The Relationship Between Fuzzy Soft and Soft Topologies

This paper attempts to forward both soft topology and fuzzy soft topology with a pioneering analysis of their mutual relationships. With each soft topology we associate a parameterized family of fuzzy soft topologies called its t-pushes. And each fuzzy soft topology defines a parameterized family of soft topologies called its t-throwbacks. Different soft topologies produce different t-pushes. But we prove by example that not all fuzzy soft topologies are characterized by their t-throwbacks. The import of these constructions is that some properties stated in one setting can be investigated in the other setting. Our conclusions should fuel future research on both fuzzy soft topology and soft topology.


Introduction
This paper lies at the crossroads of three disciplines, namely, topology, fuzzy set theory, and soft set theory.
A consensus has taken hold that topology as a welldefined mathematical discipline originates in the early 1900s, although antecedents like Euler's 1736 paper on the Seven Bridges of Königsberg can be traced back some centuries. The term ''topological space'' was coined by Felix Hausdorff [1] in 1914, although then both measure theory and topology were considered parts of set theory.
Nowadays the Dewey Decimal Classification assigns mathematics to division 510, with subdivisions for Algebra & Number theory, Arithmetic, Topology, Analysis, Geometry, Numerical analysis, and Probabilities & Applied mathematics.
The Weierstrass Extreme Value Theorem shows in its modern formulation that decision-making can enjoy the benefits of topological arguments. The result joins the forces of two topological concepts (compactness of a domain of alternatives and continuity of a real-valued function defined on it). Other disciplines like mathematical economics adopted this technique, e.g., in the form of the Bergstrom-Walker theorem [2,3] that states that any continuous acyclic relation defined on a compact topological space has a maximal element. Actually this set of sufficient conditions is also necessary [4].
It is also acknowledged that Zadeh [5] launched fuzzy set theory, and interest on fuzzy concepts mounted as new research was conducted. Topological notions were soon combined with fuzzy set theory and its extensions. Chang [6] defined fuzzy topology in 1968, and the notion was reformulated by Goguen [7] and Lowen [8]. Other models of uncertain knowledge have been combined with topological approaches too. For example, intuitionistic fuzzy topological spaces were defined in 1997 by Ç oker [9].
Topology has also exerted a gravitational pull among soft set theory researchers. Soft topology was launched by Shabir and Naz [10] in 2011 and it owes to the hybridization of soft set theory [11] with the axiomatics of a topology. Many contributions showcase the properties of soft topologies, and all too often their achievements consist of a replication of pre-existent ideas from topology. Interestingly, Das and Samanta [12], Nazmul and Samanta [13], and Zorlutuna et al. [14] set forth a notion of soft point which became crucial in the analysis of properties of soft interior points and soft neighborhood systems. Other definitions of soft point [15] and element [16] exist. The behavior of separation axioms has been translated to a soft topological framework (Shabir and Naz [10], Al-shami and El-Shafei [17], Hussain and Ahmad [18], and Terepeta [19]). The study of soft compactness owes especially to Aygünoglu and Aygün [15] and Zorlutuna et al. [14], and more recently, Al-shami et al. [20] have produced seven generalized classes of soft semi-compact spaces. Separability axioms are another remarkable element in the recent development of soft topology. Alcantud [21] used them to create well-behaved soft topological spaces. Kočinac et al. [22] have introduced and investigated soft Menger spaces.
Other authors fired off articles on a wide range of combinations of topology with the spirit of soft set theory. Fuzzy soft topological spaces were defined by Tanay and Kandemir [23]. They have been used as a valuable tool in multi-criteria group decision making (Khameneh et al. [24]), thus extending the applicability of fuzzy soft sets (e.g., in medical practice [25]) to a topological context. Other relevant references include Khameneh et al. [26] and Roy and Samanta [27]. Relatedly, Riaz and Tehrim [28] introduced bipolar fuzzy soft topologies, and in addition Riaz et al. [29] defined hesitant fuzzy soft topologies. Also, Riaz et al. [30] have used the extended approach by N-soft sets (Fatimah et al. [31]) in order to design N-soft topologies. Notably, both [29] and [30] give applications in decision-making.
However, researchers seem to have turned a blind eye to the relationships between soft topologies and fuzzy soft topologies to the extent that they have become two disparate fields of research. In order to fill this research gap we perform an extensive analysis of their relationships. We believe that this approach has its upside, even though the output is a mostly theoretical article. A strong theoretical foundation of this aspect of topological soft set theory should forward both disciplines and increase their appeal. Being a pioneering approach, it can also boost the analysis of relationships among other types of topological soft structures like those mentioned above.
This research paper consists of five sections. Section 2 gives preliminary notions from soft set theory, soft topology, fuzzy soft set theory, and fuzzy soft topology. We give a long list of yardstick examples of both soft and fuzzy soft topological spaces, some of which are new in this literature. In Sect. 3 we formally identify a soft topology with a fuzzy soft topology. In addition, with each soft topology we associate a parameterized family of fuzzy soft topologies called its t-pushes, in such way that different soft topologies produce different t-pushes. This process contains the aforementioned identification of a soft topology with a fuzzy soft topology as an extreme case, namely, its identification coincides with its 1-push. With this tool we lay the grounds of the investigation of which properties of a soft topology are transmitted to the fuzzy soft topologies thus defined. Section 4 studies the converse problems which we happen to find more interesting, plus the relationships among both processes. In particular, we investigate some properties of a fuzzy soft topology that pass on to the soft topologies called its t-throwbacks. Examples give explicit constructions that illustrate these ideas. In particular, we prove by example that different fuzzy soft topologies can produce the same collection of t-throwbacks. The goal of Sect. 5 is to conclude this paper with a succinct but precise recapitulation of our main findings, and to give some lines for future research. At the end of the paper we summarize notation and conventions used along the article to facilitate the reading of our results.

Preliminaries
In this paper X denotes a fixed nonempty set (the universe of discourse), and E denotes a set of attributes of the alternatives. For each set U, we denote by PðUÞ the set formed by all the subsets of U, i.e., the set of parts of the set U; and F ðUÞ denotes the set formed by the fuzzy subsets of U, i.e., the set of mappings U À! ½0; 1.
In Sect. 2.1, we briefly recall the fundamental concepts of soft set theory and soft topology. Section 2.2 does the same with fuzzy soft set theory and fuzzy soft topology. The reader can consult Munkres [32] and Willard [33] for the basic notions in topology.
Sometimes we need to distinguish clearly between topologies and (fuzzy) soft topologies. In this case we use the alternative terms ''crisp topology'' or ''point-set topology'' to emphasize the classical notion of topology on a set.

Elements of Soft Set Theory and Soft Topology
According to [11], a soft set on X is a pair (F, E), where E consists of all the attributes that are needed to characterize the elements of X, and F is a mapping F : E À! PðXÞ: Thus a soft set over X is a parameterized family of subsets of X. In mathematical terms, a soft set on X is a multi-function (also called correspondence, point-to-set mapping, or multi-valued mapping) F from the set of characteristics E to X. The set of all soft sets on X characterized by attributes E shall be represented by SS E ðXÞ or simply SS(X) if E is common knowledge.
For each e 2 E, the set F(e) is a subset of X that is sometimes denoted as (F, E)(e) for higher accuracy. It is the set of e-approximate elements of X, also called the subset of X approximated by e.
Suppose that (F, E) is a soft set on X such that F(e) is finite (resp., countable) for each e 2 E. Then we say that (F, E) is a finite (resp., a countable) soft set on X [12,13].
A soft set (F, E) is frequently expressed as fðe; FðeÞÞ : e 2 Eg. It can also be represented in tabular form when the sets X and E are finite [34].
Two extreme examples are the null and the absolute soft sets on X.  [10,19]: soft points in [15] are soft sets (F, E) for which there exists A E and x 2 X such that FðeÞ ¼ fxg for all e 2 A. Soft spots satisfy this definition from [15] as well.
In this messy situation, we reserve the term 'soft point' for the concept used in [10,19] that we have denoted by (x, E). This is the notion that we shall need to define important axioms in soft topology (cf., Definitions 3 and 4 below).

Remark 1
The respective extensions of the concepts of union and intersection of two soft sets to arbitrary collections of soft sets are direct.
We are in a position to define soft topology on X (with a fixed set of attributes E). It consists of a family s SS E ðXÞ that satisfies the following definition: Table 1 The tabular representation of the null soft set, a soft spot ðfo 1 g e ; EÞ, a soft point ðo 1 ; EÞ, a soft point (F, E) in the sense of [14], and the full soft set ðX ¼ fo 1 ; o 2 g; E ¼ fe; e 0 gÞ Definition 1 [10,35] A soft topology s on X is s SS E ðXÞ, a collection of soft sets on X, that satisfies: (1) U;X 2 s.
(2) The union of soft sets in s belongs to s.  Furthermore, general procedures exist that produce soft topologies from point-set topologies. First we recall a very direct construction: Table 2 The tabular representation of the concepts of soft union, intersection and complement in Example 1 (F 1 , E) e 1 e 2 e 3 (F 2 , E) e 1 e 2 e 3 (F 1 t F 2 , E) e 1 e 2 e 3 (F 1 u F 2 , E) e 1 e 2 e 3 (ðF 1 Þ c , E) e 1 e 2 e 3 Definition 2 [19,21] Suppose that R ¼ fR e g e2E is a family of point-set topologies on X. Then defines a soft topology on X. It is called the soft topology on X generated by R. We can also write sðRÞ ¼ sðRÞ if R e ¼ R e 0 ¼ R for each e; e 0 2 E.
Alcantud [21] applies the construction above to prove that s c ¼ sðR c Þ, i.e., the cofinite soft topology can be obtained from the cofinite crisp topology on X, by the process in Definition 2.
The second procedure that generates soft topologies from crisp topologies relies on the behavior of soft open bases. Let us summarize some preliminary facts.
Quite naturally, soft bases are defined as collections of soft sets that generate soft topologies by taking the set formed by the arbitrary soft unions of their elements. So B s is a soft base for the soft topology s when every ðF; EÞ 2 s can be expressed as a union of soft sets from B Notice that the two constructions of soft topologies defined above are closely related. Suppose that we fix a base for a crisp topology. We can produce both a crisp topology and a soft open base from it. Then we can apply Definition 2 to this crisp topology, and we can also produce a soft topology from the soft open base. The conclusion is that the respective soft topologies that we obtain are the same [21,Theorem 3].
Section 1 explains that the literature has provided many axioms like soft compactness, soft connectedness, soft separability, or soft separation axioms. Here we shall only refer to some soft separation axioms.
Terepeta [19,Theorem 4] proves that if R is a T 0 , resp., T 1 , T 2 , point-set topology on X, then sðRÞ is T 0 , resp., T 1 , T 2 . This is also true for other separation axioms like T 3 and T 4 that we do not refer to in this paper.

Elements of Fuzzy Soft set Theory and Fuzzy
Soft Topology According to [37], a fuzzy soft set on X is a pair (f, E), where E consists of all the attributes that are needed to characterize the elements of X, and f is a mapping f : E À! F ðXÞ: The set of all fuzzy soft sets on X with attributes E will be denoted by FSS E ðXÞ or simply FSS(X) if E is common knowledge. A fuzzy soft set on X associates with each e 2 E, the fuzzy subset f(e) of X consisting of the members that (partially) satisfy e. By convenience, the notation f e ðxÞ is frequently adopted to denote f(e)(x), the degree of membership of x to the fuzzy subset of E defined by e. For higher accuracy we sometimes denote the fuzzy subset f(e) of X by (f, E)(e). As in the case of soft set, fuzzy soft sets can also be represented in tabular form when both X and E are finite.
The null and the absolute fuzzy soft sets on X are respectively denoted byÛ ¼ ðÛ; EÞ andX ¼ ðX; EÞ. The null fuzzy soft setÛ is characterized byÛðeÞðxÞ ¼ 0 for each e 2 E and x 2 X, whereas the absolute fuzzy soft setX satisfiesXðeÞðxÞ ¼ 1 for each e 2 E and x 2 X [37].

Remark 2
The respective extensions of the concepts of union and intersection of two fuzzy soft sets to arbitrary collections of fuzzy soft sets are direct.
A special type of fuzzy soft set is formed by fuzzy soft points [26, Definition 6.2]. Let us fix x 2 X and t 2 ð0; 1. Then the fuzzy soft point ðfxg t ; EÞ is the fuzzy soft set for which For illustration, notice that the fuzzy soft set ðf 1 ; EÞ in Example 5 is the fuzzy soft point ðfo 1 g 0:7 ; EÞ.

Remark 3
The use of the term 'fuzzy soft point' is not free from difficulties either. We recommend [26, Section 6] for a discussion of earlier definitions.
Notice that ðfxg t ; EÞŶðf ; EÞ is equivalent to f ðeÞðxÞ > t for each e 2 E.
With these ideas in mind, we are ready to define fuzzy soft topology on X (the set of attributes being E). It is formed by a familyŝ FSS E ðXÞ that satisfies the properties in our next definition: Definition 5 [27] A fuzzy soft topologyŝ on X iŝ s FSS E ðXÞ, a family of fuzzy soft sets on X such that: (1)Û;X 2ŝ.
(2) The union of fuzzy soft sets inŝ belongs toŝ. Prior to this definition, a slightly different notion of fuzzy soft topology had been proposed [23]. Anyhow, both approaches owe to the procedure for defining fuzzy topologies given in [6]. Here we adopt the notion of fuzzy soft topology in the sense of [27] that satisfies the adapted version of the De Morgan's laws.
For   Table 4 The tabular representation of the concepts of fuzzy soft union and intersection in Example 5 and are respectively called the t-upper and t-lower fuzzy soft topologies on X. (iii) For each ðf ; EÞ 2 FSS E ðXÞ: defines the upper contour fuzzy soft topology derived from (f, E). (iv) For each ðf ; EÞ 2 FSS E ðXÞ: Example 7 Let us now establish some comparisons among fuzzy soft topologies: 1. Let us fix t 2 ½0; 1. Then it must be the case that for each x 2 X,ŝ t ðxÞ is finer thanŝ½t " because ðf ; EÞ 2ŝ½t " implies ðf ; EÞ 2ŝ t ðxÞ . Besides, if we fix t 2 ½0; 1, then for each x 2 X and e 2 E,ŝ t ðx;eÞ is finer thanŝ t ðxÞ because ðf ; EÞ 2ŝ t ðxÞ implies ðf ; EÞ 2ŝ t ðx;eÞ . 2. When ðf ; EÞ 2 FSS E ðXÞ is such that f ðeÞðxÞ > t for some x 2 X and e 2 E, thenŝ t ðx;eÞ is also finer than s½f " . If in fact ðf ; EÞ 2 FSS E ðXÞ is such that for some x 2 X, f ðeÞðxÞ > t for each e 2 E, thenŝ t ðxÞ is also finer thanŝ½f " . 3. When ðf ; EÞ 2 FSS E ðXÞ is such that f ðeÞðxÞ > t for all x 2 X and e 2 E, thenŝ½t " is also finer thanŝ½f " . 4. Let us fix t 2 ½0; 1. Then it must be the case that for each x 2 X,ŝ t ðxÞ is finer thanŝ½t # because ðf ; EÞ 2ŝ½t # implies ðf ; EÞ 2ŝ t ðxÞ . Besides, if we fix t 2 ½0; 1, then for each x 2 X and e 2 E,ŝ t ðx;eÞ is finer thanŝ t ðxÞ because ðf ; EÞ 2ŝ t ðxÞ implies ðf ; EÞ 2ŝ t ðx;eÞ . 5. When ðf ; EÞ 2 FSS E ðXÞ is such that f ðeÞðxÞ 6 t for some x 2 X and e 2 E, thenŝ t ðx;eÞ is also finer than s½f # . If in fact ðf ; EÞ 2 FSS E ðXÞ is such that for some x 2 X, f ðeÞðxÞ 6 t for each e 2 E, thenŝ t ðxÞ is also finer thanŝ½f # . 6. When ðf ; EÞ 2 FSS E ðXÞ is such that f ðeÞðxÞ 6 t for all x 2 X and e 2 E, thenŝ½t # is also finer thanŝ½f # . Let us now recall some definitions of fuzzy soft separation axioms. One should bear in mind that the property ðfxg t ; EÞŶðf ; EÞ is equivalent to f ðeÞðxÞ > t for each e 2 E.
(iii) T 2 when for all x; y 2 X, x 6 ¼ y, and all t; t 0 2 ð0; 1, there are disjoint ðf ; EÞ; ðg; EÞ 2ŝ with ðfxg t ; EÞŶðf ; EÞ and ðfyg t 0 ; EÞŶðg; EÞ. From each fuzzy soft topology on a set X, some related fuzzy soft topologies on X can be defined: Proposition 1 Suppose thatŝ is a fuzzy soft topology on X. Then The proof of Proposition 1 is routine.

Fuzzy Soft Topologies Generated by Soft Topological Spaces
This section is dedicated to the generation of fuzzy soft topologies on a set from a given soft topology on the same set. First we consider a predictable fact: the existence of a natural embedding that allows one to regard any soft topological space as a fuzzy soft topological space. Despite its reasonability we are not aware of any formal proof of this remarkable immersion. Then we produce a more general construction that embeds the previous one as a particular case of a collection of fuzzy soft topological spaces derived from any soft topological space. Finally in this section, we demonstrate that some properties of a soft topological space can be transferred to the fuzzy soft topological spaces constructed in that way.

Soft Topologies 'are' Fuzzy Soft Topologies
We proceed to show how one can design a natural identification of soft topologies with fuzzy soft topologies. As explained above this result will be superseded by the more general approach taken in Sect. 3.2. However the embedding is important in its own right and we believe that it deserves an explicit presentation and terminology. The constructions in this section and also in Sect. 4 rely on the following 'folk' theorem:

Theorem 1
There is an embedding of SS E ðXÞ within FSS E ðXÞ.
The statement claims that there is an embedding. By this we mean a mapping g : SS E ðXÞ À! FSS E ðXÞ that is injective and preserves unions and intersections.
Proof of Theorem 1 To define one such embedding g, we recall that for each A 2 PðXÞ, the characteristic function of With its aid we can identify subsets of X or members of PðXÞ, with fuzzy subsets of X, since we can also identify v A with the fuzzy subset v A : X À! ½0; 1.
Define the mapping g as follows: for each ðF; EÞ 2 SS E ðXÞ, gðF; EÞ ¼ ðf F ; EÞ 2 FSS E ðXÞ is such that It is now simple to check that g is injective, and also that it satisfies the following properties: 1. gðUÞ ¼Û and gðXÞ ¼X, 2. gðu i2I ðF i ; EÞÞ ¼û i2I ðgðF i ; EÞÞ, and 3. gðt i2I ðF i ; EÞÞ ¼t i2I ðgðF i ; EÞÞ. h We are ready to define an identification of soft topologies with fuzzy soft topologies on the same universe:

Definition 7
The soft topology s on X defineŝ s g ¼ fgðF; EÞjðF; EÞ 2 sg FSS E ðXÞ. We say thatŝ g is the fuzzy soft topology associated with s.
The claim in this definition leans on the following fact whose proof is routine:

Proposition 2
Suppose that s is a soft topology on X. Thenŝ g is a fuzzy soft topology on X.
Proof Theorem 1 guarantees the requirements for a fuzzy soft topology. h Thus the message that Definition 7 renders is that although soft topologies cannot be fuzzy soft topologies (they are respective subsets of different sets), they can be naturally identified with fuzzy soft topologies.
We now produce some fundamental properties that endorse the good performance of the identification of a soft topology with a fuzzy soft topology (cf., Definition 7). Proposition 3 Suppose that s 1 ; s 2 are soft topologies on X. Then 1. If s 2 is finer than s 1 , thenŝ g 2 is finer thanŝ g 1 . 2. If B is a soft base for s 1 then gðBÞ ¼ fgðF; EÞjðF; EÞ 2 Bg is a fuzzy soft base forŝ g 1 .
Proof To prove the first claim, suppose ðf ; EÞ 2ŝ g 1 . By definition, there must exist ðF; EÞ 2 s 1 such that ðf ; EÞ ¼ gðF; EÞ. Now ðF; EÞ 2 s 2 because s 2 is finer than s 1 , which yields ðf ; EÞ ¼ gðF; EÞ 2ŝ g 2 by definition. To prove the second claim, suppose ðf ; EÞ 2ŝ g 1 . By definition, there exists ðF; EÞ 2 s 1 such that ðf ; EÞ ¼ gðF; EÞ. Because B is a soft base for s 1 , there are fðF i ; EÞji 2 Ig B such that ðF; EÞ ¼ t i2I ðF i ; EÞ. Therefore ðf ; EÞ ¼ gðF; EÞ ¼ gðt i2I ðF i ; EÞÞ ¼ t i2I ðgðF i ; EÞÞ by Theorem 1. This proves the claim that gðBÞ ¼ fgðF; EÞjðF; EÞ 2 Bg is a fuzzy soft base forŝ g 1 . h Definition 2 produces the soft topology generated by a family of point-set topologies over X. In view of Proposition 2, now we can assure that the family produces a fuzzy soft topology over X as well: defines a fuzzy soft topology on X. We call it the fuzzy soft topology on X generated by R.
We also writeŝðRÞ ¼ŝðRÞ when R e ¼ R e 0 ¼ R for each e; e 0 2 E.

The t-Pushes of a Soft Topological Space
We now proceed to define a fuzzy soft topological space associated with any soft topology and t 2 ½0; 1. We shall call it the t-push of the soft topological space. In this way we can produce a parameterized family of fuzzy soft topologies associated with any fixed soft topology. Likewise, (19) shows that ðH ½t ; EÞðeÞðxÞ ¼ t when x 2 HðeÞ ¼ FðeÞ \ GðeÞ, otherwise ðH ½t ; EÞðeÞðxÞ ¼ 0.
Both expressions coincide thus the desired equality has been proven. h We take advantage of the technique of the proof of Lemma 1 in order to look into the performance of Eq. (19) with respect to unions: Likewise, (19) shows that ðG ½t ; EÞðeÞðxÞ ¼ t if x 2 GðeÞ ¼ [ i2I F i ðeÞ, and ðG ½t ; EÞðeÞðxÞ ¼ 0 otherwise.
Both expressions coincide thus the desired equality has been proven. h Now we are ready to define a family of fuzzy soft topologies from any soft topology. Notice that the universe does not vary in this process.
Theorem 2 Suppose that s SS E ðXÞ is a soft topology. Let us fix t 2 ½0; 1. Then the familŷ defines a fuzzy soft topology on X.
Definition 8 We say that the fuzzy soft topologyŝ ½t defined in Theorem 2 is the t-push associated with s.

Remark 4
Observe that because ðF ½1 ; EÞ ¼ gðF; EÞ for each ðF; EÞ 2 SS E ðXÞ andX ½1 ¼X, the 1-push of a soft topology s coincides with the fuzzy soft topology associated with it (cf., Definition 7). Formally speaking:ŝ ½1 ¼ŝ g holds true for each soft topology s on X.
Lemmas 1 and 2 are the key steps in the proof of Theorem 2: Proof of Theorem 2 Let us check the three properties in Definition 5.
(1) For each t 2 ½0; 1, U ½t ¼Û 2ŝ ½t ; andX 2ŝ ½t by definition. It is also easy to prove that the following fundamental relationship between soft topologies carries forward to all its t-pushes. The case of the 1-push has been considered before in Proposition 3: Proposition 5 Suppose that s 1 ; s 2 SS E ðXÞ are soft topologies on X. If s 2 is finer than s 1 , thenŝ ½t 2 is finer than s ½t 1 for each t 2 ½0; 1.

Separation Axioms: An Analysis of the t-Pushes Associated with a Soft Topology
Now we demonstrate that one can infer properties of the tpushes associated with a soft topology from its behavior. We do not intend to perform an exhaustive analysis, as there would be too many properties to investigate. Here we concentrate on the axioms given in Definitions 3 and 4 for illustration of this course of reasoning. They will be related to the concepts in Definition 6.
In this analysis we must be aware that except in very special cases, the t-pushes associated with a soft topology contradict the basic soft separation axioms in Definition 6.
To be more precise, the situation is strikingly different if we consider t-pushes with either t\1 or t ¼ 1.
First suppose that s is a soft topology, jXj [ 1, and t 2 ½0; 1Þ. Then it must be the case thatŝ ½t is neither T 0 nor T 1 nor T 2 . It suffices to argue for T 0 . When t\1, fix any t 0 [ t and consider two distinct fuzzy soft points ðfxg t 0 ; EÞ, ðfyg t 0 ; EÞ with x 6 ¼ y. Then it is just impossible to produce ðf ; EÞ 2ŝ ½t ¼ fðF ½t ; EÞjðF; EÞ 2 sg [ fXg such that either ðfxg t 0 ; EÞŶðf ; EÞ or ðfyg t 0 ; EÞŶðf ; EÞ, except ðf ; EÞ ¼X which does not separate one fuzzy soft point from the other.
Consider now the case of the 1-push associated with a soft topology that satisfies the axioms given in Definitions 3 and 4. Then we can prove that it satisfies the corresponding property, in the following sense: Proposition 6 Suppose that s is a soft topology that satisfies T 0 , resp., T 1 , T 2 . Thenŝ ½1 satisfies T 0 , resp., T 1 , T 2 .
Proof We prove the first claim. Therefore suppose that s is a T 0 soft topology. To prove thatŝ ½1 also satisfies T 0 , let us fix x 6 ¼ y, x; y 2 X, and t; t 0 2 ð0; 1. By assumption, either there is ðF; EÞ 2 s with x 2 ðF; EÞ and y 6 2 ðF; EÞ, or there is ðG; EÞ 2 s with y 2 ðG; EÞ and x 6 2 ðG; EÞ. We give the argument for the first case, the other being symmetrical.
The other two statements can be proven with routine modifications of the argument for T 0 . For the purpose of proving the claim about T 2 , notice that due to both Lemma 1 and U ½1 ¼Û, ðF; EÞuðG; EÞ ¼ U implies ðF ½1 ; EÞûðG ½1 ; EÞ ¼Û. h

Soft Topologies Induced by Fuzzy Soft Topologies
This section investigates soft topologies that are directly related to a given fuzzy soft topology. Our achievements are as follows. First we show that any fuzzy soft topology on X induces a collection of soft topologies on X indexed by (0, 1] that we call its t-throwbacks (cf., Sect. 4.1). Then we prove that unfortunately, this collection does not allow us to retrieve the original fuzzy soft topology (cf., Sect. 4.3). And that it is possible to guarantee that certain properties of a fuzzy soft topology are transmitted to its tthrowbacks (cf., Sect. 4.4). As in the case of Sect. 3, our fundamental construction leans on an auxiliary technical concept. This notion appeared in Feng et al. [38,Definition 4.1] under the term t-level soft set of a fuzzy soft set, and it allows us to associate a soft set on X with each fuzzy soft set on X and each membership degree t 2 ½0; 1. Hence for each ðf ; EÞ 2 FSS E ðXÞ and t 2 ½0; 1 we define ðf ½t ; EÞ 2 SS E ðXÞ by the expression Observe thatÛ ½t ¼ U when t 2 ð0; 1. The next lemma recalls some other basic properties: Lemma 3 Let us fix ðf ; EÞ; ðf 0 ; EÞ 2 FSS E ðXÞ. 3. If ðf ; EÞŶðf 0 ; EÞ then ðf ½t ; EÞYðf 0 ½t ; EÞ, for each t 2 ½0; 1.
Proof The first claim is trivial.
To prove the second claim, let us fix an arbitrary e 2 E. Then x 2 f ½t 0 ðeÞ is equivalent to f ðeÞðxÞ > t 0 , which implies f ðeÞðxÞ [ t hence x 2 f ½t ðeÞ. This proves ðf ½t 0 ; EÞYðf ½t ; EÞ.
Proof Let us fix e 2 E. In order to check the required set equality t i2I ðf i ½t ; EÞðeÞ ¼ ðg ½t ; EÞðeÞ, which boils down to [ i2I f i ½t ðeÞ ¼ g ½t ðeÞ, we observe the following string of equivalences.
The fact x 2 [ i2I f i ½t ðeÞ is equivalent to the existence of i 2 I such that x 2 f i ½t ðeÞ, which by (21), is equivalent to the existence of i 2 I such that f i ðeÞðxÞ > t. Now this is equivalent to maxff i ðeÞðxÞ : i 2 Ig > t, which is in turn equivalent to gðeÞðxÞ > t by definition of (g, E). Finally, the last fact is equivalent to x 2 g ½t ðeÞ by (21). h In addition, for all x 2 X and t 0 2 ð0; 1, the application of Eq. (21) with any t 2 ð0; t 0 to the fuzzy soft point ðfxg t 0 ; EÞ produces the soft point (x, E).

Every Fuzzy Soft Topology Generates Soft Topological Spaces
The association that Eq. (21) produces gives raise to the following construction of soft topologies from fuzzy soft topologies: We give a name to the construction above:

Definition 9
We say that s ½t defined in Theorem 3 is the t-soft topology derived from the fuzzy soft topologyŝ. We also say that s ½t is the t-throwback ofŝ.
As it turns out, the argument of the proof of Theorem 3 uses Lemmas 4 and 5: Proof of Theorem 3 Let us check the three properties in Definition 1.
( The construction of t-throwbacks has both practical and theoretical consequences. The next example shows that the t-throwbacks of some fuzzy soft topological spaces are soft topological spaces. Then Proposition 7 establishes that finer fuzzy soft topologies produce t-soft topologies that are finer.
Let us compute the family of all t-throwbacks ofŝ in the particular case where ðf 1 ; EÞYðf 2 ; EÞ are defined as in Table 5 (notice X ¼ fo 1 ; o 2 g and E ¼ fe 1 ; e 2 g).
The following fundamental relationship between fuzzy soft topologies carries forward to all its t-throwbacks: Proposition 7 Suppose thatŝ 1 ;ŝ 2 FSS E ðXÞ are fuzzy soft topologies on X. Ifŝ 2 is finer thanŝ 1 , then s 2 ½t is finer than s 1 ½t for each t 2 ð0; 1.

Some Relationships Involving t-Throwbacks and t-Pushes
The constructions we have defined in Sects. 3 and 4 are related by some fundamental properties.
First we can observe that when we start with a soft topology, then we identify it with a fuzzy soft topology (Definition 7), and afterwards we derive any t-soft topology from it (Definition 9), we end up with the original soft topology: Proposition 8 Suppose that s is a soft topology on X. Then ðŝ g Þ ½t ¼ s for each t 2 ð0; 1.
Proof The proof follows from a routine application of the concepts. Let us fix t 2 ð0; 1. Then the following string of equivalences hold true: Purely intuitively speaking, Proposition 8 states that any soft topology coincides with all the t-throwbacks that it generates, when regarded as a fuzzy soft topology. So it coincides with all the t-throwbacks of its 1-push.
Of course it is not generally true that all the t-throwbacks that a fuzzy soft topology generates coincide. We have shown this fact in Example 9.
Proposition 8 is a marked instance of a more general property whose proof is very similar. Consider the following statement: Proposition 9 Suppose that s is a soft topology on X. Then ðŝ ½t Þ ½t 0 ¼ s for each t; t 0 2 ð0; 1 such that t > t 0 .
To prove this claim along the lines of the proof of Proposition 8, we just need to notice the following special fact: Lemma 6 Suppose ðF; EÞ 2 SS E ðXÞ and fix t 2 ð0; 1. Denote ðf ; EÞ ¼ ðF ½t ; EÞ. Then for any t 0 2 ð0; 1 such that t > t 0 , it must be the case that ðf ½t 0 ; EÞ ¼ ðF; EÞ.
A relationship in the other direction is ruled out because Example 10 below asssures that the following weak relationship cannot be improved: Lemma 7 Suppose ðf ; EÞ 2 FSS E ðXÞ and fix t 2 ð0; 1. Denote ðF; EÞ ¼ ðf ½t ; EÞ. Then for any t 0 2 ð0; 1 such that t > t 0 , it must be the case that ðF ½t 0 ; EÞŶðf ; EÞ.
Proof Let us fix e 2 E. In order to check the required inequality F ½t 0 ðeÞðxÞ 6 f ðeÞðxÞ for each x 2 X, we only need to observe that h The next counterexample confirms that even if we restrict ourselves to the case t ¼ t 0 , the conditions of Lemma 7 do not guarantee the conclusion ðF ½t 0 ; EÞ ¼ ðf ; EÞ: Example 10 Consider ðf 1 ; EÞ 2 FSS E ðXÞ defined in Example 9. There we checked that its 0.6-throwback is ðF 1 ; EÞ defined in Example 9. However the 0.6-push of ðF 1 ; EÞ does not coincide with ðf 1 ; EÞ.

Non-uniqueness of the t-Throwbacks Associated with a Fuzzy Soft Topology
We now provide a counterexample that shows that unfortunately, the information that the t-throwbacks associated with a fuzzy soft topology embody is not sufficient to retrieve such fuzzy soft topology.
To this purpose we need to show that two different fuzzy soft topologies produce exactly the same collection of tthrowbacks. We do this in the following example:  We can go through the computations producing the tthrowbacks associated withŝ 0 as we did in Example 9. And we shall conclude that they coincide with the t-throwbacks associated withŝ defined in Example 9, for each t 2 ½0; 1. However,ŝ andŝ 0 are different.

Properties of a Fuzzy Soft Topology that Transfer to its t-Throwbacks
It is natural to wonder which properties of a fuzzy soft topology can be inherited by the t-soft topologies that it generates, i.e., by its t-throwbacks. The study parallels the analysis performed in Section 3.3 for property shifts in the other direction, i.e., from a soft topology to its t-pushes. Thus like we did in that case, for the purpose of illustration of the technique we concentrate on the implications of the fundamental separation axioms (cf., Definition 6).
Proof We prove the first claim. Therefore suppose thatŝ is a T 0 fuzzy soft topology and fix t 2 ð0; 1. To prove that s ½t also satisfies T 0 , let us fix x 6 ¼ y, x; y 2 X. Consider the distinct fuzzy soft points ðfxg t ; EÞ and ðfyg t ; EÞ. By assumption, either there is ðf ; EÞ 2ŝ with ðfxg t ; EÞŶðf ; EÞ such that ðfyg t ; EÞŶðf ; EÞ is false, or there is ðg; EÞ 2ŝ with ðfyg t ; EÞŶðg; EÞ such that ðfxg t ; EÞŶðg; EÞ is false. We give the argument for the first case, the other being symmetrical.
The fact that ðf ; EÞ 2ŝ does not satisfy ðfyg t ; EÞŶðf ; EÞ is equivalent to the existence of e 2 E such that f ðeÞðyÞ\t. This means that y 2 f ½t ðeÞ for all e 2 E is false, hence y 6 2 ðf ½t ; EÞ, which concludes the proof that s ½t satisfies T 0 .
The other statements can be proven with routine modifications of the argument for T 0 . For the purpose of proving the claim about T 2 , notice that due to Lemma 3 and U ½t ¼ U because t 2 ð0; 1, the fact that ðf ; EÞûðg; EÞ ¼Û implies ðf ½t ; EÞuðg ½t ; EÞ ¼ U. h

Conclusion
To the best of our knowledge this paper conducts the first systematic analysis of the relationships between soft topologies and fuzzy soft topologies. We have shown that this long-forgotten problem produces a rich variety of fundamental results thus it deserves further investigation. Let us summarize our main findings. Consider first a soft topology s. study can inspire comparable analysis concerning for example bipolar fuzzy soft topologies, hesitant fuzzy soft topologies, or N-soft topologies. We expect to return to these issues in the future.
6 Notation and Conventions u, t, Y denote the soft intersection, union, and inclusion.
u,t,Ŷ denote the fuzzy soft intersection, union, and inclusion.
( U;X 2 s denote the null and absolute soft sets, respectively.
U;X 2 s denote the null and absolute fuzzy soft sets, respectively.
s g is the fuzzy soft topology associated with s (cf., Definition 7).
s ½t is the t-push associated with the soft topology s (cf., Theorem 2). It is a fuzzy soft topology.
s ½t is the t-throwback of the fuzzy soft topologyŝ (cf., Theorem 3). It is a soft topology.
Funding Open Access funding provided thanks to the CRUE-CSIC agreement with Springer Nature. The author is grateful to the Junta de Castilla y León and the European Regional Development Fund (Grant CLU-2019-03) for the financial support to the Research Unit of Excellence ''Economic Management for Sustainability'' (GECOS).

Data Availability Not applicable.
Code Availability Not applicable.

Declarations
Conflict of interest The author declares no conflict of interest.
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