# Infra Soft Compact Spaces and Application to Fixed Point Theorem

Authors: Tareq M. Al-shami
Journal: Journal of Function Spaces
 (2021)

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## Abstract

Infra soft topology is one of the recent generalizations of soft topology which is closed under finite intersection. Herein, we contribute to this structure by presenting two kinds of soft covering properties, namely, infra soft compact and infra soft Lindelöf spaces. We describe them using a family of infra soft closed sets and display their main properties. With the assistance of examples, we mention some classical topological properties that are invalid in the frame of infra soft topology and determine under which condition they are valid. We focus on studying the “transmission” of these concepts between infra soft topology and classical infra topology which helps us to discover the behaviours of these concepts in infra soft topology using their counterparts in classical infra topology and vice versa. Among the obtained results, these concepts are closed under infra soft homeomorphisms and finite product of soft spaces. Finally, we introduce the concept of fixed soft points and reveal main characterizations, especially those induced from infra soft compact spaces.


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## Body

## 1. Introduction

In our daily life, we face many types of uncertain phenomena and problems which require looking for adequate approaches to deal with them. The researchers’ efforts in this regard lead to proposing various convenient tools to address uncertainty and vagueness. One of the notable tools related to our interest is the soft set which was introduced in 1999 by Molodtsov [1]. He discussed its different applications like smoothness of functions, game theory, theory of measurement, and Riemann integration. Then, soft sets have been applied to several scopes like medical science [2], computer science [3], and decision-making problems [4].After three years of the emergence of soft sets, Maji with his coauthors [5] put forward the basic concepts of soft set theory. They explored primary operations like the intersection of two soft sets and their union and difference; also, they defined a complement of soft set. Later on, many scholars and researchers interested in soft set theory redefined the concepts given by Maji et al. and displayed new types of soft operations and operators. To keep some properties and results of crisp set theory, Ali et al. [6] proposed new types of these operations and operators. The improvements and contributions to soft set theory have been continued which lead to defining many sorts of soft equality like lower and upper soft equality [7], gf-soft equality [8], and T-soft equality [9].As it is well known, topologists employed the generalizations of crisp sets to construct novel extensions of topologies. As a continuation of this path, Shabir and Naz [10] and Çag˘man et al. [11] hybridized classical topology and soft sets to formulate soft topological spaces. In fact, they differently formulated the concept of soft topology. Shabir and Naz stipulated the constant of the universal crisp set and set of parameters which the members of soft topology defined over them, whereas Çag˘man et al. did not impose any conditions on the universal crisp set and set of parameters. Our approach in this paper goes according to the definition of Shabir and Naz. Special kinds of soft topologies called enriched and extended soft topologies were introduced in [12]. Al-Shami and Kocinac [13] scrutinized the role of extended soft topology to link the concepts in soft topologies with their counterparts in classical topologies.It was noted that several properties of topological concepts are preserved under some generalizations of soft topology such as supra soft topology [14], soft bitopology [15], and infra soft topology [16]. This means that we can consider that some topology’s conditions are superfluous in some cases. In fact, this matter was applied in crisp settings to describe some real-life issues (see [17, 18]). Another merit of generalizations of soft topology is the ease of building counterexamples that show the relationships between the concepts under study. For these reasons, we are interested to study infra soft topology which is one of the recent interesting developments of soft topology.Compactness and Lindelöfness are one of the interesting concepts in soft topologies. They were studied in some pioneer articles such as [19–21]. Our contribution herein is to analyze the properties of these two concepts in the frame of infra soft topological spaces. We note the validity of some properties of (classical) soft compactness and Lindelöf spaces via infra soft topological structures. This helps us to discuss many topological concepts and reveal the relationships among them in this frame instead of soft topology, so we target in this manuscript to perform an exhaustive analysis of infra soft topological structures.The fixed point theorem is a hot topic in recent years. It has been investigated in many papers such as [22–25]. In this work, we put forward the basis of fixed soft points in the introduced frame. Some results that associated fixed soft points with infra soft compact spaces and infra parametric infra topological spaces are studied in detail.The layout followed in the rest of this manuscript is as follows. In Section2, we recall the definitions and results that we need to comprehend this work. In Section 3, we introduce the concepts of infra soft compact and infra soft Lindelöf spaces and characterize them. We establish their master features and reveal some of their counterparts’ properties that are lost. In Section 4, we study fixed soft points in the frame of infra soft topological spaces and explore the role of infra soft compactness to obtain a fixed soft point. Finally, we epitomize the paper’s fulfillments and suggest some future works in Section 5.


## 2. Preliminaries

Through this section, we mention the materials that make this study self-contained. We divide it into two subsections.


### 2.1. Soft Set Theory

Definition 1 (see [1]).
LetB be a set of parameters, A a universal set, and 2A the power set of A. A soft set over A is an ordered pair δ,B, where δ:B⟶2A is a crisp map. We express the soft set as follows: δ,B=β,δβ:β∈Bandδβ∈2A.
A family of all soft sets overA under a set of parameters B is symbolized by SAB.Definition 2 (see [6]).
A soft setδc,B is called a complement of δ,B provided that a map δc:B⟶2A such that δcβ=A\δβ for each β∈B.Definition 3 (see [5]).
If the image of each parameter ofB under a map δ:B⟶2A is the universal set A, then δ,B is called the absolute soft set over A. Its complement is called the null soft set. The absolute and null soft sets are symbolized by A~ and Φ, respectively.Definition 4 (see [26, 27]).
If all components of a soft set are equal (resp., finite, countable), then we called it a stable (resp., finite, countable) soft set. Otherwise, it is called unstable (resp., infinite, uncountable).Definition 5 (see [12, 27]).
If the image of one parameter, sayβ, under a map P:B⟶2A is a singleton set, say α, and the image of each parameter β′∈B\β is the empty set, then a soft set P,B is called a soft point over A. It is briefly symbolized by Pβα.Definition 6 (see [10, 27]).
There are two belong and two nonbelong relations between an elementα∈A and a soft set δ,B defined as follows.(i)
α∈δ,B if α∈δβ for all β∈B(ii)
α∈δ,B if α∈δβ for some β∈B(iii)
α⋐δ,B if α∈δβ for some β∈B(iv)
α⋐δ,B if α∈δβ for all β∈BDefinition 7 (see [6]).
The intersection of two soft setsδ,Bandξ,CoverA, symbolized by δ,B⋂~ξ,C, is a soft set λ,D, where D=B∩C≠∅, and a map λ:D⟶2A is given by λβ=δβ∩ξβ for each β∈D.Definition 8 (see [5]).
The union of two soft setsδ,B and ξ,CoverA, symbolized by δ,B⋃~ξ,C, is a soft set λ,D, where D=B∪C and a map λ:D⟶2A is given as follows:(1)λβ=δβ:β∈B\C,ξβ:β∈C\B,δβ∪ξβ:β∈B∩C.Definition 9 (see [28]).
A soft setδ,B is a subset of a soft set ξ,C, symbolized by δ,B⊆~ξ,C, ifB⊆C and δβ⊆ξβ for all β∈B. The soft sets δ,B and ξ,C are called soft equal if each is a subset of the other.Definition 10 (see [19]).
A family of soft sets is said to have the finite (resp., countable) intersection property if the finite (resp., countable) intersection of any members of this family is nonnull.Definition 11 (see [20]).
The Cartesian product ofδ,B and ξ,C, symbolized by δ×ξ,B×C, is defined as δ×ξβ,β′=δβ×ξβ′ for each β,β′∈B×C.Definition 12 (see [29]).
A soft mapfφ from SAB to SZC is a pair of crisp maps f and φ, where f:A⟶Z, φ:B⟶C. Let δ,M and ξ,N be, respectively, subsets of SAB and SZC. Then, the image of δ,M and preimage of ξ,N are given by the following.(i)
fφδ,M=fδ,C is a soft set in SZC such that(2)fδω=⋃~β∈φ−1ω⋂Mfδβ:φ−1ω≠∅,∅:φ−1ω=∅,for each ω∈C(ii)
fφ−1ξ,N=f−1ξ,B is a soft set in SAB such that(3)f−1ξβ=f−1ξφβ:φβ∈N,∅:φβ∈N,for each β∈BDefinition 13 (see [20, 30]).
A softmapfφ:SAB⟶SZC is said to be injective (resp., surjective, bijective) if both f and φ are injective (resp., surjective, bijective).


### 2.2. Infra Soft Topological Spaces

Definition 14 (see [16]).
A familyΩ of soft sets over A with B as a parameter set is said to be an infra soft topology on A if it is closed under finite intersection and Φ is a member of Ω.The tripleA,Ω,B is called an infra soft topological space (briefly, ISTS). We called a member of Ω an infra soft open set and called its complement an infra soft closed set. We called A,Ω,B stable if all its infra soft open sets are stable and called finite (resp., countable) if A is finite (resp., countable).Proposition 15 (see [16]).
LetA,Ω,B be an ISTS. Then, the collection Ωβ=δβ:δ,B∈Ω forms an infra topology on A for each β∈B.
We calledΩβ a parametric infra topology.Proposition 16 (see [16]).
Suppose thatΨ=Ωββ∈B is a class of crisp infra topologies on A. Then,(4)ΩΨ=β,Fβ:β∈B∈SABsuchthatFβ∈Ωβforeachβ∈Bdefines an infra soft topology on A.The ISTS given in the above proposition is called an extended infra soft topology onA or an infra soft topology on A generated by Ψ.Definition 17.
An ISTSA,Ω,B is called an infra pp-soft T2 (or an infra pp-soft Hausdorff)-space if for every α≠ε∈A, there exist disjoint infra soft open sets δ,B and ξ,B such that α⋐δ,B,ε⋐ξ,B and ε∈δ,B,α∈ξ,B.In the above definition, if we replace the relations⋐,∈ by ⋐,⋐ (resp., ∈,∈∈,⋐), then we called A,Ω,B an infra pt-soft T2 (resp., infra tp-soft T2, infra tt-soft T2)-space.Definition 18 (see [16]).
LetA,Ω,B be an ISTS and ξ,B be a nonnull subset of A~. Then, Ωξ,B=ξ,B⋂~δ,B:δ,B∈Ω is called an infra soft relative topology on ξ,Bandξ,B,Ωξ,B,B is called an infra soft subspace of A,Ω,B.Theorem 19 (see [16]).
Letξ,B,Ωξ,B,B be an infra soft subspace of A,Ω,B. Then, λ,B is an infra soft closed subset of ξ,B,Ωξ,B,B iff there exists an infra soft closed subset δ,BofA,Ω,B such that λ,B=ξ,B⋂~δ,B.Definition 20 (see [31]).
A soft mappingfφ:A1,Ω1,B1⟶A2,Ω2,B2is said to be(i)
infra soft continuous provided that the preimage of any infra soft open set is an infra soft open set(ii)
infra soft open (resp., infra soft closed) if the image of any infra soft open (resp., infra soft closed) set is an infra soft open (resp., infra soft closed) set(iii)
an infra soft homeomorphism if it is infra soft continuous, infra soft open, and bijectiveA property which is kept by any infra soft homeomorphism is said to be an infra soft topological property.Proposition 21 (see [31]).
LetAk,Ωk,Bk:k∈K be a family of ISTSs. Then, Ω=∏k∈Kδk,Bk:δk,Bk∈τk is an infra soft topology on A=∏k∈KAk under a set of parameters B=∏k∈KBk.We calledΩ given in the proposition above a product of infra soft topologies and A,Ω,B a product of infra soft spaces.


## 2.1. Soft Set Theory

Definition 1 (see [1]).
LetB be a set of parameters, A a universal set, and 2A the power set of A. A soft set over A is an ordered pair δ,B, where δ:B⟶2A is a crisp map. We express the soft set as follows: δ,B=β,δβ:β∈Bandδβ∈2A.
A family of all soft sets overA under a set of parameters B is symbolized by SAB.Definition 2 (see [6]).
A soft setδc,B is called a complement of δ,B provided that a map δc:B⟶2A such that δcβ=A\δβ for each β∈B.Definition 3 (see [5]).
If the image of each parameter ofB under a map δ:B⟶2A is the universal set A, then δ,B is called the absolute soft set over A. Its complement is called the null soft set. The absolute and null soft sets are symbolized by A~ and Φ, respectively.Definition 4 (see [26, 27]).
If all components of a soft set are equal (resp., finite, countable), then we called it a stable (resp., finite, countable) soft set. Otherwise, it is called unstable (resp., infinite, uncountable).Definition 5 (see [12, 27]).
If the image of one parameter, sayβ, under a map P:B⟶2A is a singleton set, say α, and the image of each parameter β′∈B\β is the empty set, then a soft set P,B is called a soft point over A. It is briefly symbolized by Pβα.Definition 6 (see [10, 27]).
There are two belong and two nonbelong relations between an elementα∈A and a soft set δ,B defined as follows.(i)
α∈δ,B if α∈δβ for all β∈B(ii)
α∈δ,B if α∈δβ for some β∈B(iii)
α⋐δ,B if α∈δβ for some β∈B(iv)
α⋐δ,B if α∈δβ for all β∈BDefinition 7 (see [6]).
The intersection of two soft setsδ,Bandξ,CoverA, symbolized by δ,B⋂~ξ,C, is a soft set λ,D, where D=B∩C≠∅, and a map λ:D⟶2A is given by λβ=δβ∩ξβ for each β∈D.Definition 8 (see [5]).
The union of two soft setsδ,B and ξ,CoverA, symbolized by δ,B⋃~ξ,C, is a soft set λ,D, where D=B∪C and a map λ:D⟶2A is given as follows:(1)λβ=δβ:β∈B\C,ξβ:β∈C\B,δβ∪ξβ:β∈B∩C.Definition 9 (see [28]).
A soft setδ,B is a subset of a soft set ξ,C, symbolized by δ,B⊆~ξ,C, ifB⊆C and δβ⊆ξβ for all β∈B. The soft sets δ,B and ξ,C are called soft equal if each is a subset of the other.Definition 10 (see [19]).
A family of soft sets is said to have the finite (resp., countable) intersection property if the finite (resp., countable) intersection of any members of this family is nonnull.Definition 11 (see [20]).
The Cartesian product ofδ,B and ξ,C, symbolized by δ×ξ,B×C, is defined as δ×ξβ,β′=δβ×ξβ′ for each β,β′∈B×C.Definition 12 (see [29]).
A soft mapfφ from SAB to SZC is a pair of crisp maps f and φ, where f:A⟶Z, φ:B⟶C. Let δ,M and ξ,N be, respectively, subsets of SAB and SZC. Then, the image of δ,M and preimage of ξ,N are given by the following.(i)
fφδ,M=fδ,C is a soft set in SZC such that(2)fδω=⋃~β∈φ−1ω⋂Mfδβ:φ−1ω≠∅,∅:φ−1ω=∅,for each ω∈C(ii)
fφ−1ξ,N=f−1ξ,B is a soft set in SAB such that(3)f−1ξβ=f−1ξφβ:φβ∈N,∅:φβ∈N,for each β∈BDefinition 13 (see [20, 30]).
A softmapfφ:SAB⟶SZC is said to be injective (resp., surjective, bijective) if both f and φ are injective (resp., surjective, bijective).


## 2.2. Infra Soft Topological Spaces

Definition 14 (see [16]).
A familyΩ of soft sets over A with B as a parameter set is said to be an infra soft topology on A if it is closed under finite intersection and Φ is a member of Ω.The tripleA,Ω,B is called an infra soft topological space (briefly, ISTS). We called a member of Ω an infra soft open set and called its complement an infra soft closed set. We called A,Ω,B stable if all its infra soft open sets are stable and called finite (resp., countable) if A is finite (resp., countable).Proposition 15 (see [16]).
LetA,Ω,B be an ISTS. Then, the collection Ωβ=δβ:δ,B∈Ω forms an infra topology on A for each β∈B.
We calledΩβ a parametric infra topology.Proposition 16 (see [16]).
Suppose thatΨ=Ωββ∈B is a class of crisp infra topologies on A. Then,(4)ΩΨ=β,Fβ:β∈B∈SABsuchthatFβ∈Ωβforeachβ∈Bdefines an infra soft topology on A.The ISTS given in the above proposition is called an extended infra soft topology onA or an infra soft topology on A generated by Ψ.Definition 17.
An ISTSA,Ω,B is called an infra pp-soft T2 (or an infra pp-soft Hausdorff)-space if for every α≠ε∈A, there exist disjoint infra soft open sets δ,B and ξ,B such that α⋐δ,B,ε⋐ξ,B and ε∈δ,B,α∈ξ,B.In the above definition, if we replace the relations⋐,∈ by ⋐,⋐ (resp., ∈,∈∈,⋐), then we called A,Ω,B an infra pt-soft T2 (resp., infra tp-soft T2, infra tt-soft T2)-space.Definition 18 (see [16]).
LetA,Ω,B be an ISTS and ξ,B be a nonnull subset of A~. Then, Ωξ,B=ξ,B⋂~δ,B:δ,B∈Ω is called an infra soft relative topology on ξ,Bandξ,B,Ωξ,B,B is called an infra soft subspace of A,Ω,B.Theorem 19 (see [16]).
Letξ,B,Ωξ,B,B be an infra soft subspace of A,Ω,B. Then, λ,B is an infra soft closed subset of ξ,B,Ωξ,B,B iff there exists an infra soft closed subset δ,BofA,Ω,B such that λ,B=ξ,B⋂~δ,B.Definition 20 (see [31]).
A soft mappingfφ:A1,Ω1,B1⟶A2,Ω2,B2is said to be(i)
infra soft continuous provided that the preimage of any infra soft open set is an infra soft open set(ii)
infra soft open (resp., infra soft closed) if the image of any infra soft open (resp., infra soft closed) set is an infra soft open (resp., infra soft closed) set(iii)
an infra soft homeomorphism if it is infra soft continuous, infra soft open, and bijectiveA property which is kept by any infra soft homeomorphism is said to be an infra soft topological property.Proposition 21 (see [31]).
LetAk,Ωk,Bk:k∈K be a family of ISTSs. Then, Ω=∏k∈Kδk,Bk:δk,Bk∈τk is an infra soft topology on A=∏k∈KAk under a set of parameters B=∏k∈KBk.We calledΩ given in the proposition above a product of infra soft topologies and A,Ω,B a product of infra soft spaces.


## 3. Infra Soft Compact and Infra Soft Lindelöf Spaces

This section is devoted to investigating compactness and Lindelöfness in infra soft topological spaces. We scrutinize their main properties and bring to light some celebrated results of classical compactness and Lindelöfness that are invalid in the frame of ISTS. For illustration and validation, various examples are offered.Definition 22.
(i)
A family of infra soft open setsδk,B:k∈K is said to be an infra soft open cover of an ISTS A,Ω,B provided that the union of its elements covers A~, i.e., A~=⋃~k∈Kδk,B(ii)
An ISTSA,Ω,B is said to be infra soft compact (resp., infra soft Lindelöf) provided that every infra soft open cover of A~ has a finite (resp., countable) subcover(iii)
A soft subsetδ,B of an ISTS A,Ω,B is said to be infra soft compact (resp., infra soft Lindelöf) provided that every infra soft open cover of δ,B has a finite (resp., countable) subcoverBy the two examples below, we explain the existence and uniqueness of infra soft compact and infra soft Lindelöf spaces.Example 23.
LetΩ=ℝ~⋃δ,B⊆~ℝ~:3⋐δ,B be an infra soft topology on the real number set ℝ with B as an arbitrary set of parameters. It is easy to check that ℝ,Ω,B is infra soft compact.Example 24.
LetΩ=ℝ~⋃δ,B⊆~ℝ~:3∈δ,B be an infra soft topology on the real number set ℝ with B as an arbitrary set of parameters such that ∣B∣≥2. Now, the family δ,B⊆~ℝ~: there is only one parameter β∈B such that 3∈δβ forms an infra soft open cover of ℝ~. It is easy to check that this infra soft open cover does not have a countable subcover; hence, ℝ,Ω,B is not infra soft Lindelöf.The proofs of the next two results are easy, so they will be canceled.Proposition 25.
Every infra soft compact space is infra soft Lindelöf.Proposition 26.
A family of infra soft compact (resp., infra soft Lindelöf) sets is closed under a finite (resp., countable) union.By the example below, we explain that Proposition25 is not converse.Example 27.
A familyΩ=ℕ~,δ,B⊆~ℕ~:δ,Bisfinite represents an infra soft topology on the set of natural numbers ℕ with B=β1,β2 as a set of parameters. It is easy to check that ℕ,Ω,B is an infra soft Lindelöf space, but not infra soft compact.Proposition 28.
Every infra soft closed subsetξ,B of an infra soft compact (resp., infra soft Lindelöf) space A,Ω,B is infra soft compact (resp., infra soft Lindelöf).Proof.
Considerδk,B:k∈K as an infra soft open cover of ξ,B which is a subset of an infra soft compact space A,Ω,B. Then, ⋃~k∈Kδk,B⋃~ξc,B is an infra soft open cover of A~. By hypothesis, A~=⋃~k=1nδk,B⋃~ξc,B. Consequently, we obtain ξ,B⊆~⋃~k=1nδk,B. This ends the proof that ξ,B is infra soft compact.☐Following a similar technique, one can prove the case between parentheses.The converse of the above proposition fails as illustrated in the next example.Example 29.
Consider an ISTSℝ,Ω,B as given in Example 23. Let B=β1,β2. Then, δ,B=β1,5,β2,5 is infra soft compact, but not infra soft closed.Corollary 30.
The intersection of infra soft compact (resp., infra soft Lindelöf) and infra soft closed sets is infra soft compact (resp., infra soft Lindelöf).One of the celebrated results in classical topology reports that the finite (resp., countable) topological space is compact (resp., Lindelöf); this result evaporates in ISTSs as the next example elucidates.Example 31.
LetΩ1 and Ω2 be two discrete infra soft topologies on a finite set A1 and a countable set A2, respectively. Let the sets of natural numbers ℕ and real numbers ℝ be sets of parameters. It is clear that A1,Ω1,ℕ is not infra soft compact in spite of A1 being finite, and A2,Ω2,ℝ is not infra soft Lindelöf in spite of A2 being countable.Note that the intersection of two infra soft compact (resp., infra soft Lindelöf) sets needs not be infra soft compact (resp., infra soft Lindelöf). The example given below confirms this fact.Example 32.
Consider an ISTSℝ,Ω,B as given in Example 23. Let B=β1,β2. Note that the two soft sets δ1,B=β1,ℝ\3,β2,ℝandδ2,B=β1,ℝ,β2,ℝ\3 are infra soft compact. But their intersection is the soft set β1,ℝ\3,β2,ℝ\3 which is not infra soft Lindelöf.Now, we give a complete description for infra soft compact and infra soft Lindelöf spaces using a family of infra soft closed sets.Theorem 33.
An ISTSA,Ω,B is infra soft compact (resp., infra soft Lindelöf) iff every family of infra soft closed subsets of A,Ω,B, satisfying the finite (resp., countable) intersection property, has, itself, a nonnull intersection.Proof.
Necessity: consider Σ=δk,B:k∈K as a family of infra soft closed subsets of A,Ω,B which is an infra soft compact space. Suppose that ⋂~k∈Kδk,B=Φ. Then, A~=⋃~k∈Kδkc,B. By the hypothesis of infra soft compactness, we obtain A~=⋃~k=1nδkc,B. Hence, ⋂k=1nδk,B=Φ, as required.
Sufficiency: consider Σ=δk,B:k∈K as an infra soft open cover of A,Ω,B. Suppose that Σ does not have a finite subcover of A~. Then A~\⋃k∈Iδk,B≠Φ for any finite set I⊆K. Therefore, ⋂~k∈Iδkc,B≠Φ. This means that δkc,B:k∈I is a family of infra soft closed subsets of A~ which has the finite intersection property. Thus, ⋂~k∈Kδkc,B≠Φ. Consequently, A~≠⋃~k∈Iδk,B. But this contradicts that Σ is an infra soft open cover of A~. Hence, A,Ω,B is infra soft compact.
Following a similar technique, one can prove the case between parentheses.☐In the next example, we show that there is no relationship between infra soft closed sets and the different types of infra softT2-spaces.Example 34.
Consider an ISTSN,Ω,B as given in Example 27. Note that a soft set δ1,B=β1,ℕ\1,2,β2,ℕ\2,3 is infra soft closed. But it is not infra soft compact in spite of N,Ω,B being an infra tt-soft T2 (infra tp-softT2, infrapt-softT2, infrapp-softT2)-space.Now, we investigate under which conditions the well-known relationship between closed sets andT2-spaces are satisfied in the frame of ISTSs.Proposition 35.
LetA,Ω,B be an infra tt-soft Hausdorff space such that Ω is closed under a finite union. Then, every stable infra soft compact subset of A,Ω,B is infra soft closed.Proof.
Letξ,B be a stable infra soft compact subset of an infra tt-soft Hausdorff space A,Ω,B, and let Pβα∈ξ,Bc. Since ξ,B is stable, we get α≠γk for each Pβγk∈ξ,B. Therefore, there are two disjoint infra soft open sets δk,B and λk,B such that α∈δk,B and γk∈λk,B. Now, λk,B:k∈K forms an infra soft open cover of ξ,B. By hypothesis, ξ,B⊆~⋃~k=1nλk,B. Since Ω is an infra soft topology closing under a finite union, ⋂k=1nδk,B and ⋃~k=1nλk,B are disjoint infra soft open sets. Note that Pβα∈⋂~k=1nδk,B⊆~ξ,Bc. Thus, ξ,Bc is infra soft open set which automatically means that ξ,B is infra soft closed.☐Corollary 36.
LetA,Ω,B be an infra tp-soft Hausdorff space such that Ω is closed under a finite union. Then every stable infra soft compact subset of A,Ω,B is infra soft closed.Corollary 37.
LetA,Ω,B be a stable infra pp-soft Hausdorff space such that Ω is closed under a finite union. Then every stable infra soft compact subset of A,Ω,B is infra soft closed.Theorem 38.
Letξ,B be an infra soft compact subset of an infra tt-soft Hausdorff space A,Ω,B such that Ω is closed under a finite union. Then for each α⋐ξ,B, there are disjoint infra soft open sets δ,B and λ,B such that α∈δ,B and ξ,B⊆~λ,B.Proof.
Letα⋐ξ,B. Then α≠γk for each γk⋐ξ,B. By hypothesis, there exist infra soft open sets δk,B and λk,B such that α∈δk,B and γk∈λk,B. Now, λk,B:k∈K forms an infra soft open cover of ξ,B. By the infra soft compactness of ξ,B we obtain ξ,B⊆⋃~k=1nλk,B. Since Ω is an infra soft topology closing under a finite union, ⋃~k=1nλk,B and ⋂~k=1nδk,B are the required disjoint infra soft open sets. Hence, the proof is complete.☐Theorem 39.
LetA,Ω,B be an infra tt-soft Hausdorff space which is infra soft compact and closed under a finite union. Then it is infra tt-soft regular.Proof.
Letξ,B be an infra soft closed set such that α⋐ξ,B. Then, ξ,B is infra soft compact. According to Theorem 38, there exist disjoint infra soft open sets δ,B and λ,B such that α∈δ,B and ξ,B⊆~λ,B. Hence, A,Ω,B is infra tt-soft regular.☐Corollary 40.
LetA,Ω,B be an infra tt-soft Hausdorff space which is infra soft compact and closed under a finite union. Then it is infra tt-soft T3.Definition 41.
An ISTSA,Ω,B is said to be infra soft T2′-space if for every Pβ1α1≠Pβ2α2α1≠α2orβ1≠β2, there are two disjoint infra soft open sets δ,B and λ,B such that Pβ1α1∈δ,B and Pβ2α2∈λ,B.Proposition 42.
LetA,Ω,B be an infra soft T2′-space such that Ω is closed under a finite union. Then every infra soft compact subset of A,Ω,B is infra soft closed.Proof.
Letξ,B be an infra soft compact subset of A,Ω,B which is an infra soft T2′-space. Let Pβα∈ξ,Bc. Then for each Pρkγk∈ξ,B, we get Pρkγk≠Pβα. Therefore, there are two disjoint infra soft open sets δk,B and λk,B such that Pρkγk∈δk,B and Pβα∈λk,B. Now, δk,B:k∈K forms an infra soft open cover of ξ,B. By hypothesis, ξ,B⊆~⋃~k=1nδk,B. Since Ω is an infra soft topology closing under a finite union, ⋂~k=1nλk,B and ⋃~k=1nδk,B are disjoint infra soft open sets. Note that Pβα∈⋂~k=1nλk,B⊆~ξ,Bc. Thus, ξ,Bc is infra soft open set which automatically means that ξ,B is infra soft closed.☐Proposition 43.
The infra soft continuous image of an infra soft compact (resp., infra soft Lindelöf) set is infra soft compact (resp., infra soft Lindelöf).Proof.
Considerfφ:A,Ω,B⟶C,℧,D as an infra soft continuous map and let ξ,B be an infra soft compact subset of A~. Suppose that δk,B:k∈K is an infra soft open cover of fφξ,B. Then ξ,B⊆~⋃~k∈Kfφ−1δk,B and fφ−1δk,B is an infra soft open set for each k∈K. By hypotheses of infra soft compactness of ξ,B, we obtain ξ,B⊆~⋃~k=1nfφ−1δk,B. Therefore, fφξ,B⊆~⋃~k=1nfφfφ−1δk,B⊆~⋃~k=1nδk,B. This ends the proof that fφξ,B is infra soft compact.☐Following similar arguments, one can prove the case between parentheses.Corollary 44.
The property of being an infra soft compact (infra soft Lindelöf) space is preserved under an infra soft homeomorphism, i.e., it is an infra soft topological property.In the next result, we explain that the properties of infra soft compactness and infra soft Lindelöfness transmit from infra soft topology to its parametric infra topologies under an extended condition.Theorem 45.
IfA,Ω,B is an extended infra soft compact (resp., extended infra soft Lindelöf) space, then A,Ωβ is infra compact (resp., infra Lindelöf) for each β∈B.Proof.
LetHk:k∈K be an infra open cover of A,Ωβ. Consider Θ=δk,B:k∈K as a family of soft set which is defined as δkβ=Hk and δkβ′=A for each β′≠β. Since Ω is extended, Θ is an infra soft open cover of A~. By the hypothesis of infra soft compactness, we obtain A~=⋃~k=1nδk,B. This implies that A=⋃k=1nδkβ=⋃k=1nHk which proves that A,Ωβ is an infra compact space.☐Following similar arguments, one can prove the case between parentheses.The converse of the theorem above fails as the next example illustrates.Example 46.
Consider ISTSsA1,Ω1,ℕ and A2,Ω2,ℝ as given in Example 31. It is clear that A1,Ω1n is an infra compact space for all n∈ℕ and A2,Ω2r is an infra Lindelöf space for all r∈ℝ, whereas A1,Ω1,ℕ is not infra soft compact and A2,Ω2,ℝ is not infra soft Lindelöf.Note that an extended infra soft compactness (resp., extended infra soft Lindelöfness) ofA,Ω,B implies that B is finite (resp., countable).In the next result, we determine under which conditions the converse of Theorem45 hold.Theorem 47.
LetA,Ω,B be extended such that B is finite (resp., countable). Then, A,Ω,B is infra soft compact (resp., extended infra soft Lindelöf) space iff A,Ωβ is infra compact (resp., infra Lindelöf) for each β∈B.Proof.
The necessary part was proved in Theorem45. To prove the sufficient part, consider Θ=δk,B:k∈K as an infra soft open cover of A,Ω,B. Then, A=⋃k∈Kδkβ for each β∈B. By hypothesis of infra compactness (resp., resp., infra Lindelöfness), we obtain A=⋃k∈Iδkβ for each β∈B, where I is finite (resp., countable). Now, we take from Θ all infra soft open sets which (at least) one of their β-components belong to δkβ:k∈I. Hence, we obtain the desired result.☐Proposition 48.
LetA,Ω,B be stable. Then, A,Ω,B is infra soft compact (resp., infra soft Lindelöf) space iff A,Ωβ is infra compact (resp., infra Lindelöf).Proof.
Straightforward.☐Theorem 49.
ξ,B is infra soft compact (resp., infra soft Lindelöf) subset of A,Ω,B iff a soft subspace ξ,B,Ωξ,B,B is infra soft compact (resp., infra soft Lindelöf).Proof.
Necessity: considerδk,B:k∈K as an infra soft open cover of ξ,B,Ωξ,B,B. Then, for each δk,B∈Ωξ,B, there exists πk,B∈Ω such that δk,B=πk,B⋂~ξ,B. Now, πk,B:k∈K is an infra soft open cover of ξ,B. By hypothesis, ξ,B⊆~⋃~k=1nπk,B. Therefore, ξ,B⊆~⋃~k=1nπk,B⋂~ξ,B=⋃~k=1nδk,B. Thus, ξ,B,Ωξ,B,B is infra soft compact.
Sufficiency: considerδk,B∈Ω:k∈K as an infra soft open cover of ξ,B. Then, δk,B⋂~ξ,B:k∈K is an infra soft open cover of ξ,B,Ωξ,B,B. By hypothesis, ξ,B⊆~⋃~k=1nδk,B⋂~ξ,B. Therefore, ξ,B⊆~⋃~k=1nδk,B. Thus, ξ,B is infra soft compact.
Following similar arguments, one can prove the case between parentheses.☐Theorem 50.
The finite product of infra soft compact (resp., infra soft Lindelöf) spaces is infra soft compact (resp., infra soft Lindelöf).Proof.
LetA1,Ω1,B1 and A2,Ω2,B2 be two infra soft Lindelöf spaces. Consider ξk,B1×B2:k∈K as an infra soft open cover of A1~×A2~. Now, for each k∈Kξk,B1×B2=δk,B1×λk,B2, where δk,B1∈Ω1 and λk,B2∈Ω2. Then A1~×A2~=⋃~k∈Kξk,B1×B2⊆~⋃~k∈Kδk,B1×⋃~k∈Kλk,B2. That is, A1~=⋃~k∈Kδk,B1 and A2~=⋃~k∈Kλk,B2. By hypothesis, there are two countable sets I and J such that A1~=⋃~i∈Iδi,B1 and A2~=⋃~j∈Jλj,B2. Thus, A1~×A2~=⋃~i∈Iδi,B1×⋃~j∈Jλj,B2=⋃~k∈I⋃Jξk,B1×B2. Since I⋃J is a countable set, we obtain the desired result.☐Following similar arguments, one can prove the theorem in the case of infra soft compact.Theorem 51.
Every uncountable (resp., infinite) subset of an infra soft Lindelöf (resp., infra soft compact) space has an infra soft limit point.Proof.
Letδ,B be an uncountable subset of A,Ω,B which is an infra soft Lindelöf space. Suppose that no soft point of A~ is an infra soft limit point of δ,B. Then for each Pβα∈A~, there is an infra soft open set λαk,B containing Pβα such that λαk,B⋂~δ,B\Pβα=Φ. Now, a family Λ=λαk,B:k∈K forms an infra soft open cover of A~. Since A is infra soft Lindelöf, there is a countable set I such that A~=⋃~k∈Iλαk,B. Therefore, A has at most countable soft points of δ,B. This means that δ,B is countable which contradicts the uncountability of δ,B. Hence, δ,B has an infra soft limit point.☐Following similar arguments, one can prove the case between parentheses.


## 4. Fixed Point Theorem in Infra Soft Topological Spaces

Through this portion, we aim to introduce the concept of fixed soft points in the frame of ISTSs and establish its master properties. We present interesting findings that associated fixed soft points with infra soft compact spaces. Finally, the transmission of fixed soft points from infra soft topology to classical infra topology and vice versa is investigated.Theorem 52.
LetCn:n∈ℕ be a family of soft sets in an infra soft compact space A,Ω,B. Then ⋂~n∈ℕCn≠Φ~ provided that the following three conditions are satisfied:(i)
Cn≠Φ~ for each n∈ℕ(ii)
Cn is an infra soft closed set for each n∈ℕ(iii)
Cn+1 is a subset of Cn for each n∈ℕProof.
Suppose that⋂~n∈ℕCn=Φ~. Then, ⋃~n∈ℕCnc=A~. According to (ii), we obtain Cnc:n∈ℕ forms an infra soft open cover of A~. By the hypothesis of infra soft compactness, there exist k1,k2,⋯,kj∈ℕ, k1<k2<⋯<kj such that A~=Ck1c⋃~Ck2c⋃~⋯⋃~Ckjc. It comes from (iii) that Ckj⊆~A~=Ck1c⋃~Ck2c⋃~⋯⋃~Ckjc=Ck1⋂~Ck2⋂~⋯⋂~Ckjc=Ckjc. This is a contradiction. Hence, ⋂~n∈ℕCn≠Φ~, as required.☐Proposition 53.
LetA,Ω,B be an infra soft compact and infra soft T2-space such that Ω is closed under finite union. If fφ:A,Ω,B⟶A,Ω,B is infra soft continuous, then fφ has a unique soft point Pβα∈A~.Proof.
LetCn=fφCn−1=fφnA~ for each n∈ℕ, where C1=fφA~ be a family of soft sets in A,Ω,B. Obviously, Cn+1⊆~Cn for each n∈ℕ. Since fφ is an infra soft continuous map, Cn is an infra soft compact set for each n∈ℕ, and since A,Ω,B is an infra soft T2-space such that Ω is closed under finite union, Cn is an infra soft closed set for each n∈ℕ. Putting δ,B=⋂~n∈ℕCn, according to Theorem 52, we find δ,B is a nonnull soft set. Note that fφδ,B=fφ⋂~n∈ℕfφnA~⊆~⋂~n∈ℕfφn+1A~⊆~⋂~n∈ℕfφnA~=δ,B. To prove that δ,B⊆~fφδ,B, suppose that there exists a Pβα∈δ,B such that Pβα∈fφδ,B. Let Dn=fφ−1Pβα⋂~Cn. It is clear that Dn≠Φ~ and Dn⊆~Dn−1 for each n∈ℕ. Now, Dn is an infra soft closed set for each n∈ℕ, and by Theorem 52, there is a soft point Pργ such that Pργ∈fφ−1Pβα⋂~Cn. Therefore, Pβα=fφPργ∈fφδ,B; this is a contradiction. Hence, fφδ,B=δ,B which means we obtain the desired result.☐Definition 54.
An ISTSA,Ω,B is said to have a fixed soft point property if every infra soft continuous map fφ:A,Ω,B⟶A,Ω,B has a fixed soft point, i.e., there exists Pβα∈A such that fφPβα=Pβα.Proposition 55.
The property of being a fixed soft point is an infra soft topological property.Proof.
LetA1,Ω1,B1 and A2,Ω2,B2 be an infra soft homeomorphism. Then there is a bijective soft map fφ:A1,Ω1,B1⟶A2,Ω2,B2 such that fφ and fφ−1 are infra soft continuous. Consider that A1,Ω1,B1 has a fixed soft point property, i.e., every infra soft continuous map fφ:A1,Ω1,B1⟶A1,Ω1,B1 has a fixed soft point. Now, let hφ:A2,Ω2,B2⟶A2,Ω2,B2 be an infra soft continuous map. Obviously, hφ∘fφ:A1,Ω1,B1⟶A2,Ω2,B2 is an infra soft continuous map. Also, fφ−1∘hφ∘fφ:A1,Ω1,B1⟶A1,Ω1,B1 is an infra soft continuous. Since A1,Ω1,B1 has a fixed soft point property, fφ−1hφfφPβα=Pβα for some Pβα∈A~, so fφfφ−1hφfφPβα=fφPβα. This leads to hφfφPβα=fφPβα. Thus, fφPβα is a fixed soft point of hφ. This ends the proof that A2,Ω2,B2 has a fixed soft point property.☐Theorem 56.
ConsiderΩ as an extended infra soft topology over A1. A soft map fφ:A1,Ω,B1⟶A2,℧,B2 is infra soft continuous iff a crisp map f:A1,Ωβ⟶A2,℧φβ is infra continuous.Proof.
⇒: let H be an infra open set in A2,℧φβ. Then there is an infra soft open set δ,B2 in A2,℧,B2 such that δφβ=H. Since fφ is an infra soft continuous map, fφ−1δ,B2 is an infra soft open set. Therefore, a soft set fφ−1δ,B2=f−1δ,B1 in A1,Ω,B1 is given by fφ−1δβ=f−1δφβ for each β∈B1. By hypothesis, Ω is extended, a set f−1δφβ=f−1H in A1,Ωβ is infra open. Hence, f is an infra continuous map.
⇐: let ξ,B2 be an infra soft open set in A2,℧,B2. Then a soft set fϕ−1ξ,B2=fφ−1ξ,B1 in A1,Ω,B1 is given by fφ−1ξβ=f−1ξφβ for each β∈B1. Since f is an infra continuous map, a subset f−1ξφβ in A1,Ωβ is infra open. By hypothesis, Ω is extended, we obtain fφ−1ξ,B2 is an infra soft open set in A1,Ω,B1. Hence, fφ is an infra soft continuous map.☐Proposition 57.
If an ISTSA,Ω,B has a fixed soft point, then A,Ωβ has a fixed point property for all β∈B.Proof.
Consider an ISTSA,Ω,B as a fixed soft point property. This means that any infra soft continuous map fφ:A,Ω,B⟶A,Ω,B has a fixed soft point. Say, Pβα. It comes from Theorem 56 that f:A,Ωβ⟶A,Ωφβ is infra continuous. Since Pβα is a fixed soft point of fφ, we obtain fα=α. Thus, f has a fixed point.☐


## 5. Conclusion and Future Work

This article is aimed at completely presenting and scrutinizing the concepts and notions in the frame of ISTSs. So, we have initiated two sorts of covering properties in the frame of ISTSs which consider two classifications of soft spaces. Also, we have established the concept of fixed soft points and elucidated essential properties, in particular, those induced from infra soft compact spaces.We sum our accomplishments through this work in the following:(1)
Introduce the concepts of infra soft compact and infra soft Lindelöf spaces(2)
Offer some illustrative examples to point out the relationships between these two spaces and validate the obtained findings and notes(3)
Explain the interchangeable property of these concepts from infra soft topology to classical infra topology and vice versa(4)
Investigate these concepts under finite product of soft spaces and infra soft homeomorphism(5)
Display the concept of fixed soft points and reveal its basic featuresOur blueprint in the forthcoming papers is as follows.(1)
Establish new sorts of covering properties like almost infra soft compact (Lindelöf) and nearly infra soft compact (Lindelöf) spaces(2)
Investigate metric spaces in the content of ISTSs which will open a door for more studies on fixed soft points(3)
Carry out further investigations in the areas of rough set theory and infra topologies(4)
Generalize ISTSs to infra soft bitopological and infra soft ordered topological spaces



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