Weak Forms of Soft Separation Axioms and Fixed Soft Points

Realizing the importance of separation axioms in classifications of topological spaces and studying certain properties of fixed points, we formulate new soft separation axioms, namely tt-soft and tt-soft b-regular spaces. Their definitions depend on three factors: soft b-open sets, total belong and total non-belong relations. In fact, they are genuine generalizations of p-soft -spaces in the cases of i = 0, 1, 2. With the help of examples, we study the relationships between them as well as with soft and soft b-regular spaces. Some interesting properties of them are obtained under the conditions of soft hyperconnected and extended soft topological spaces. Also, we show that they are preserved under finite product soft spaces and soft -homeomorphism mappings. Finally, we introduce a concept of b-fixed soft points and investigate its main properties.


Introduction
Molotdsov's soft set [1] was established in 1999 as a new technique for tackling real-life problems that suffer from imprecision and uncertainty. Molotdsov [1] investigated the merits of soft sets in comparison to probability theory and fuzzy set theory. The soft settheoretic concepts were then introduced and investigated by a number of researchers, and many applications of soft sets were made in different disciplines such as decision-making [2], engineering [3] and medical science [2].
In 2011, Shabir and Naz [4] used soft sets defined over an initial universal set with a fixed set of parameters to introduce the concept of soft topological space. Researchers then studied several concepts of classical topological spaces through soft topological spaces and discussed the validity of some known topological results in soft topological spaces. Soft compactness was defined and studied by [27], in 2012. Hida [6] distinguished between two types of soft compactness depending on the belong relation. Al-shami et al. [7] studied almost soft compact and approximate soft compact spaces as extensions of a soft compact space. [8] utilized soft b-open sets to generalize soft compactness. The behavior of soft closed sets in a soft Hausdorff was revised in [9] and many of the allegation results of soft separation axioms were corrected in [10] with the help of concrete examples. Al-shami and Kočinac [11] proved that the enriched and extended soft topologies are coincide. This result is very important in the study of the interrelationships between soft topological space and its parametric topological spaces.
The relations of belong and non-belong given in [4] were utilized in the studies of soft set and soft topology. However, the authors of [12], in 2018, came up new relations of belong and non-belong between an ordinary points and soft set, namely partial belong and total non-belong relations. In fact, these relations widely open the door to study and redefine many soft topological notions. This leads to obtain many fruitful properties and changes that can be seen significantly on the study of soft separation axioms as it was shown in [12][13][14]. These relations were studied in the contents of bipolar soft sets [15] and double framed soft sets [16]. Das and Samanta [17] studied the concept of a soft metric based on the soft real set and soft real numbers given in [18]. Wardowski [19] tackled the fixed point in the setup of soft topological spaces. Abbas et al. [20] presented soft contraction mappings and established a soft Banach fixed point theorem in the framework of soft metric spaces. Recently, many researchers explored fixed point findings in soft metric type spaces, see, for example, [21][22][23][24].
One of the main ideas that helps to prove some properties and eliminate some problems on soft topology is the concept of a soft point. It was first defined by Zorlutuna et al. [25] in order to study the interior points of a soft set and soft neighborhood systems. Then [18] and [26] redefined soft points concurrently to discuss soft metric spaces. In fact, the recent definition of a soft point shows similarity between many set-theoretic properties and their counterparts on soft setting. Two types of soft topologies, namely enriched soft topology and extended soft topology were introduced in [27] and [26], respectively. The equivalence between these two topologies have been recently proved in [11]. Recently, Al-shami [28,29] has presented some practical applications of soft compact and soft separation axioms, and Kočinac et al. [30] have studied Menger spaces in soft setting.
We organized this paper as follows: Section 2 recalls the basic principles of soft sets and soft topologies. In Section 4, we introduce the concepts of tt-soft bT i (i = 0, 1, 2, 3, 4) and ttsoft b-regular spaces with respect to the ordinary points by using total belong and total nonbelong relations. The relationships between them and their main properties are discussed with the help of interesting examples. In Section 5, we explore a b-fixed soft point theorem and study some main properties. In particular, we conclude under what conditions b-fixed soft points are preserved between a soft topological space and its parametric topological spaces. Section 6 concludes the paper.

Preliminaries
To well understand the results obtained in this study, we shall recall some basic concepts, definitions and properties from the literature.

Definition 2.1 ([1]):
Let X be the universal set and M be a set of parameters. A pair (G, M) is said to be a soft set over X provided that G is a map of M into the power set 2 X .
In this study, we use a symbol G M to refer a soft set instead of (G, M) and we identify it as ordered pairs G M = {(m, G(m)) : m ∈ M and G(m) ∈ 2 X }.
A family of all soft sets defined over X with M is denoted by S(X M ).

Definition 2.2 ([31]):
A soft set G M is said to be a subset of a soft set H M , denoted by The soft sets G M and H M are said to be soft equal if each one of them is a subset of the other.

Definition 2.3 ([1, 12]):
Let G M be a soft set over X and x ∈ X. We say that: Remark 2.1: Let G M be a soft set over X and x ∈ X. We say that: Otherwise, it is said to be uncountable (resp. infinite). (

Definition 2.10 ([4]):
A family τ of soft sets over X under a fixed set of parameters M is said to be a soft topology on X if it satisfies the following.   To study the properties that preserved under soft b -homeomorphism maps, the concept of a soft b-irresolute map will be presented in this work under the name of a soft b -continuous map.

Definition 2.19:
A soft topology τ on X is said to be: (i) an enriched soft topology [27] if all soft sets G M such that G(m) = ∅ or X are members of τ ; (ii) an extended soft topology [26] Al-shami and Kočinac [11] proved the equivalence of enriched and extended soft topologies and obtained many useful results that help to study the relationships between soft topological spaces and their parametric topological spaces.
defines a soft topology on X with a fixed set of parameters M.

Definition 2.20 ([5]):
The soft topological space (X, τ , M) given in the above proposition is said to be the sum of soft topological spaces and is denoted by (⊕ i∈I X i , τ , M).

Further Properties of Soft b-open Sets
In the following results, we prove under what condition the family of soft b-open subsets of (X, τ , M) forms a soft topology over X that is finer than τ . In fact, it will help us to study some properties of soft b-separation axioms and soft b-compact spaces, see, for example, Theorem (4.7) and Proposition (4.13).

b-soft Separation Axioms
By making use of the relations of total belong and total non-belong, we define new type of soft separation axioms, namely tt-soft bT i (i = 0, 1, 2, 3, 4). We provide some examples to elucidate the relationships between them and to show some of their properties. Furthermore, we study the interrelations of them and topological and additive properties.
First of all, we see that it is necessary to classify containment into several categories as it is shown in remark below. Factually, this classification will play a vital role in redefining many soft theoretic-set and soft topological concepts, in particular, the concepts of soft interior and closure operators, soft compactness and soft separation axioms.  As a direct consequence, we infer that every soft b-closed and soft b-open subsets of a soft b-regular space must be stable. However, this matter does not hold on the tt-soft b-regular spaces because we replace a partial non-belong relation by a total non-belong relation. Therefore a tt-soft b-regular space need not be stable.

Proof:
The proofs of (i) and (ii) follow from the fact that a total non-belong relation implies a partial non-belong relation ∈.
To prove (iii), it suffices to prove that a soft The following examples clarify that the converse of the above proposition is not always true. Before we show the relationship between tt-soft bT i -spaces, we need to prove the following useful lemma.  The following examples show that the converse of the above proposition is not always true. In what follows, we establish some properties of tt-soft bT i and tt-soft b-regular. Proof:       In the following examples, we show that there is no a relationship between soft topological space and their parametric topological spaces in terms of separation axioms if a condition of an extended soft topological space given in the above theorem does not exist.
Let f ϕ : (X, τ , A) → (Y, τ , B) be a soft b-continuous map and v = w ∈ X. Since f is injective, then there are two distinct points x and y in Y such that f (v) = x and f (w) = y. Since (Y, τ , B) is a p-soft T 2 -space, then there are two disjoint soft open sets G B and F B such that x ∈ G B and y ∈ F B . Now, In a similar way, one can prove the following result.
In a similar way, one can prove the following result.  We complete this section by discussing some interrelations between tt-soft bT i -spaces (i = 2, 3, 4) and soft b-compact spaces.

Proposition 4.24: A stable soft b-compact subset of a tt-soft bT 2 -space is soft b-closed.
Proof: It follows from Proposition (3.5) and Remark (4.1).   Proof: Let the given conditions be satisfied. Then for each P is soft bT 3 , then it is tt-soft bT 1 . Hence, it is tt-soft bT 4 .

b-fixed Soft Points of Soft Mappings
In this section, we introduce a b-fixed soft point property and investigate some main features, in particular, those are related to parametric topological spaces. (i) B n = for each n ∈ N; (ii) B n is a soft b-closed set for each n ∈ N; (iii) B n+1 ⊆B n for each n ∈ N.
Then n∈N B n = .
Proof: Suppose that n∈N B n = . Then n∈N B c n = X. It follows from (ii) that {B c n : n ∈ N} is a soft b-open cover of X. By hypothesis of soft b-compactness, there exist i 1 This yields a contradiction. Thus we obtain the proof that n∈N B n = . Proof: Let {B 1 = g ϕ ( X) and B n = g ϕ (B n−1 ) = g n ϕ ( X) for each n ∈ N} be a family of soft subsets of (X, τ , M). It is clear that B n+1 ⊆B n for each n ∈ N. Since g ϕ is soft b -continuous, then B n is a soft b-compact set for each n ∈ N and since (X, τ , M) is soft bT 2 , then B n is also a soft b-closed set for each n ∈ N. It follows from Theorem (5.1) that (H, M) = n∈N B n is a non null soft set. Note that g ϕ (H, M) = g ϕ ( n∈N g n ϕ ( X)) ⊆ n∈N g n+1 Obviously, C n = and C n ⊆C n−1 for each n ∈ N. By Theorem C n is a soft b-closed set for each n ∈ N; and by Theorem (5.1), there exists a soft point P y m such that P y m ∈ g −1 ϕ (P x m ) B n . Therefore P x m = g ϕ (P Before we investigate a relationship between soft topological space and their parametric topological spaces in terms of possessing a fixed (soft) point, we need to prove the following result.  (G(φ(m))) for each m ∈ M. By hypothesis, τ is an extended soft topology on X, we obtain from Theorem (2.5) that a subset g −1 (G(φ(m))) = g −1 (U) of (X, τ m ) is b-open. Hence, a map g is b -continuous.
Sufficiency: Let G M be a soft b-open subset of (Y, τ , M). Then from Definition (2.8), it follows that a soft subset g −1 φ (G M ) = (g −1 φ (G)) M of (X, τ , M) is given by g −1 φ (G)(m) = g −1 (G(φ(m))) for each m ∈ M. Since a map g is b -continuous, then a subset g −1 (G(φ(m))) of (X, τ m ) is b-open. By hypothesis, τ is an extended soft topology on X, we obtain from Theorem (2.5) that g −1 φ (G M ) is a soft b-open subset of (X, τ , M). Hence, a soft map g ϕ is soft b -continuous. It follows from the above theorem that g m : (X, τ m ) → (X, τ φ(m) ) is b -continuous. Since P x m is a fixed soft point of g ϕ , then it must be that g m (x) = x. Thus g m has a fixed point. Hence, we obtain the desired result.
Sufficiency: Let (X, τ m ) has the property of an b-fixed point for each m ∈ M. Then every b -continuous map g m : (X, τ m ) → (X, τ φ(m) ) has a fixed point. Say, x. It follows from the above theorem that g ϕ : (X, τ , M) → (X, τ , M) is soft b -continuous. Since x is a fixed point of g m , then it must be that g ϕ (P x m ) = P x m . Thus, g ϕ has a fixed soft point. Hence, we obtain the desired result.

Conclusion
One of the reasons of diversity of soft separation axioms is the variety of belong and nonbelong relations between ordinary points and soft set. This article is devoted to studying separation axioms and fixed points in soft setting. First, we have introduced new soft separation axioms with respect to ordinary points by using total belong and total non-belong relations. This way of definition helps us to generalize existing comparable properties via general topology and to remove a strict condition of the shape of soft open and closed subsets of soft b-regular spaces. In general, we have studied their main properties and showed the interrelations between them with help of interesting examples. Second, we have defined b-fixed soft point theorem and investigated its basic properties. Finally, we hope that the concepts initiated herein will find their applications in many fields soon.

Human participants
This article does not contain any studies with human participants performed by any of the authors.