1. Introduction

The topological structures of set theories dealing with uncertainties were first discussed by Chang [2]. E. F. Lashin et al. [6] generalized rough set theory in the framework of topological spaces. Shabir and Naz [20] are the first persons who introduce the concept of soft topological spaces which are defined over an initial universe with a fixed parameters. Zorlutuna, et al. [21] introduced some new concepts in soft topological spaces and give some new properties about soft topological spaces. Molodtsov [17] introduced the concept of a soft set as a mathematical tool for dealing with uncertainties. Soft set theory has rich potential for practical applications in several domains. In recently years, soft set theory has been researched in many fields see ([8]-[16]). Compact spaces are one of the most important classes in general topological spaces [3]. They have many well-known properties which can be used in many disciplines. Next, the notion of compact soft spaces around a soft topology is introduced see [21]. Later some other properties of soft topological spaces were also examined: soft countability axioms [19]. In 2017, the notion of soft sequentially absolutely closed space is introduced by S. Mahmood. We introduce in this work the new notation is called soft closed space and we prove in this work the soft closed space is a soft topological property, also we study its relation with soft Lindelof space, we show that every soft Lindelof spaces is a soft closed space but the converse is not true in general, also we show that every soft space satisfies the soft second axiom of countability that is soft closed space and every soft compact space or soft absolutely closed space is soft closed space.

2. Preliminaries

We now begin by recalling some definitions and some of the basic prosperities of the soft sets.

Definition 2.1: ([17])

Let be an initial universe set and let be a set of parameters. A pair is called a soft set (over ) where and is a multivalued function . In other words, the soft set is a parameterized family of subsets of the set . Every set from this family may be considered as the set of e-elements of the soft set or as the set of approximate elements of the soft set. Clearly, a soft set is not a set. For two soft sets and over the common universe , we say that is a soft subset of if and for all We write is said to be a soft superset of if is a soft subset of Two soft sets and over a common universe are said to be soft equal if is a soft subset of and is a soft subset of . A soft set over is called a null soft set, denoted by if for each . Similarly, it is called universal soft set, denoted by , if for each . The collection of all soft sets over a universe and the parameter set is a family of soft sets denoted by . Also, the collection of all soft sets over a universe and the parameter set is a family of soft sets denoted by

Definition 2.2: ([7]) The union of two soft sets and over is the soft set , where and for all if if if We write . The intersection of and over , denoted , is defined as , and for all .

Definition 2.3: ([21]) The soft set is called a soft point in , denoted by , if there exist and and for all . The soft point is said to be in the soft set , denoted by , if for the element and .

Definition 2.4: ([20]) The difference of two soft sets and over , denoted by , is defined as for all .

Definition 2.5: ([20]) Let be a soft set over . The complement of with respect to the universal soft set , denoted by , is defined as , where and for all

Proposition 2.6: ([20]) Let and be the soft sets over . Then

(1)

(2)

Definition 2.7: ([20])

Let be the collection of soft sets over . Then is called a soft topology on if satisfies the following axioms:

(i) belong to . (ii) The union of any number of soft sets in belongs to .

(iii) The intersection of any two soft sets in belongs to .

The triplet is called a soft topological space over . The members of are called soft open sets in and complements of them are called soft closed sets in .

Definition 2.8: ([4])

The soft closure of is the intersection of all soft closed sets containing . (i.e) The smallest soft closed set containing and is denoted by . The soft interior of is the union of all soft open set is contained in and is denoted by

Definition 2.9: ([1]) Let be a soft topological space. A sub-collection of is said to be a base for if every member of can be expressed as a union of members of .

Definition 2.10: ([1]) Let be a soft topological space, and let be a family a soft neighborhood of some soft point . If, for each soft neighborhood of , there exists a such that then we say that is a soft neighborhoods base at .

Definition 2.11: ([18]) A family of soft sets is a cover of a soft set (F, A) if . If each member of is a soft open set, then is called a soft open cover.

Definition 2.12: ([18]) A soft topological space is called soft compact space if each soft open cover of has a finite subcover.

Definition 2.14 ([1]) Let be a soft topological space. Then is called soft absolutely closed iff for each soft open cover of , there exist such that

Definition 2.15 ([1]) Let be a soft topological space. If has a countable soft base, then we say that is soft second-countable.

Definition 2.16: ([1]) A soft space is soft Lindelof if each soft open covering of has a countable subcover.

Definition 2.17 ([5]) Let and be soft classes and let and

be mappings. Then a mapping is defined as: for a soft set in , is a soft set in given by for is called a soft image of a soft set . If B = K, then we shall write as .

Definition 2.18 ([5]) Let be a mapping from a soft class to another soft class and a soft set in soft class where Let and be mappings. Then is a soft set in the soft classes defined as: for is called a soft inverse image of . Hereafter, we shall write as

Theorem 2.19 ([5]) Let and be mappings. Then for soft sets and a family of soft sets in the soft class we have:

(1) ,

(2) in general

(3) in general

(4) If then ,

(5) ,

(6) in general

(7) in general

(8) If then

Definition 2.20 ([5]) A soft mapping is said to be soft continuous (briefly s-continuous) if the soft inverse image of each soft open set of is a soft open set in .

Definition 2.21 ([5]) A soft mapping is said to be soft open (briefly s-open) if soft image of each soft open set of is a soft open set in .

Definition 2.22 ([5]) A soft mapping is said to be soft homeomorphism if f is onto and one to one.

3. Soft Lindelof Closed Space

Definition 3.1 Let be a topological space we say that is a soft Lindelof closed space (soft closed space) iff each soft open cover of has a soft countable subfamily whose soft closure covers [i.e. ].

Example 3.2 Let be real line, and let . Then be an uncountable set. Let , and let be the soft topology generated by as a base. Hence is soft closed space. Indeed, for each soft point , it is easy to see that is a soft neighborhoods base at . Let . Thus is closed space.

Theorem: 3.3 If is a soft Lindelof space, then is a soft closed space.

Proof:

Let be a soft Lindelof space. Then for each soft open cover of there is a countable sub-collection of which also covers [i.e. , where ]. However, This implies that Hence is a soft closed space.

Remark 3.4: The converse of theorem (3.3) is not true in general. So we will show in following theorem when will be hold.

Theorem: 3.5

Let be a soft discrete space. Then is soft Lindelof space if and only if is closed.

Proof:

Assume that is soft Lindelof space. Then by [theorem (3.3)] we consider that is soft closed space.

Conversely, suppose that is a soft closed space and let be a soft open cover of . Then there is a countable sub family whose closure covers [i.e. where ]. However, for each soft open set in we have [since is soft discrete space]. Therefore [ where ], Then (X,T) is a soft Lindelof space.

Theorem 3.6: Every soft compact space is a soft closed space.

Proof:

Assume that is a soft compact space, then for each soft open cover of there exists a finite sub-collection of which also covers , However, each finite family is countable family. Therefore is a soft Lindelof space. Hence is a soft closed space by [Theorem, (3.3)].

Theorem 3.7:

Every soft absolutely closed space is a soft closed space.

Proof:

Assume that is a soft topological space where is absolutely closed, let be a soft open cover of . Then there is a finite sub-collection of whose soft closure covers [since is soft absolutely closed space], thus there is a countable sub-collection of whose soft closure covers [since each finite family is a countable family]. Then is a soft closed space.

Theorem 3.8:

If is soft second-countable, then is a soft closed space.

Proof:

Assume that is a soft second-countable, then has a countable soft base. Therefore for each soft open cover there is a countable sub-cover of . Then is a soft Lindelof space but by [Theorem, (3,3)] we have is a soft closed space.

Remark 3.9: The converse of the theorem (3, 8) is not true in general.

Example 3.10

Let be a soft closed topological space in example (3.2) we shall prove that is not soft second-countable. Let be an arbitrary soft base for For arbitrary soft point and soft open neighborhood , there exists a soft open set such that , and hence . If , then . Therefore, the set is an uncountable subfamily of , and thus is an uncountable family.

Proposition 3.11:

A soft closeness is a topological property.

Proof:

Assume that is a soft homeomorphism and is a soft closed space, we went to show that is a soft closed space. Suppose that is a soft open cover of [i.e. ]. However, [since is on to] and since is soft continuous function we have is a soft open cover of but is a soft closed space, then there is a countable sub-collection of whose soft closure covers [i.e. where ], therefore Thus , where . Then is soft closed space.

Theorem 3.12: The cartesian product of countably many soft closed spaces is soft closed.

Proof: Let be a family of countable many soft closed spaces. For each , let be asoft open cover of . Then there is a soft countable subfamily of whose soft closure covers [i.e. ]. Put , for countable many values . Hencev is a countable. Further, Then for any soft open cover for , there is a soft countable subfamily of whose soft closure covers .

Remark 3.13: By the above results we have the following diagram:

Figure 1. Diagram showing relationships among some of the soft spaces

4. Conclusions

In this paper, the concept of soft closed spaces is introduced. Further, in this work we describe its relation with soft Lindelof space, soft compact space, and soft absolutely closed space. Assume is a soft strongly open map (soft generality open) from soft closed space into soft countable closed space (soft sequentially closed space). The question we are concerned with is: what is the possible property of soft map need to provide that is soft countable closed (soft sequentially closed) subspace of

ACKNOWLEDGEMENTS

The authors would like to thank from the anonymous reviewers for carefully reading of the manuscript and giving useful comments, which will help to improve the paper.