Fuzzy parametrized fuzzy soft topology

: Recently, researches have contributed a lot towards fuzziﬁcation of Soft Set Theory. In this paper, we introduce the topological structure of fuzzyfying soft sets called fuzzy parametrized fuzzy soft sets. We deﬁne the notion of quasi-coincidence for fuzzy parametrized fuzzy soft sets and investigated basic properties of it. We study the closure, interior, base, continuity and compactness in the content of fuzzy parametrized fuzzy soft topological spaces


Introduction
In 1965, Zadeh [32] generalized the usual notion of a set with the introduction of fuzzy set. The theory of fuzzy set has been successfully applied to many areas such as many real life problems in uncertain, ambiguous environment. Chang [15] defined the fuzzy topology and introduced many topological notions in fuzzy setting, in 1968.
In 1999, Molodtsov [26] introduced the soft set theory which is a new approach for modelling uncertainty and presented that soft set can be applied to several areas, such as game theory, perron integrations, smoothness of functions and so on.
Recently, researchers have combined fuzzy set and soft set to generalize the spaces and to solve more complicated problems. By this way, many interesting applications of soft set theory have been expanded. First combination of fuzzy set and soft set is fuzzy soft set and it was given by Maji and et al [24]. Then fuzzy soft set theory has been applied in several directions, such as topology (e.g. [3,5,29,30]), various algebraic structures (e.g. [4,20]) and especially decision making (e.g. [18,22,28,31]). Second combination of fuzzy set and soft set was given by Ç agman and et al. [8] and called it as fuzzy parametrized soft set (as shortly FPS set). In that paper, Ç agman and et al. defined operations on FPS sets and improved several results. After that, Ç agman and Deli [9,11] applied FPS sets to define some decision making methods and applied these methods to problems that contain uncertainties and fuzzy object. The third and the last one was also given by Ç agman and et al. [10] and it is called fuzzy parametrized fuzzy soft set (as shortly FPFS set). Then they defined operations on FPFS sets and improved an method to solve some decision making problems.
In the present paper, we consider the topological structure of FPFS sets. Firstly, we give some basic ideas of FPFS sets and also studied results. We define FPFS quasi-coincidence, as a generalization of quasi-coincidence in fuzzy manner c ⃝ 2016 BISKA Bilisim Technology [25] and use this notion to characterize concepts of FPFS closure and FPFS base in FPFS topological spaces. We also introduce the notion of mapping on FPFS classes and investigate the properties of FPFS images and FPFS inverse images of FPFS sets. We define FPFS topology in Chang's sense. We study the FPFS closure and FPFS interior operators and properties of these concepts. Lastly we define FPFS continuous mappings and we show that image of a FPFS compact space is also FPFS compact.
This paper is the fundamental study on FPFS topological spaces. One can use results deducted from this paper in the theory topological structures.

Preliminaries
Throughout this paper X denotes initial universe, E denotes the set of all possible parameters which are attributes, characteristic or properties of the objects in X, and the set of all subsets of X will be denoted by P(X).
In the other words, a soft set is a parametrized family of subsets of the set X. For each e ∈ E, the set F(e) may be considered as the set of e-elements of the soft set (F, E).

Definition 3.
[10] Let A be a fuzzy set over E. A fuzzy parametrized fuzzy soft set (FPFS) F A on the universe X is defined as follows: where the function f A : E → I X is called approximate function of F A such that f A (e) = 0 if µ A (e) = 0.
From now on, the set of all FPFS sets over X will be denoted by FPFS(X, E).
(1) F A is called the empty FPFS set if µ A (e) = 0 and f A (e) = 0 for all every e ∈ E, denoted by F ∅ .
[10] Let F A , F B and F C ∈ FPFS(X, E). Then 3 Some properties of FPFS sets and FPFS mappings Definition 7. Let J be an arbitrary index set and F A i ∈ FPFS(X, E) for all i ∈ J.

is the FPFS set, defined by the membership and approximate functions
for every e ∈ E, respectively.
(2) The intersection of F A i 's, denoted by ∩ i∈J F A i , is the FPFS set, defined by the membership and approximate functions for every e ∈ E, respectively.

Proposition 2. Let J be an arbitrary index set and F A i ∈ FPFS(X, E) for all i ∈ J. Then
Proof.
This completes the proof. The other can be proved similarly Proof. This follows from the fact that any fuzzy set is the union of fuzzy points which belong to it [25].
, Then the following are true.

1) for F A ∈ FPFS(X, E), the image of F A under the f up is the FPFS set G S over Y defined by the approximate function,
(2) for G S ∈ FPFS(Y, K), then the pre-image of G S under the f up is the FPFS set F A over X defined by the approximate If u and p is injective, then the FPFS mapping f up is said to be injective. If u and p is surjective, then the FPFS mapping f up is said to be surjective. The FPFS mapping f up is called constant, if u and p are constant.

Theorem 1. Let X and Y crips sets F A , F
the equality holds if f up is injective.
This completes the proof.
This completes the proof.
This completes the proof. where p −1 (S) and p −1 (S c ) are fuzzy sets in E. This shows that the approximate functions of F B and F c A are equal. This completes the proof.
Then for all e ∈ E and x ∈ X, f A (e)(x) = g K (p(e))(u(x))) = 1. This shows that F A = F E .

FPFS topological spaces
Definition 13. A FPFS topological space is a pair (X, τ) where X is a nonempty set and τ is a family of FPFS sets over X satisfying the following properties: Theorem 2. Let (X, τ) be a FPFS topological space and τ ′ be family of all FPFS closed sets. Then; Proof. Straightforward.

Clearly, F A is the smallest FPFS closed set over X which contains F A .
Theorem 3. Let (X, τ) be a FPFS topological space and F A , F B ∈ FPFS(X, E). Then,

is a FPFS open set if and only if F
Proof. Similar to that of Theorem 3.
Theorem 6. Let (X, τ) be a FPFS topological space and F A ∈ FPFS(X, E). Then, Proof. We only prove (1). The other is similar. Example 3. If we consider the FPFS topology τ in Example 2, then one easily see that the family Conversely, If B is not a base for τ, then there exists a F A ∈ τ such that Then in both case, we obtain that e β x α qF A and e β x α qF C . Therefore, we have e β x α qF B for all F B ∈ B which contained in F A . This is a contradiction.
is a FPFS open and so f up is FPFS continuous. Proof. Straightforward. Proof. If C is a family of FPFS sets in a FPFS topological space (X, τ), then C is a cover of F E if and only if one of the following conditions holds:

Definition 21. A family C of FPFS sets is a cover of a FPFS set F A if F A ⊆ ∪ {F
Hence this shows that FPFS topological space is FPFS compact if and only if each family of FPFS open sets over X such that no finite subfamily covers F E , fails to be a cover, and this is true if and only if each family of FPFS closed sets which has the finite intersection property has a nonempty FPFS intersection.