Infra Soft Compact Spaces and Application to Fixed Point Theorem

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.


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 Çagman 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][20][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][23][24][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 Section 2, 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.

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

Soft Set Theory
Definition 1 (see [1]). Let B be a set of parameters, A a universal set, and 2 A the power set of A. A soft set over A is an ordered pair ðδ, BÞ, where δ : B ⟶ 2 A is a crisp map. We express the soft set as follows: ðδ, BÞ = fðβ, δðβÞÞ: β ∈ B and δðβÞ ∈ 2 A g.
A family of all soft sets over A under a set of parameters B is symbolized by SðA B Þ.
Definition 2 (see [6]). A soft set ðδ c , BÞ is called a complement of ðδ, BÞ provided that a map δ c : B ⟶ 2 A such that δ c ðβÞ = A \ δðβÞ for each β ∈ B.
Definition 3 (see [5]). If the image of each parameter of B under a map δ : B ⟶ 2 A is the universal set A, then ðδ, BÞ is called the absolute soft set over A. Its complement is called the null soft set. The absolute and null soft sets are symbolized byÃ and Φ, respectively.
Definition 5 (see [12,27]). If the image of one parameter, say β, under a map P : B ⟶ 2 A is a singleton set, say fαg, and the image of each parameter β ′ ∈ B \ fβg is the empty set, then a soft set ðP, BÞ is called a soft point over A. It is briefly symbolized by P α β .
Definition 6 (see [10,27]). There are two belong and two nonbelong relations between an element α ∈ A and a soft set ðδ, BÞ defined as follows.
Definition 8 (see [5]). The union of two soft sets ðδ, BÞ and ðξ, CÞ over A, symbolized by ðδ,BÞ S~ð ξ, CÞ, is a soft set ðλ, DÞ, where D = B ∪ C and a map λ : D ⟶ 2 A is given as follows: Definition 9 (see [28] 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.

Infra Soft Topological Spaces
Definition 14 (see [16]). A family Ω 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 triple ðA, Ω, BÞ 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 ðA, Ω, BÞ stable if all its infra soft open sets are stable and called finite (resp., countable) if A is finite (resp., countable).
We called Ω β a parametric infra topology.
Proposition 16 (see [16]). Suppose that Ψ = fΩ β g β∈B is a class of crisp infra topologies on A. Then, defines an infra soft topology on A.
The ISTS given in the above proposition is called an extended infra soft topology on A or an infra soft topology on A generated by Ψ.
Definition 20 (see [31] A property which is kept by any infra soft homeomorphism is said to be an infra soft topological property.
We called Ω given in the proposition above a product of infra soft topologies and ðA, Ω, BÞ a product of infra soft spaces.

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 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 Proposition 25 is not converse.
This ends the proof that ðξ, BÞ 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.
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.
Let Ω 1 and Ω 2 be two discrete infra soft topologies on a finite set A 1 and a countable set A 2 , respectively. Let the sets of natural numbers ℕ and real numbers ℝ be sets of parameters. It is clear that ðA 1 , Ω 1 , ℕÞ is not infra soft compact in spite of A 1 being finite, and ðA 2 , Ω 2 , ℝÞ is not infra soft Lindelöf in spite of A 2 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.
Theorem 33. An ISTS ðA, Ω, BÞ is infra soft compact (resp., infra soft Lindelöf) iff every family of infra soft closed subsets of ðA, Ω, BÞ, satisfying the finite (resp., countable) intersection property, has, itself, a nonnull intersection. gÞg is infra soft closed. But it is not infra soft compact in spite of ðN , Ω, BÞ being an infra tt-soft T 2 (infra tp -soft T 2 , infra pt -soft T 2 , infra pp -soft T 2 )-space. Now, we investigate under which conditions the wellknown relationship between closed sets and T 2 -spaces are satisfied in the frame of ISTSs.
Proposition 35. Let ðA, Ω, BÞ be an infra tt-soft Hausdorff space such that Ω is closed under a finite union. Then, every stable infra soft compact subset of ðA, Ω, BÞ is infra soft closed.

Journal of Function Spaces
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).
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.
Proof. Let fH k : k ∈ Kg be an infra open cover of ðA, Ω β Þ. Consider Θ = fðδ k , BÞ: k ∈ Kg as a family of soft set which is defined as δ k ðβÞ = H k and δ k ðβ ′ Þ = A for each β ′ ≠ β.
Since Ω is extended, Θ is an infra soft open cover ofÃ. By the hypothesis of infra soft compactness, we obtainÃ = S~n k=1 ðδ k , BÞ. This implies that A = S n k=1 δ k ðβÞ = S n k=1 H k which proves that ðA, Ω β Þ 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.
In the next result, we determine under which conditions the converse of Theorem 45 hold.
Proof. Straightforward. 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. Let ðA 1 , Ω 1 , B 1 Þ and ðA 2 , Ω 2 , B 2 Þ be two infra soft Lindelöf spaces. Consider fðξ k , B 1 × B 2 Þ: k ∈ Kg as an infra soft open cover of f By hypothesis, there are two countable sets I and J such that f Following similar arguments, one can prove the theorem in the case of infra soft compact.
Proof. Let ðδ, BÞ be an uncountable subset of ðA, Ω, BÞ which is an infra soft Lindelöf space. Suppose that no soft point ofÃ is an infra soft limit point of ðδ, BÞ. Then for each P α β ∈Ã, there is an infra soft open set ðλ α k , BÞ containing P α β such that ðλ α k ,BÞ T~ð δ, BÞ \ P α β = Φ. Now, a family Λ = fðλ α k , BÞ: k ∈ Kg forms an infra soft open cover ofÃ. Since A is infra soft Lindelöf, there is a countable set I such thatÃ = S~k ∈I ðλ α k , BÞ. Therefore, A has at most countable soft points of ðδ, BÞ. This means that ðδ, BÞ is countable which contradicts the uncountability of ðδ, BÞ. Hence, ðδ, BÞ has an infra soft limit point. ☐ Following similar arguments, one can prove the case between parentheses.

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. Let fC n : n ∈ ℕg be a family of soft sets in an infra soft compact space ðA, Ω, BÞ. Then T~n ∈ℕ C n ≠Φ provided that the following three conditions are satisfied: (i) C n ≠Φ for each n ∈ ℕ (ii) C n is an infra soft closed set for each n ∈ ℕ (iii) C n+1 is a subset of C n for each n ∈ ℕ Proof. Suppose that T~n ∈ℕ C n =Φ. Then, S~n ∈ℕ C c n =Ã. According to (ii), we obtain fC c n : n ∈ ℕg forms an infra soft open cover ofÃ. By the hypothesis of infra soft compactness, there exist k 1 , k 2 , ⋯, k j ∈ ℕ, k 1 < k 2 < ⋯ < k j such that Proof. Let fC n = f φ ðC n−1 Þ = f n φ ðÃÞ for each n ∈ ℕ, where C 1 = f φ ðÃÞg be a family of soft sets in ðA, Ω, BÞ. Obviously, C n+1 ⊆~C n for each n ∈ ℕ. Since f φ is an infra soft continuous map, C n is an infra soft compact set for each n ∈ ℕ, and since ðA, Ω, BÞ is an infra soft T 2 -space such that Ω is closed under finite union, C n is an infra soft closed set for each n ∈ ℕ. Putting ðδ, BÞ = T~n ∈ℕ C n , according to Theorem 52, we find ðδ, BÞ is a nonnull soft set. Note that f φ ðδ, BÞ = f φ ð T~n ∈ℕ f n φ ðA~ÞÞ ⊆~T~n ∈ℕ f n+1 φ ðA~Þ ⊆~T~n ∈ℕ f n φ ðÃÞ = ðδ, BÞ. To prove that ðδ,BÞ ⊆~f φ ðδ, BÞ, suppose that there exists a P α β ∈ ðδ, BÞ such that P α β ∈f φ ðδ, BÞ.
It is clear that D n ≠Φ and D n ⊆~D n−1 for each n ∈ ℕ. Now, D n is an infra soft closed set for each n ∈ ℕ, and by Theorem 52, there is a soft point P   Proof. Consider an ISTS ðA, Ω, BÞ as a fixed soft point property. This means that any infra soft continuous map f φ : ðA, Ω, BÞ ⟶ ðA, Ω, BÞ has a fixed soft point. Say, P α β . It comes from Theorem 56 that f : ðA, Ω β Þ ⟶ ðA, Ω φðβÞ Þ is infra continuous. Since P α β is a fixed soft point of f φ , we obtain f ðαÞ = α. Thus, f has a fixed point. ☐

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 features Our 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

Data Availability
No data were used to support this study.

Conflicts of Interest
The author declares no conflicts of interest.