Nearly Soft Menger Spaces

Al-shami, Tareq M.; Kočinac, Ljubiša D. R.

doi:https://doi.org/10.1155/2020/3807418

Journal of Mathematics

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Abstract Introduction Preliminaries Conclusion Data Availability Conflicts of Interest References Copyright Related Articles

Research Article | Open Access

Volume 2020 | Article ID 3807418 | https://doi.org/10.1155/2020/3807418

Nearly Soft Menger Spaces

Tareq M. Al-shami¹and Ljubiša D. R. Kočinac²

Academic Editor: Ali Jaballah

Received18 Mar 2020

Accepted05 May 2020

Published26 May 2020

Abstract

In this paper, we define a weak type of soft Menger spaces, namely, nearly soft Menger spaces. We give their complete description using soft -regular open covers and prove that they coincide with soft Menger spaces in the class of soft regular spaces. Also, we study the role of enriched and soft regular spaces in preserving nearly soft Mengerness between soft topological spaces and their parametric topological spaces. Finally, we establish some properties of nearly soft Menger spaces with respect to hereditary and topological properties and product spaces.

1. Introduction

The theory of selection principles is an area of mathematics that studies the possibility of generating a mathematical object of one kind from a sequence of objects of the same or different kind. The beginnings of this theory are going back to Borel, Hurewicz, Menger, Rothberger, and Sierpiński. This theory is one of the important tools of numerous subareas of mathematics such as set theory and general topology, Ramsey theory, game theory, hyperspaces, function spaces, uniform structures, cardinal invariants, and dimension theory.

In 1924, Menger [1] studied selection property under the name Menger basis property and Hurewicz [2], in 1925, reformulated it in the present form. Menger’s property is strictly between -compactness and Lindelöfness. The papers [3, 4] carried out a systematic study of selection principles in topology and then research in this field expanded immensely and attracted many researchers (see survey papers [5–7] and references therein). Some types of selection principles (so-called weak selection principles) have been formulated by applying the interior and closure operators in the definition of a selection property (see [8–21]) and the other types have been explored by replacing sequences of open covers by sequences of covers by some generalized open sets (see [22–24]). In this paper, we apply the ideas from selection principles theory to soft topological spaces. In fact, we are focused on a weaker form of the classical Menger covering property in the soft topology settings.

Soft sets were established by Molotdsov [25], in 1999, as a new technique to approach real-life problems which suffer vague and uncertainties. He investigated merits of soft sets compared with probability theory and fuzzy set theory. Many applications of soft sets have been recently given on the different areas such as decision-making problem, information theory, computer sciences, engineering, and medical sciences. In 2011, Shabir and Naz [26] employed soft sets that are defined over an initial universe set with a fixed set of parameters to introduce the concept of soft topological space. Then, researchers have studied several concepts of classical topological spaces through soft topological spaces. Soft compactness [27] and some weak variants of it [28–32] have been established and investigated. One of divergences between soft topological spaces and classical topological spaces was discussed in [33].

This paper is organized as follows. Section 2 provides basic definitions and results which are used in this paper. In Section 3, we establish some properties of soft semiopen and soft s-regular open sets which will help us to prove some results in the next sections. Section 4 introduces the concept of nearly soft Menger spaces and investigates its fundamental properties with the help of examples. Section 5 concludes the paper.

2. Preliminaries

In what follows, we recall the main definitions and results which shall be used throughout this work.

2.1. Selection Principles

Definition 1. (see [3]). Let and be given families of sets. Then, denotes the selection hypothesis: for each sequence of elements of , there is a sequence such that is a finite subset of for each and .
If denotes the collection of all open covers of a topological space , then spaces satisfying are said to have the Menger (covering) property (or is a Menger space).

Definition 2. (see [22]). A topological space is said to be nearly Menger if for each sequence of open covers of there is a sequence such that, for each , is a finite subset of and .
Evidently, every nearly compact space [34] (every open cover has a finite subcover such that the interiors of closures of members of cover the space) is nearly Menger, and every nearly Menger space is nearly Lindelöf [35] (every open cover has a countable subcover such that the interiors of closures of members of cover the space).

2.2. Soft Sets

In this paper, will be a nonempty set (called the initial universal set), its power set, a fixed set (called the set of parameters), and subsets of .

Definition 3. (see [25]). A pair is said to be a soft set over provided that is a mapping of a set of parameters into .
A soft set is identified with the set of ordered pairs . The collection of all soft sets defined over under a parameter set is denoted by .

Definition 4. (see [36]). A soft set over is said to be finite (resp. countable) if is finite (resp. countable) for each . Otherwise, it is infinite (resp. uncountable).

Definition 5. (see [36]). A soft set over is called a soft point if there exist and such that and for each .
Throughout this study such soft point is briefly denoted by .
In a similar way, we define as a soft set such that and for each .

Definition 6. (see [37]). The relative complement of a soft set , denoted by , is given by , where is the mapping defined by for each .

Definition 7. (see [38]). A soft set over is said to be a null soft set, denoted by , if for each . is said to the absolute soft set, denoted by , if for each .

Definition 8. (see [38]). The union of two soft sets and over , denoted by , is a soft set , where and the mapping is given as follows:If , , is an indexed family of soft sets over , then , where .

Definition 9. (see [38]). The intersection of two soft sets and over , denoted by , is the soft set , where , and the mapping is given by for every .
For a family , , of soft sets over , one defines , where for every .

Definition 10. A soft set is a soft subset of a soft set , denoted by , if and for all , we have .The soft sets and are soft equal if each one of them is a soft subset of the other.

Definition 11. (see [39]). A soft mapping between and is a pair , denoted also by , of mappings such that and and the image of and preimage of are defined by(i), (ii). A soft mapping is said to be injective (resp. surjective and bijective) if and are injective (resp. surjective and bijective).

2.3. Soft Topological Spaces

Definition 12. (see [26]). A triple is said to be a soft topological space if is a collection of soft sets over satisfying the following axioms: (ST.1) and belong to (ST.2) If and , then (ST.3) If is any subset of , then Elements of are called soft open sets, and their relative complements are called soft closed.
If (ST.1) is replaced by (ST.1′) , whenever or for each , then is called the enriched soft topology (see [27]).
Throughout this paper, the ordered triple indicates a soft topological space.

Proposition 1. (see [26]). Let be a soft topological space. Then, the collection defines a topology on for each . This collection is called a parametric topology on .

The family,is a soft topology on finer than .

The soft topology given in the proposition above is called an extended soft topology.

The equivalence between the enriched and extended soft topologieswas proved in [40].

Definition 13. (see [29, 41]). Let be a soft set in . Then,(i)The interior of , denoted by , is the union of all soft open sets in contained in ; the closure of , denoted by , is the intersection of all soft closed sets containing (ii) is soft semiopen in if ; the relative complement of a semi open set is called semiclosed(iii)The semi interior of , denoted by , is the union of all soft semiopen sets contained in ; the semi closure of , denoted by , is the intersection of all soft semi closed sets containing

Definition 14. (see [26]). For a subset of , the family is called a soft relative topology on and the triple is called a soft subspace of .

Definition 15. (see [26, 42]). Let be a soft set over and . We write(i) if for some ; if for every .(ii) if for every ; if for some .

Definition 16. (see [26, 43]). is said to be(i)Soft regular if for every soft closed set and such that , there are disjoint soft open sets and containing and , respectively(ii)Soft regular if for every soft closed set and such that , there are disjoint soft open sets and , containing and , respectively.

Definition 17. (see [42]). A soft set over is called stable provided that there is such that for each ; is called stable provided that all proper nonnull soft open sets are stable.
A family of soft sets in is said to be a soft cover of (or a soft cover of ) if . A soft cover is said to be locally finite if for each soft point has a soft neighbourhood intersecting only finitely many . A soft cover is a soft refinement of a soft cover if for each there is such that .

Definition 18. A soft space is said to be(i)Soft compact (resp. soft Lindelöf) [27] provided that every soft open cover of has a finite (resp. countable) subcover(ii)Soft paracompact [30, 44] if every soft open cover has a soft open, locally finite refinement

Definition 19. (see [36]). A soft map is said to be(i)Soft continuous if the inverse image of each soft open set is soft open(ii)Soft open (resp. soft closed) if the image of each soft open (resp. soft closed) set is soft open (resp. soft closed)(iii)Soft homeomorphism if it is bijective, soft continuous, and open

Definition 20. Let and be soft sets over and , respectively. The cartesian product of and is a soft set over such that for each .

Theorem 1. (see [40]). is extended if and only if and for any soft subset of .

Theorem 2. (see [40]). If is a soft regular space, then and for any stable soft subset of .

3. Further Properties of Soft Semiopen Sets

This section is devoted to investigation of some properties of soft semiopen and soft s-regular open sets, which we need to prove some results in Section 4.

Proposition 2.. (see [45]). If is a soft open set, thenfor every soft subset of .

Proposition 3. If is a soft closed set, thenfor every soft subset of .

Proof. Let . Then, and . Therefore, there is a soft open set such that . Suppose that . Then, there is a soft open set such that . Now, is a soft open set containing such that and . This implies that . This is a contradiction. Thus, , as required.

Proposition 4. for every soft subset of .

Proof. Since is soft semiclosed, then . Obviously,Therefore,Conversely, it can be observed that (using Proposition 3)Now, we haveThus, is a soft semiclosed set. Hence,as required.

Corollary 1. for every soft open subset of .

Definition 21. A soft subset of is said to be soft s-regular open if .

Proposition 5. Every soft -regular open set is soft open and soft semiclosed.

Proof. Let be a soft s-regular open set. Then, . Obviously, is soft regular open. It follows, from Corollary 1, that . Thus, and . Hence, the desired result is proved.

Proposition 6. is a soft s-regular open set for every soft subset of .

Proof. Since , thenConversely, , thenThus, , as required.

4. Nearly Soft Menger Spaces

Definition 22. (see [46]). A soft topological space is said to be soft Menger if for each sequence of soft open covers of there is a sequence such that is a finite subset of for each and is a soft open cover of .
In this section, we formulate the concept of nearly soft Menger spaces and show its relationships with soft compact, soft Lindelöf, and soft Menger spaces with the help of examples. Also, we characterize this concept in terms of soft s-regular open covers and study under what conditions the concepts of soft Menger and nearly soft Menger spaces are equivalent.

Definition 23. A soft topological space is said to be nearly soft Menger if for each sequence of soft open covers of there is a sequence such that, for every , is a finite subset of and .

Definition 24. A soft space is said to be nearly soft compact (resp. nearly soft Lindelöf) if every soft open cover of has a finite (resp. countable) subcover such that the soft interiors of soft closures of whose members cover .
Clearly, every nearly soft compact space and every soft Menger space are nearly soft Menger, and every nearly soft Menger space is nearly soft Lindelöf.
The three examples below show that the above implications are not reversible.

Example 1. Let be a set of parameters. Consider the soft topological space , where is the set of integers, and is the discrete soft topology on . Then, is a nearly soft Menger space, but it is not nearly soft compact.

Example 2. Let be a parameters set and be a soft topology on the set of real numbers . Then, is not a soft Lindelöf space, so it is not soft Menger. To show that is a nearly soft Menger space, let be a nonnull soft open set. Then, is the only soft closed set containing . This implies that . Therefore, for any sequence of soft open covers of , we choose only one element of . Set , . We have the sequence of finite subsets of , , such that . Hence, is a nearly soft Menger space.

Example 3. It is well known that a soft topological space is the classical topological space if is a singleton. Then, we can consider the Sorgenfrey line as an example of a soft Lindelöf, hence a soft nearly Lindelöf space. Since for any open subset of we have , one concludes that it is not a Menger space.
An immediate consequence of Corollary 1 gives a characterization of nearly soft Menger spaces in terms of soft semiclosed sets.

Proposition 7. A soft space is nearly soft Menger if for each sequence of soft open covers of there is a sequence such that is a finite subset of for each , and .

The following result is a consequence of Corollary 1 and Definition 23 and characterizes a nearly soft Menger space in terms of soft -regular open covers.

Corollary 2. A soft space is a nearly soft Menger space if and only if it satisfies , where denotes the family of all soft -regular open covers of .

We investigate in the following result under what condition the concepts of soft Menger and nearly soft Menger spaces are equivalent.

Theorem 3. The following two properties are equivalent if is a soft regular space:(i) is a soft Menger space(ii) is a nearly soft Menger space

Proof. (i) (ii) It follows from Proposition 7.(ii) (i) Let be a sequence of soft open covers of . Then, for every , there is a sequence of soft open covers such that forms a soft open refinement of (because is soft regular). Since is a nearly soft Menger space, there is a sequence such that is a finite subset of for each , and . For every and every , there is such that . Let . To prove that covers , let . Then, there are and such that . This means that there is such that . Thus, . Hence, .

Corollary 3. The following two properties are equivalent if is a soft paracompact:(i) is a soft Menger space(ii) is a nearly soft Menger spaceOne of topological results states that Mengerness and near Mengerness coincide in spaces in which is a finite set for any . This result does not valid for soft topological spaces as the following example shows.

Example 4. Let the set of parameters be the set of real numbers and or is the null soft set be a soft topology on . Following the similar arguments given in Example 2, we infer that is a nearly soft Menger space, but it is not soft Menger.

Definition 25. A soft subset of is said to be nearly soft Menger in (or nearly soft Menger relative to ) if for every sequence of covers by soft open subsets of there is a sequence such that is a finite subset of for each and .

Proposition 8. The soft union of two nearly soft Menger sets in a soft space is also nearly soft Menger in .

Proof. Let and be two nearly soft Menger sets in and let be a sequence of soft open covers of . Then, there is a sequence of soft open covers of and a sequence of soft open covers of such that and are finite subsets of for each , and and are soft covers of and , respectively. Thus, the sequence witnesses for that is nearly soft Menger in .

Proposition 9. The property of being a nearly soft Menger is hereditary by soft clopen subspaces.

Proof. Let be a soft clopen subspace of a nearly soft Menger space . Suppose that is a sequence of soft open covers of . Then, for each and each , there is a soft open subset of such that . Set . Now, is a sequence of soft open covers of . By hypothesis, there is a sequence such that is a finite subset of for each and is a soft cover of . By taking for each , we find that is a finite subset of for each , and is a soft cover of . Hence, is a nearly soft Menger space.
Example 2 illustrates that the nearly soft Menger property is not soft closed hereditary. The following example illustrates that the nearly soft Menger property is not soft open hereditary as well.

Example 5. Let be a set of parameters and be a soft topology on the set of real numbers . Since is soft compact, it is a nearly soft Menger space. On the contrary, let . Then, is an uncountable soft open set and is the discrete soft topology. Hence, is not a nearly soft Menger space.

Proposition 10. If is an enriched nearly soft Menger space, then is a countable set.

Proof. Let be enriched. Then, and for each is a soft open cover of . We construct a sequence of soft open covers of as follows: for each . By nearly soft Mengerness, there is a sequence such that is a finite subset of for each and is a soft cover of . Since is a soft clopen set, then is also a soft cover of . Hence, must be countable.
Example below shows that the converse of the abovementioned proposition fails.

Example 6. Let be the discrete soft topology on the real numbers set and let be a set of parameters. Then, is enriched and is countable. However, it is not a nearly soft Menger space.
The following results investigate nearly soft Mengerness between a soft topological space and its parametric topological spaces.

Proposition 11. If is an enriched nearly soft Menger space, then is a nearly Menger space for each .

Proof. Let be a sequence of open covers of , and let for each , . Set , where and for each , and consider the sequence of soft open covers of . By near soft Mengerness of , there is a sequence such that, for each , is a finite subset of and is a soft cover of . It follows from Theorem 1 that is also a soft cover of . Denote . Then, the sequence witnesses for that is a nearly Menger space.

Proposition 12. If all , , are nearly Menger spaces, then is a nearly soft Menger space provided that is enriched and is countable.

Proof. Let be a sequence of soft open covers of , where . Fix and let . Then, is a sequence of open covers of . By near Mengerness of , there is a sequence such that, for each , is a finite subset of and is an open cover of . Taking a sequence such that . It follows from Theorem 1 that . Since is finite, then is a finite set and since is countable, then the sequence witnesses for that is a nearly soft Menger space.
The following example elucidates that the condition of countability of given in the abovementioned proposition is essential.

Example 7. Let be a soft topology on , and be an uncountable set of parameters. Then, all are the indiscrete topologies on . Therefore, all are Menger spaces. On the contrary, it follows from Proposition 10 that is not a soft Menger space.

Proposition 13. Let be a soft regular space. Then, it is nearly soft Menger if and only if is nearly Menger for all .

Proof (necessity). Let be a sequence of open covers of such that . Then, such that for each is a soft cover of . Since is soft regular, then every soft open set is stable. Therefore, is a sequence of soft open covers of . By nearly soft Mengerness of , there is a sequence such that for each , is a finite subset of and is a soft open cover of . From Theorem 2, it follows that is also a soft open cover of . Set . Then, the sequence testifies for that is a nearly Menger space.
Sufficiency: let be a sequence of soft open covers of such that . Since is stable, then is an open covers of for each and each . By the nearly Menger property of , there is a sequence such that, for each , is a finite subset of and is an open cover of . Consider the sequence such that for each , of soft open covers of . Since for each , we conclude that is a nearly soft Menger space.
In what follows, we examined some features of a nearly soft Menger space under some types of soft mappings.

Proposition 14. The soft continuous open surjective image of a nearly soft Menger space is a nearly soft Menger space.

Proof. Let be a soft continuous open mapping of a nearly soft Menger space onto and be a sequence of soft open covers of . From the soft continuity of , we obtain that is a sequence of soft open covers of . Therefore, there is a sequence such that, for each , is a finite subset of and . Since is surjective, then and since is soft open, then . Thus, is a soft cover of . Hence, the sequence witnesses for that is a nearly soft Menger space.

Corollary 4. The property of being a nearly soft Menger is a soft topological property.

Example 2.15 in [9] shows that the product of two nearly Menger spaces is not always a nearly Menger space. Since a classical topological space is a special case of a soft topological space when a parameters set is a singleton, then this result is still valid on soft topological spaces.

In the rest of this section, we prove that the product of a nearly soft Menger space and a nearly soft compact space is a nearly soft Menger.

Lemma 1. Let and be two subsets of and , respectively. Then,(i)(ii)

Proof. (i)Suppose that . Then, there exists a soft semiopen subset of containing such that . So. or . This means that or . Thus, . Hence, . Suppose now that or . Without loss of generality, let . Then, there exists a soft semiopen subset of containing such that . Now, is a soft semiopen subset of containing satisfies that . This means . Hence, .(ii)By using a similar argumentation, one can prove item (ii).

Theorem 4. The product of a nearly soft Menger space and a nearly soft compact space is a nearly soft Menger.

Proof. Let be a nearly soft Menger space and be a nearly soft compact space, and let is a sequence of soft open covers of . One may assume that, for every , , where all are soft open covers of and all are soft open covers of . Since is nearly soft compact there is a sequence such that, for each , is a finite subset of and . As is nearly soft Menger there are finite sets such that . Let for each . Then, for every , is a finite subset of . By using the previous lemma, it is easy now to see that which means that is nearly soft Menger.

5. Conclusion

This study is devoted to introducing and investigating the concept of nearly soft Menger spaces. We provide several examples to discuss its relationships with soft Menger and soft Lindelöf spaces, and to show the interchangeability of nearly Mengerness between a soft space and its parametric spaces. In general, we have initiated many properties of it parallel to their corresponding properties from classical topology. Our hope is that the introduced concepts will be beneficial for the researchers to further promote and advance the study selection principles in soft topologies.

Data Availability

The data used to support the findings of this study are cited at relevant places within the text as references. They are also available from the corresponding author upon request.

Conflicts of Interest

The authors declare that there are no conflicts of interests regarding the publication of this article.

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Copyright © 2020 Tareq M. Al-shami and Ljubiša D. R. Kočinac. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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