that since the condition "preopen" in Proposition 3.13 is essential, the sufficiency of Proof. We will see the case of μ-semi-Lindelöf, the other case is similar.
The following corollary includes a correction of the necessity of [16, Corollary 2.10].

ω μ -semi-open sets
, be a space and A be a subset of X. , be a space and A be an ω μ -semi-closed subset of X. Then ⊂ ∪ A B C for some μ-semi-closed subset B of X and some countable subset C of X.
The following easy example shows that the converse of Proposition 4.4(iii) need not be correct in general even for topological spaces.
is a disjoint family of sets. The collection μ of subsets of ⋃X α is defined as follows: : Λ α is a disjoint family of sets, and let μ be as in Definition 5.6. Then μ is a GT on ⋃X α . The GTS ( ) ⋃X μ , α will be called the generalized topological sum of ∈ X α , Λ α and will be denoted by ⊕X α .
, be a μ-space. Then the countable union of subsets of X, each of which μ-semi-Lindelöf, is μ-semi-Lindelöf.
Then ⊕X α is μ-semi-Lindelöf if and only if X α is μ-semi-Lindelöf for each ∈ α Λ and Λ is countable.

Mapping properties
This section is mainly devoted to study several mapping properties of μ-semi-Lindelöf sets in μ-spaces.
Proof. Observe first by Remark 6.9 that since ( ) X μ , is a μ-space and ( ) Y κ , is a κ-space, ( ) × X Y λ , is a λspace. The result follows from Corollary 6.3 and Lemma 6.10. □