CATS of soft topological spaces

Abstract

The goal of this study is to consider the natural interrelations between all of the (fuzzy) soft topological spaces in the categorical viewpoint. For this reason, the categories of (fuzzy) soft topological spaces are constructed and then the relations between these categories are observed by using the defined functors.

Keywords

(Fuzzy) Soft set topology (fuzzy) soft topology functor category

1 Introduction

General topology was one of the first branches of pure mathematics to which fuzzy sets have been applied. Three years after 1965, Zadeh’s paper [5] appeared, Chang first gave the definition of fuzzy topology which is a family of fuzzy sets satisfying three certain axioms [23]. In his paper, he made an attempt to develop basic topological notions for such spaces. This paper was followed by other mathematicians and intensity of research in the area of fuzzy topology has increased sharply. At present there are hundreds of publications in this area. In 1976, Lowen [14] proposed a new notion of fuzzy topological space which differs from Chang’s definition. He replaced the first axiom of Chang’s definition which is “(1) empty fuzzy set and universal fuzzy set are in the fuzzy topology” by a sharper condition “(1)^′ all constant fuzzy sets are in the fuzzy topology”. An essential feature of Lowen type fuzzy topological spaces is that as easily verified, the constant mappings of such spaces are a priori continuous (that is, they are morphisms in the category of fuzzy topological spaces FTOP) which need not to be continuous in Chang type fuzzy topological spaces.

In 1999, Molodtsov [17] initiated a novel concept of soft set theory, which is a completely new approach for modeling vagueness and uncertainty. He showed several applications of this theory in solving many practical problems in economics, engineering, social science, medical science, etc. Later Maji et al. [15] introduced the concept of fuzzy soft sets which combines fuzzy sets and soft sets. From then on, many authors have contributed to (fuzzy) soft set theory in the different fields such as algebra, topology and etc., see [1–4 , 22].

Soft topology is a relatively new and promising domain which can lead to the development of new mathematical models and innovative approaches that will significantly contribute to the solution of complex problems in natural sciences. According to this idea, in this paper, we present the notions, constructions and some results of (fuzzy) soft topological spaces. We begin with recalling the definitions of such spaces and deal with their basic properties by some examples. Then we construct categories of (fuzzy) soft topological spaces and we investigate the interrelations between the categories of these topological spaces by constructing functors between such categories.

2 Preliminaries

In this section we give some fundamental properties of soft sets and fuzzy soft sets. Throughout this paper, X refers to an initial universe, E is the set of all parameters for X, I^X is the set of all fuzzy sets on X (where for λ ∈ I = [0, 1] , $\bar{λ} (x) = λ$ , for all x ∈ X) .

2.1 Soft sets

Definition 2.1. [16] A soft set F_A over the universe X is a mapping from the parameter set E to the power set P (X) =2^X of X, i.e., F_A : E → 2^X, where F_A (e)¬ = ∅, if e ∈ A ⊆ E and F_A (e) =∅ if e ∉ A.

The subscript A in the notation F_A indicates where the image of F_A is non-empty.

A soft set can also be defined by the set of ordered pairs

F_A = {(e, F_A (e)) : e ∈ E, F_A (e) ∈ P (X)} [6].

The value F_A (e) is a set called e-component of the soft set F_A, for all e ∈ E.

From now on, the family of all soft sets over X will be denoted by (2^X) ^E .

Example 2.1. [18] For a topological space (X, τ) , if F (x) is the family of all open neighborhoods of a point x ∈ X, i.e., F (x) = {V ∈ τ ∣ x ∈ V} , then the ordered pair (F, X) is indeed a soft set over 2^X .

Definition 2.2. [6] Let F_A, G_B be two soft sets over X, then

(1) F_A is a soft subset of G_B, denoted F_A ⊑ G_B if F_A (e) ⊆ G_B (e), for each e ∈ E. F_A and G_B are soft equal, denoted by F_A = G_B if F_A ⊑ G_B and G_B ⊑ F_A.

(2) union of F_A and G_B is a soft set H_C defined by H_C (e) = F_A (e) ∪ G_B (e), ∀e ∈ E, where C = A ∪ B.

(3) intersection of F_A and G_B is a soft set H_C defined by H_C (e) = F_A (e) ∩ G_B (e), ∀e ∈ E, where C = A ∩ B.

(4) the soft set F_∅ is called a null soft set, denoted by Φ. Here, F_∅ (e) =∅ for every e ∈ E.

(5) if F_E (e) = X, ∀e ∈ E, then F_E is called an absolute soft set, denoted by $\tilde{E}$ .

(6) if F_A (e) = X, ∀e ∈ A, then F_A is called an A-absolute soft set, denoted by $\tilde{A}$ .

(7) the complement of F_A, denoted by $F_{A}^{c}$ is defined by $F_{A}^{c} (e) = X \ F_{A} (e)$ , for all e ∈ E.

Definition 2.3. [13] Let φ : X → Y and ψ : E → F be two functions, where E and F are parameter sets for the crisp sets X and Y, respectively. Then the pair φ_ψ is called a soft mapping from (2^X) ^E into (2^Y) ^F and denoted by φ_ψ : (2^X) ^E → (2^Y) ^F.

Definition 2.4. [13] Let F_A and G_B be two soft sets over X and Y, respectively and let φ_ψ be a soft mapping from (2^X) ^E into (2^Y) ^F.

(1) The image of F_A under the soft mapping φ_ψ is the soft set over Y, denoted by φ_ψ (F_A) and defined by $φ_{ψ} (F_{A}) (k) = {\begin{matrix} ⋃_{e \in ψ^{- 1} (k) \cap A} φ (F_{A} (e)), & if ψ^{- 1} (k) \neg = \emptyset; \\ \emptyset, & otherwise . \end{matrix}$

(2) The pre-image of G_B under the soft mapping φ_ψ is the soft set over X denoted by $φ_{ψ}^{- 1} (G_{B})$ and defined by $φ_{ψ}^{- 1} (G_{B}) (e) = {\begin{matrix} φ^{- 1} (G_{B} (ψ (e))), & if ψ (e) \in B; \\ \emptyset, & otherwise . \end{matrix}$

The soft mapping φ_ψ is called injective, if φ and ψ are injective. The soft mapping φ_ψ is called surjective, if φ and ψ are surjective.

Definition 2.5. [2] Let φ_ψ be a soft mapping from (2^X) ^E into (2^Y) ^F and $φ_{ψ^{*}}^{*}$ be a soft mapping from (2^Y) ^F into (2^Z) ^L. Then the composition of these mappings from (2^X) ^E into (2^Z) ^L is defined as follows: $φ_{ψ^{*}}^{*} \circ φ_{ψ} : = (φ^{*} \circ φ)_{ψ^{*} \circ ψ},$ where ψ : E → F and ψ^* : F → L .

2.2 Fuzzy soft sets

Definition 2.6. [1, 15] f_A is called a fuzzy soft set on X, where f is a mapping from E into I^X, i.e., (f_A) _e : = f_e is a fuzzy set on X, for each e ∈ E and $f_{e} = \bar{0}$ , if e ∉ A, where $\bar{0}$ is zero function on X. f_e, for each e ∈ E, is called e-component of the fuzzy soft set f_A.

A fuzzy soft set can also be denoted by $f_{A} = {f_{e} : e \in E} .$

From now on, the family of all fuzzy soft sets over X will be denoted by (I^X) ^E.

Definition 2.7. [15] Let f_A, g_B ∈ (I^X) ^E, then

(1) we say that f_A is a fuzzy soft subset of g_B and write f_A ⊑ g_B if f_e ≤ g_e, for each e ∈ E. f_A and g_B on X are called equal if f_A ⊑ g_B and g_B ⊑ f_A.

(2) union of f_A and g_B on X is the fuzzy soft set h_C = f_A ⊔ g_B, where C = A ∪ B and h_e = f_e ∨ g_e, for each e ∈ E. That is, $h_{e} = f_{e} \lor \bar{0} = f_{e}$ for each e ∈ A \ B, $h_{e} = \bar{0} \lor g_{e} = g_{e}$ for each e ∈ B \ A and h_e = f_e ∨ g_e, for each e ∈ A ∩ B.

(3) intersection of f_A and g_B on X is the fuzzy soft set h_C = f_A ⊓ g_B, where C = A ∩ B and h_e = f_e ∧ g_e, for each e ∈ E.

(4) the complement of f_A is denoted by $f_{A}^{c}$ , where f^c : E → I^X is a mapping given by $f_{e}^{c} = \bar{1} - f_{e}$ , for each e ∈ E. Clearly ${(f_{A}^{c})}^{c} = f_{A}$ .

(5) (Null fuzzy soft set) f_E on X is called a null fuzzy soft set and denoted by Φ, if $f_{e} = \bar{0}$ , for each e ∈ E.

(6) (Absolute fuzzy soft set) f_E on X is called an absolute fuzzy soft set and denoted by $\tilde{E}$ , if $f_{e} = \bar{1}$ , for each e ∈ E. Clearly $(\tilde{E})^{c} = Φ$ and $Φ^{c} = \tilde{E}$ .

(7) (λ-absolute fuzzy soft set) f_E is called a λ-absolute fuzzy soft set and denoted by ${\tilde{E}}^{λ}$ , if $f_{e} = \bar{λ}$ , for each e ∈ E. Clearly, $({\tilde{E}}^{λ})^{c} = {\tilde{E}}^{1 - λ}$ .

Definition 2.8. [3, 12] Let f_A and g_B be two fuzzy soft sets over X and Y, respectively and let φ_ψ be a fuzzy soft mapping from (I^X) ^E into (I^Y) ^F.

(1) The image of f_A under the soft mapping φ_ψ, denoted by φ_ψ (f_A), is the fuzzy soft set on Y defined by follows: if φ^-1 (y)¬ = ∅ , ψ^-1 (k) ∩ A ¬ = ∅, then $φ (f_{A})_{k} (y) = ⋁_{x \in φ^{- 1} (y)} (⋁_{a \in ψ^{- 1} (k) \cap A} f_{a} (x)),$

otherwise φ (f_A) _k (y) =0 for all k ∈ F and y ∈ Y.

(2) The pre-image of g_B under the soft mapping φ_ψ, denoted by $φ_{ψ}^{- 1} (g_{B})$ , is the fuzzy soft set on X defined by $φ^{- 1} (g_{B})_{e} (x) = {\begin{matrix} g_{ψ (e)} (φ (x)), & if ψ (e) \in B \\ 0, & otherwise \end{matrix},$ for all e ∈ E and x ∈ X.

3 (Fuzzy) Soft Topological Spaces

In this section we recall the definitions of (fuzzy) soft topological spaces and construct related categories. From now on, E and K denote arbitrary nonempty sets viewed on the sets of parameters, otherwise stated.

3.1 Soft topological spaces

Definition 3.1. [20] A parametrized family T = {T_k} _k∈K of T_k ⊆ (2^X) ^E which satisfies the following properties for each k ∈ K is called the (E, K)-soft topology on X:

(1) $Φ, \tilde{E} \in T_{k} .$

(2) If F_A, G_B ∈ T_k, then F_A ⊓ G_B ∈ T_k .

(3) If {(F_A) _i} _i∈Γ ⊆ T_k, then ⨆_i∈Γ (F_A) _i ∈ T_k.

The pair (X, T) is called an (E, K)-soft topological space. A soft set F_A is called open according to k ∈ K if F_A ∈ T_k, and it is called closed if $F_{A}^{c} \in T_{k}$ w.r.tk ∈ K.

We note that if K = {k} is a singleton, then the above definition coincides with Shabir and Naz’s definition [20].

Example 3.1. Let R be the set of real numbers, E = R⁺ be the set of positive real numbers and (F_E) _λ ={ (x, (x - λ, x + λ)) : x ∈ E }. Consider the family $τ = {(F_{E})_{λ} : λ \in E} \cup {Φ, \tilde{E}}$ . Then the pair (R, τ) is a soft topological space.

Definition 3.2. Let (X, T¹) and (Y, T²) be an (E₁, K₁)-soft topological space and (E₂, K₂)-soft topological space, respectively. Let φ : X → Y, ψ : E₁ → E₂ and η : K₁ → K₂ be functions. A soft mapping φ_ψ,η : (X, T¹) → (Y, T²) is called continuous if $φ_{ψ}^{- 1} (G_{B}) \in T_{k}^{1}$ , for each $G_{B} \in T_{η (k)}^{2}$ .

(E, K)-soft topological spaces and continuous mappings between these spaces form a category and it is denoted by STop (E, K). (In the case of the parameter set K being a singleton, it is denoted by STop.)

Example 3.2. Let (X, T) be an (E, K)-soft topological spaces, where T_k = {(F_A) _λ | λ ∈ Δ} for each k ∈ K. Then a family of classical topologies can be constructed by using parameters in the following way:

Let T_k (e) ={ (F_A) _λ (e) | (F_A) _λ ∈ T_k }, for all e ∈ E, then T_k is a classical topology on X and it is called e-parameter topology on X.

Proposition 3.1. Let (X, T¹) and (Y, T²) be an (E₁, K₁)-soft topological space and (E₂, K₂)-soft topological space, respectively. If φ_ψ,η : (X, T¹) → (Y, T²) is continuous, then $φ : (X, T_{k}^{1} (e)) \to (Y, T_{η (k)}^{2} (ψ (e)))$ , is continuous for each e ∈ E and k ∈ K.

Proof. Let φ_ψ,η : (X, T¹) → (Y, T²) be continuous between (E₁, K₁)-soft topological space and (E₂, K₂)-soft topological space.

Let $G_{B} (ψ (e)) \in T_{η (k)}^{2} (ψ (e)),$ where $G_{B} \in T_{η (k)}^{2} .$ By the assumption $φ_{ψ}^{- 1} (G_{B}) \in T_{k}^{1}$ and so $φ^{- 1} (G_{B} (ψ (e))) = φ_{ψ}^{- 1} (G_{B}) (e) \in T_{k}^{1} (e) .$

Definition 3.3. A stratified (E, K)-soft topology on X, T = {T_k} _k∈K where T_k ⊆ (2^X) ^E satisfies the following properties, for each k ∈ K.

(1) $\tilde{A} \in T_{k}$ , for all A ⊆ E.

(2) T_k is closed under finite intersections.

(3) T_k is closed under arbitrary unions.

The pair (X, T) is called a stratified (E, K)-soft topological space. The category of stratified (E, K)-soft topological spaces and its continuous mappings is denoted by LSTop (E, K) . (If K is one pointed, it is denoted by LSTop.)

A primary feature of stratified soft topological spaces is that, as it is shown in [20], the constant soft mappings of such spaces are continuous. Another essential difference of such spaces is that there is only one stratified soft topology on a one-point set, while different soft topologies can be defined if the parameter set is not singleton. For example, for such set and parameter set, trivial soft topology and discrete soft topology are different. According to this, LSTop is a topological category in the sense of Herrlich [10].

3.2 Fuzzy soft topological spaces

The definition of fuzzy soft topology was first given in [21] and the basic idea of this definition depends on Chang’s definition of fuzzy topology [5] which has some disadvantages mentioned by Lowen [14]. In this subsection, we attempt to develop fuzzy soft topological spaces with respect to Lowen’s, Shostak’s and also Ying’s ideas, respectively, as follows.

In the following we first give the definition of quasi fuzzy soft topology on an initial universe X.

Definition 3.4. [7] A parametrized family $T = {T_{k}}_{k \in K}$ of $T_{k} \subseteq (I^{X})^{E}$ which satisfies the following properties for each k ∈ K is called the quasi (E, K)-soft I-topology on X .

(1) $Φ, \tilde{E} \in T_{k}$ .

(2) If $f_{A}, g_{B} \in T_{k}$ , then $f_{A} ⊓ g_{B} \in T_{k}$ .

(3) If ${(f_{A})_{i}}_{i \in Γ} \subseteq T_{k}$ , then $⨆_{i \in Γ} (f_{A})_{i} \in T_{k} .$

The pair $(X, T)$ is called a quasi (E, K)-softI-topological space.

Example 3.3. [7] (1) The family $T^{t} = {T_{k}^{t}}$ is called trivial quasi (E, K)-soft I-topology on X where for each k ∈ K, $T_{k}^{t} = {Φ_{X}, {\tilde{E}}_{X}} .$

(2) The family $T^{D} = {T_{k}^{D}}$ is called discrete quasi (E, K)-soft I-topology on X where for each k ∈ K, $T_{k}^{D} = (I^{X})^{E} .$

Definition 3.5. [7] Let $(X_{1}, T^{1})$ be a quasi (E₁, K₁)-soft I-topological space and $(X_{2}, T^{2})$ be a quasi (E₂, K₂)-soft I-topological space, φ : X₁ → X₂, ψ : E₁ → E₂ and η : K₁ → K₂ be functions. Then the soft mapping $φ_{ψ, η} : (X_{1}, T^{1}) \to (X_{2}, T^{2})$ is called continuous if $g_{B} \in T_{η (k)}^{2}$ implies $φ_{ψ}^{- 1} (g_{B}) \in T_{k}^{1},$ for each k ∈ K.

It is noted that if K = {k} is a one-pointed set, then the definition corresponds to the definition of fuzzy soft topology given by Tanay and Kandemir [21].

The category of quasi (E, K)-soft I-topological spaces and its continuous mappings is denoted by QFSTop (E, K) (In the case of K being a singleton, it is denoted by QFSTop).

Example 3.4. Let X = [0, 1] and E = (0, 1). Consider fuzzy soft sets F_E ={ f_e : e ∈ E } and G_E ={ g_e : e ∈ E } on X defined as for each x ∈ X and e ∈ E, $\begin{matrix} f_{e} (x) & = & {\begin{matrix} - \frac{1}{e} x + 1, & if 0 \leq x \leq e; \\ 0, & if e \leq x \leq 1 . \end{matrix} \\ g_{e} (x) & = & {\begin{matrix} 0, & if 0 \leq x \leq e; \\ \frac{1}{x} e - 1, & if e \leq x \leq 1 . \end{matrix} . \end{matrix}$

Then $Γ = {Φ, \tilde{E}, F_{E}, G_{E}, F_{E} ⊔ G_{E}}$ is a member of QFSTop.

Definition 3.6. A fuzzy soft set f_A ∈ (I^X) ^E is called an α-constant fuzzy soft set on X, denoted by $\bar{α}$ if f_A ={ α_e : e ∈ E }, where α_e : X → I, α_e (x) = α (e), for each x ∈ X, and α : E → I is an arbitrary mapping.

Example 3.5.Φ and $\tilde{E}$ are $\tilde{0}$ -constant and $\tilde{1}$ -constant fuzzy soft sets on X, respectively, where $\tilde{0} : E \to I$ , $\tilde{0} (e) = 0$ and $\tilde{1} : E \to I$ , $\tilde{1} (e) = 1$ , for each e ∈ E.

Definition 3.7. A parametrized family $T = {T_{k}}_{k \in K}$ of $T_{k} \subseteq (I^{X})^{E}$ which satisfies the following conditions for each k ∈ K is called the (E, K)-soft I-topology (or Lowen type soft I-topology) on X:

(1) $\bar{α} \in T_{k}$ , for each α : E ⟶ I.

(2) $T_{k}$ is closed under finite intersections.

(3) $T_{k}$ is closed under arbitrary unions.

The pair $(X, T)$ is called an (E, K)-soft I-topological space. The elements of $T_{k}$ are called open fuzzy soft sets w.r.t k ∈ K. A fuzzy soft set g_B is called closed fuzzy soft set w.r.t k ∈ K if $g_{B}^{c} \in T_{k}$ .

It is obvious that every (E, K)-soft I-topological space is a quasi (E, K)-soft I-topological space.

We denote LFSTop (E, K) as the category of (E, K)-soft I-topological spaces and their continuous soft mappings (if K is one-pointed, then it is denoted by LFSTop).

Example 3.6. Constant mappings are fuzzy soft continuous. Indeed, let us consider fuzzy soft topological spaces $(X, T, E)$ and $(Y, T^{'}, K)$ and let φ_ψ be a constant soft mapping from X to Y, where for fixed y₀ ∈ Y, φ (x) = y₀, ∀x ∈ X and ψ : E → K be a mapping. Then for each $G_{B} \in T^{'}$ , e ∈ E and x ∈ X, $\begin{matrix} {(φ_{ψ}^{- 1} (G_{B}))}_{e} (x) & = & (G_{B})_{ψ (e)} (φ (x)) \\ = & (G_{B})_{ψ (e)} (y_{0}) . \end{matrix}$

Let us define α : E ⟶ I as α (e) = (G_B) _ψ(e) (y₀). Then $φ_{ψ}^{- 1} (G_{B}) = \bar{α} \in T$ . Hence constant mappings are fuzzy soft continuous.

However, this is not satisfied for quasi fuzzy soft topological spaces in general. This is shown by the following example.

Example 3.7. Let (X, Γ) be the quasi-fuzzy soft topological space in Example 3.1 and let φ_ψ be a constant soft mapping from X to X, where for fixed y₀ ∈ Y, φ (x) = y₀, ∀x ∈ X and ψ : E ⟶ E be the identity mapping. For F_E ∈ Γ, $\begin{matrix} {(φ_{ψ}^{- 1} (F_{E}))}_{e} (x) & = & (F_{E})_{ψ (e)} (φ (x)) \\ = & (F_{E})_{e} (y_{0}) \\ : = & α (e), \end{matrix}$

where for each e ∈ E, $α (e) = {\begin{matrix} 0, & if 0 < e \leq y_{0}; \\ - \frac{1}{e} y_{0} + 1, & if y_{0} \leq e < 1 . \end{matrix}$

Then $φ_{ψ}^{- 1} (F_{E})$ is an α-constant fuzzy soft set which is not contained in Γ. Hence the constant mapping φ_ψ is not continuous.

According to the above examples constant mappings are morphism in LFSTop, but not in QFSTop.

We also note that there is a continuum of different fuzzy soft topologies on a one-point set, while there is only one stratified fuzzy soft topology. This makes LFSTop, unlike QFSTop, a topological category in the sense of Herrlich [10].

In the following, the definition of fuzzy soft topology is given in Shostak sense.

Definition 3.8. [4] A mapping τ : K → I^{(I^X)^E} is called a fuzzy (E, K)-soft topology on X if it satisfies the following conditions for each k ∈ K (where τ_k : = τ (k) : (I^X) ^E → I is a mapping for each k ∈ K).

(O1) $τ_{k} (Φ) = τ_{k} (\tilde{E}) = 1 .$

(O2) τ_k (f_A ⊓ g_B) ≥ τ_k (f_A) ∧ τ_k (g_B) , for each f_A, g_B ∈ (I^X) ^E .

(O3) τ_k (⨆ _i∈Δ (f_A) _i) ≥ ⋀ _i∈Δτ_k ((f_A) _i) , for each (f_A) _i ∈ (I^X) ^E, i ∈ Δ .

Then the pair (X, τ) is called a fuzzy (E, K)-soft topological space. The value τ_k (f_A) is interpreted as the degree of a fuzzy soft set f_A w.r.t the parameter k ∈ K .

Example 3.8. [7] Let X = {x, y} , K = {k₁, k₂} and E be a nonempty arbitrary parameter set for X. Define the mapping, τ : {k₁, k₂} → I^{(I^X)^E} as follows: for each e ∈ E,

$τ_{k_{1}} (f) = {\begin{matrix} \frac{1}{4}, & if f_{e} (x) > f_{e} (y) \\ \frac{1}{5}, & if f_{e} (x) < f_{e} (y) \\ 1, & if f_{e} (x) = f_{e} (y) \end{matrix},$

$τ_{k_{2}} (f) = {\begin{matrix} \frac{9}{10}, & if f_{e} (x) > f_{e} (y) \\ \frac{8}{10}, & if f_{e} (x) < f_{e} (y) \\ 1, & if f_{e} (x) = f_{e} (y) \end{matrix} .$

It is easy to verify that τ is a fuzzy (E, K)-soft topology on X .

Definition 3.9. [4] Let (X₁, τ¹) and (X₂, τ²) be fuzzy (E₁, K₁)-soft topological space and fuzzy (E₂, K₂)-soft topological space, respectively. Let φ : X₁ → X₂, ψ : E₁ → E₂ and η : K₁ → K₂ be functions. Then the fuzzy soft mapping φ_ψ,η : (X₁, τ¹) → (X₂, τ²) is called a continuous map if

$τ_{k}^{1} (φ_{ψ}^{- 1} (g_{B})) \geq τ_{η (k)}^{2} (g_{B})$ for all g_B ∈ (I^{X
₂}) ^{E
₂}, k ∈ K₁ .

The category of fuzzy (E, K)-soft topological spaces and continuous mappings between them is denoted by FSTOP (E, K) .

Theorem 3.1. [4] The category FSTOP (E, K) is a topological category over SET³ w.r.t the forgetful functor V : FSTOP (E, K) → SET³ which is defined by V (X, τ) = (X, E, K) and V (φ_ψ,η) = (φ, ψ, η) .

Definition 3.10. [9] If a mapping τ : K → I^{(I^X)^E} fulfills the conditions (O2), (O3) and

(O1)^′ $τ_{k} (\bar{α}) = 1$ , for each α-constant fuzzy soft set and k ∈ K

then the pair (X, τ) is called a stratified fuzzy (E, K)-soft topological space.

The category of stratified fuzzy (E, K)-soft topological spaces and their continuous mappings is denoted as SFSTOP (E, K) .

It is clear that constant soft mappings are fuzzy soft continuous, since $τ_{k} (φ_{ψ}^{- 1} (g_{B})) = τ_{k} (\bar{α}) = 1$ , for some α : E ⟶ I, for each g_B ∈ (I^{X
₂}) ^{E
₂}, k ∈ K.

This makes constant mapping morphism in SFSTOP (E, K) .

Definition 3.11. [9] A mapping ζ : K → I^{(2^X)^E} is called a fuzzifying (E, K)-soft topology on X if it satisfies the followings for each k ∈ K,

(1) $ζ_{k} (Φ) = ζ_{k} (\tilde{E}) = 1 .$

(2) ζ_k (F_A ⊓ G_B) ≥ ζ_k (F_A) ∧ ζ_k (G_B) , for all F_A, G_B ∈ (2^X) ^E .

(3) ζ_k (⨆ _i∈Γ (F_A) _i) ≥ ⋀ _i∈Γζ_k ((F_A) _i) for all {(F_A) _i} _i∈Γ ⊆ (2^X) ^E .

The pair (X, ζ) is called a fuzzifying (E, K)-soft topological space. Let (X₁, ζ¹) and (X₂, ζ²) be fuzzifying (E₁, K₁)-soft topological space and fuzzifying (E₂, K₂)-soft topological space, respectively. Then the mapping φ_ψ,η : (X₁, ζ¹) → (X₂, ζ²) is called continuous if $ζ_{k}^{1} (φ_{ψ}^{- 1} (G_{B})) \geq ζ_{η (k)}^{2} (G_{B})$ for all G_B ∈ (2^{X
₂}) ^{E
₂}, k ∈ K₁ .

Theorem 3.2. [9] The category of fuzzifying (E, K)-soft topological spaces and their continuous mappings, FYSTOP (E, K) is a topological category over SET³ .

4 Categorical interrelations between soft topological spaces

In this section we construct functors between categories of topological spaces, soft topological spaces and fuzzy soft topological spaces.

4.1 The category of topological spaces TOP as a subcategory of STop

Let (X, T) be a topological space and E be a fixed parameter set. Let us identify a subset G of X with a soft set G_E : = {(e, G) : e ∈ E} which is called soft characteristic of G, and given a mapping φ with a soft mapping φ_{id
_E}, where id_E is the identity function of E. It is easy to show that $\tilde{T} : = {G_{E} : G \in T}$ is a soft topology on X. Since, for a soft set G_E, $φ_{{id}_{E}}^{- 1} (G_{E}) (e) = φ^{- 1} (G_{E} ({id}_{E} (e))) = φ^{- 1} (G_{E} (e)) = φ^{- 1} (G)$ , the continuity of a mapping φ : (X, T) → (Y, T^*) implies soft continuity of the mapping $φ_{{id}_{E}} : (X, \tilde{T}) ⟶ (Y, \tilde{T^{*}})$ . Thus, there is an inclusion functor $I : TOP \to STop$ mapping TOP isomorphically to a full subcategory of STop whose objects are consisted of soft characteristics of some sets.

4.2 The functor $J :$ STop(E, K)→ TOP

Let (X, T) be an (E, K)-soft topological space and let k ∈ K be arbitrary. Then $(X, J (T_{k}))$ is a classical topological space where $J (T_{k})$ is a topology on X whose subbase is {⋂ _e∈EF_A (e) : F_A ∈ T_k }.

Proposition 4.1. If ψ is surjective and φ_ψ,η : (X₁, T¹) → (X₂, T²) is continuous between (E₁, K₁)-soft and (E₂, K₂)-soft topological spaces, then the function $φ : (X_{1}, J (T_{k}^{1})) \to (X_{2}, J (T_{η (k)}^{2}))$ is continuous.

Proof. Let ⋂_e^*∈E₂F_A (e^*) be a sub-basic element of $J (T_{η (k)}^{2})$ , where $F_{A} \in T_{η (k)}^{2}$ .

Since φ_ψ,η : (X₁, T¹) → (X₂, T²) is soft continuous, $φ_{ψ}^{- 1} (F_{A}) \in T_{k}^{1}$ .

By the surjectivity of the function ψ, $\begin{matrix} φ^{- 1} (⋂_{e^{*} \in E_{2}} F_{A} (e^{*})) & = & ⋂_{e^{*} \in E_{2}} φ^{- 1} (F_{A} (e^{*})) \\ = & ⋂_{e \in E_{1}} φ^{- 1} (F_{A} (ψ (e))) \\ = & ⋂_{e \in E_{1}} φ_{ψ}^{- 1} (F_{A}) (e) . \end{matrix}$

So, $φ^{- 1} (⋂_{e^{*} \in E_{2}} F_{A} (e^{*})) \in J (T_{k}^{1})$ .

Thus, $J : STop (E, K) \to TOP$ is a covariant functor and it is the right inverse of the functor $I$ , that is, $J \circ I (T) = T$ , for each topology T.

4.3 STop(E, K) as a subcategory of QFSTop(E, K)

Let us identify a soft set F_A ∈ (2^X) ^E with $X_{F_{A}} : = {X_{F_{A} (e)} : e \in E}$ , where $X_{F_{A} (e)}$ is a characteristic function of F_A (e) ⊆ X, and $X_{F_{A}}$ is called characteristic mapping of the soft set F_A. In this way, it can be considered that an (E, K)-soft topological space (X, T) as an object of QFSTop (E, K) and naturally an inclusion functor $I^{'} : STop (E, K) \to QFSTop (E, K)$ arises, which identifies STop (E, K) isomorphically to a full subcategory of QFSTop (E, K) whose objects are consisted of characteristic mappings of some soft sets.

4.4 The functor j :QFSTop(E, K)→ STop(E, K)

If $(X, T)$ is a quasi (E, K)-soft I-topological space, then $j (T)$ , where $j (T_{k}) : = T_{k} \cap (2^{X})^{E}$ for each k ∈ K, is an (E, K)-soft topology on X. Also if $φ_{ψ, η} : (X, T^{X}) \to (Y, T^{Y})$ is continuous, then $φ_{ψ, η} : (X, j (T^{X})) \to (Y, j (T^{Y}))$ is also continuous (where $j (T^{X}) = {j (T_{k}^{X})}_{k \in K}$ ). So, j is a functor from QFSTop (E, K) into STop (E, K) .

4.5 STop(E, K) as a subcategory of LFSTop(E, K)

We associate with any (E, K)-soft topological space (X, T) the stratified (E, K)-soft I-topological space $(X, W (T))$ , where for each k ∈ K, $W (T_{k})$ is the family of all fuzzy soft sets, whose e-components are lower semi-continuous from (X, T_k) to the unit interval.

Proposition 4.2. If φ_ψ,η : (X₁, T¹) → (X₂, T²) is continuous, then $φ_{ψ, η} : (X_{1}, W (T^{1})) \to (X_{2}, W (T^{2}))$ is continuous, too.

Proof. Let φ_ψ,η : (X₁, T¹) → (X₂, T²) be continuous and $g_{B} \in W (T_{η (k)}^{2})$ . It follows that (g_B) _{e
^*} is lower semi-continuous from $(X_{2}, T_{η (k)}^{2})$ to [0, 1], for each e^* ∈ E₂. It is enough to show that the mapping $φ_{ψ}^{- 1} (g_{B})_{e}$ from $(X, T_{k}^{1})$ to [0, 1] is lower semi-continuous, $\begin{matrix} {(φ_{ψ}^{- 1} (g_{B})_{e})}^{- 1} ((α, 1]) & = & {((g_{B})_{ψ (e)} \circ φ)}^{- 1} ((α, 1]) \\ = & φ^{- 1} ((g_{B})_{ψ (e)}^{- 1} ((α, 1])) . \end{matrix}$

So, ${(φ_{ψ}^{- 1} (g_{B})_{e})}^{- 1} ((α, 1]) \in T_{k}^{1} .$

Hence, $φ_{ψ}^{- 1} (g_{B})_{e}$ is lower semi-continuous.

It implies that STop (E, K) can be viewed as a full subcategory of LFSTop (E, K).

4.6 The functor $I^{″} :$ LFSTop(E, K)→ LSTop(E, K)

We associate any (E, K)-soft I-topological space (X, τ) with the (E, K)-soft topological space $(X, I^{″} (τ))$ , where

$I^{″} (τ_{k}) : = {(e, f^{- 1} ((α_{e}, 1])) : e \in E}$ , such that α_e ∈ [0, 1) and f_A ∈ τ_k .

It is easy to show that $I^{″} (τ) = {I^{″} (τ_{k})}_{k \in K}$ is a stratified (E, K)-soft topology on X. So we have the following proposition.

Proposition 4.3. If φ_ψ,η : (X₁, τ¹) → (X₂, τ²) is continuous between (E₁, K₁)-soft topological space and (E₂, K₂)-soft topological space, respectively, then $φ_{ψ, η} : (X_{1}, I^{″} (τ^{1})) \to (X_{2}, I^{″} (τ^{2}))$ is continuous.

Proof. Let $F_{A} = {(e, f^{- 1} ((α_{e}, 1])) : e \in E_{2}} \in I^{″} (τ_{η (k)}^{2})$ , α_e ∈ [0, 1), for each e ∈ E, where $f_{A} = {f_{e} : e \in E_{2}} \in τ_{η (k)}^{2}$ .

Since φ_ψ,η is continuous, $φ_{ψ}^{- 1} (f_{A}) \in τ_{k}^{1}$ . On the other hand, $\begin{matrix} φ_{ψ}^{- 1} (F_{A}) (e) & = & φ^{- 1} (F_{A} (ψ (e))) \\ = & φ^{- 1} (f_{ψ (e)}^{- 1} ((α_{ψ (e)}, 1])) \\ = & {(φ^{- 1} (f_{ψ (e)}))}^{- 1} ((α_{ψ (e)}, 1]) . \end{matrix}$

By the definition of $I^{″} (τ_{k})$ , $φ_{ψ}^{- 1} (f_{A})_{e}$ is lower semi-continuous for each e ∈ E₁, it follows that $φ_{ψ}^{- 1} (F_{A}) \in I^{″} (τ_{k})$ . Hence, $φ_{ψ, η} : (X_{1}, I^{″} (τ^{1})) \to (X_{2}, I^{″} (τ^{2}))$ is soft continuous.

Evidently, $I^{″} : LFSTop (E, K) \to LSTop (E, K)$ is a covariant functor and it is the right inverse of the functor $W$ , that is, $I^{″} \circ W (τ) = τ$ , for each τ ∈ LSTop (E, K).

4.7 QFSTop(E, K) as a subcategory of FSTOP(E, K)

We associate any quasi (E, K)-soft I-topological space $(X, T)$ with $(X, ℑ (T))$ , where $ℑ (T) : = τ : K \to I^{(I^{X})^{E}}$ , defined by for each f_A ∈ (I^X) ^E and k ∈ K, $τ_{k} (f_{A}) = {\begin{matrix} 1, & if f_{A} \in T_{k} \\ 0, & otherwise \end{matrix} .$

It is easy to show that $(X, ℑ (T))$ is a fuzzy (E, K)-soft topological space.

Proposition 4.4. If $φ_{ψ, η} : (X_{1}, T^{1}) \to (X_{2}, T^{2})$ is continuous, then $φ_{ψ, η} : (X_{1}, ℑ (T^{1})) \to (X_{2}, ℑ (T^{2}))$ is also continuous.

Proof. Let g_B ∈ (I^{X
₂}) ^{E
₂}. If $g_{B} \in T_{η (k)}^{2}$ , then by the continuity of φ_ψ,η, it is clear that $φ_{ψ}^{- 1} (g_{B}) \in T_{k}^{1}$ . It follows that $τ_{k}^{1} (φ_{ψ}^{- 1} (g_{B})) = 1$ , for each k ∈ K, and if $g_{B} \notin T_{η (k)}^{2}$ , it is clear that $τ_{η (k)}^{2} (g_{B}) = 0$ . This concludes the proof.

Evidently, ℑ : QFSTop (E, K) → FSTOP (E, K) is a covariant functor.

4.8 The functor I′ :FSTOP(E, K)→ QFSTop(E, K)

We associate any fuzzy (E, K)-soft topological space (X, τ) with (X, I′ (τ)), where I′ (τ) _k : = {f_A ∣ τ_k (f_A) =1}. Clearly, (X, I′ (τ)) is a quasi (E, K)-soft I-topological space.

Proposition 4.5.If φ_ψ,η : (X₁, τ¹) → (X₂, τ²) is continuous between fuzzy (E₁, K₁)-soft and fuzzy (E₂, K₂)-soft topological spaces, then

φ_ψ,η : (X₁, I′ (τ¹)) → (X₂, I′ (τ²)) is continuous, too.

Proof. Let g_B ∈ (I′ (τ²)) _η(k), then $τ_{η (k)}^{2} (g_{B}) = 1$ . Since φ_ψ,η is continuous, then

$τ_{k}^{1} (φ_{ψ}^{- 1} (g_{B})) \geq τ_{η (k)}^{2} (g_{B}) = 1$ . It implies that $τ_{k}^{1} (φ_{ψ}^{- 1} (g_{B})) = 1$ , then $φ_{ψ}^{- 1} (g_{B}) \in (I^{'} (τ^{1}))_{k}$ . Hence, φ_ψ,η is continuous between generated spaces.

Thus, I′ : FSTOP (E, K) → QFSTop (E, K) is a covariant functor and it is the right inverse of the functor ℑ, that is, I′ ∘ ℑ = id_QFSTop(E,K).

4.9 Functors between LFSTop(E, K), SFSTOP(E, K) and I_a - FSTOP (E, K)

Let (X, τ) be a stratified fuzzy (E, K)-soft topological space and define τ (a) = {τ (a) _k} _k∈K, where τ (a) _k ⊆ (I^X) ^E as follows: $τ (a)_{k} = {f_{A} \in (I^{X})^{E} ∣ τ_{k} (f_{A}) \geq a}, a \in (0, 1] .$

Proposition 4.6. Let (X, τ) be a stratified fuzzy (E, K)-soft topological space. Then for each α ∈ (0, 1] , τ (a) is an (E, K)-soft I-topology on X . Moreover τ (a) ⊂ τ (b) if a ≥ b .

Proof. (1) For any α-constant fuzzy soft set $\bar{α}$ and for each k ∈ K, $τ_{k} (\bar{α}) = 1 .$ So $\bar{α} \in τ (a)_{k},$ for each k ∈ K .

(2) Let f_A, g_B ∈ τ (a) _k. Then τ_k (f_A) ≥ a and τ_k (g_B) ≥ a, for all k ∈ K . So, τ_k (f_A ⊓ g_B) ≥ τ_k (f_A) ∧ τ_k (g_B) ≥ a, k ∈ K . Hence f_A ⊓ g_B ∈ τ (a) _k, for each k ∈ K .

(3) Let (f_A) _i ∈ τ (a) _k, for all i ∈ Γ . Then τ_k ((f_A) _i) ≥ a for each i ∈ Γ and k ∈ K . So, τ_k (⨆ _i∈Γ (f_A) _i) ≥ ⋀ _i∈Γτ_k ((f_A) _i) ≥ a, for each k ∈ K . Hence, ⨆_i∈Γ ((f_A) _i) ∈ τ (a) _k .

Proposition 4.7. Let (X, τ) be a stratified fuzzy (E, K)-soft topological space, then

τ_k (f_A) = ⋁ {a ∈ (0, 1] ∣ f_A ∈ τ (a) _k} .

Proof. ⋁ {a ∈ (0, 1] ∣ f_A ∈ τ (a) _k} = ⋁ {a ∈ (0, 1] ∣ τ_k (f_A) ≥ a} = τ_k (f_A) .

Corollary 4.1.Two fuzzy (E, K)-soft topologies τ and ξ are equal if and only if τ (a) = ξ (a) for each a ∈ (0, 1] .

Proposition 4.8.Let (X, T) be an (E, K)-soft I-topological space. Define for each a ∈ (0, 1] , a map T^a : K → I^{(I^X)^E} by the rule: for each k ∈ K, $T_{k}^{a} (f_{A}) = {\begin{matrix} 1, & if f_{A} = \bar{α} constant fuzzy soft set, \\ a, & if f_{A} \in T_{k} and non - constant, \\ 0, & otherwise . \end{matrix}$

Then T^a is a stratified fuzzy (E, K)-soft topology on X such that T^a (a) = T .

Proof. (1) $T_{k}^{a} (\bar{α}) = 1,$ for any constant fuzzy soft set $\bar{α} .$

(2) Note that $T_{k}^{a} (f_{A}) \land T_{k}^{a} (g_{B}) = 0, a or 1 .$ If $T_{k}^{a} (f_{A}) \land T_{k}^{a} (g_{B}) = 0$ , it is obvious. If $T_{k}^{a} (f_{A}) \land T_{k}^{a} (g_{B}) = 1$ , then f_A and g_B are constant and hence f_A ⊓ g_B is also constant. Thus $T_{k}^{a} (f_{A} ⊓ g_{B}) = 1 \geq T_{k}^{a} (f_{A}) \land T_{k}^{a} (g_{B}) .$ If $T_{k}^{a} (f_{A}) \land T_{k}^{a} (g_{B}) = a,$ then $T_{k}^{a} (f_{A}) \geq a$ and $T_{k}^{a} (g_{B}) \geq a .$ Then f_A ∈ T_k and g_B ∈ T_k, and hence f_A ⊓ g_B ∈ T_k .

So $T_{k}^{a} (f_{A} ⊓ g_{B}) \geq T_{k}^{a} (f_{A}) \land T_{k}^{a} (g_{B}),$ for all k ∈ K .

(3) If $⋀_{i \in Γ} T_{k}^{a} ((f_{A})_{i}) = 0,$ it is obvious.

If $⋀_{i \in Γ} T_{k}^{a} ((f_{A})_{i}) = 1,$ then $T_{k}^{a} ((f_{A})_{i}) = 1,$ for all i ∈ Γ and k ∈ K . Thus (f_A) _i is constant for all i ∈ Γ . Hence ⨆_i∈Γ (f_A) _i is constant. So, $T_{k}^{a} (⨆_{i \in Γ} (f_{A})_{i}) = 1 \geq ⋀_{i \in Γ} T_{k}^{a} ((f_{A})_{i}) .$

If $⋀_{i \in Γ} T_{k}^{a} ((f_{A})_{i}) = a,$ then $T_{k}^{a} ((f_{A})_{i}) \geq a$ for all i ∈ Γ . Thus (f_A) _i ∈ T_k, and hence ⨆_i∈Γ (f_A) _i ∈ T_k .

So, $T_{k}^{a} (⨆_{i \in Γ} (f_{A})_{i}) \geq a = ⋀_{i \in Γ} T_{k}^{a} ((f_{A})_{i}) .$

Hence the following is obtained for each a ∈ (0, 1] , $T_{k}^{a} (⨆_{i \in Γ} (f_{A})_{i}) \geq ⋀_{i \in Γ} T_{k}^{a} ((f_{A})_{i}) .$ Moreover f_A ∈ (T^a (a)) _k if and only if $T_{k}^{a} (f_{A}) \geq a$ if and only if f_A ∈ T_k, for each k ∈ K . HenceT^a (a) = T .

Definition 4.1. Let T ∈ LFSTop (E, K) . Then the stratified fuzzy (E, K)-soft topology T^a on X is said to be the a-th graded.

Proposition 4.9.φ_ψ,η : (X, τ) → (Y, ξ) is continuous if and only if φ_ψ,η : (X, τ (a)) → (Y, ξ (a)) is continuous for all a ∈ (0, 1] .

Proof. Let φ_ψ,η : (X, τ) → (Y, ξ) be continuous and a ∈ (0, 1] be fixed. Take g_B ∈ (ξ (a)) _η(k), then $a \leq ξ_{η (k)} (g_{B}) \leq τ_{k} (φ_{ψ}^{- 1} (g_{B}))$ for all k ∈ K . Hence $φ_{ψ}^{- 1} (g_{B}) \in τ (a)_{k} .$ That is φ_ψ,η : (X, τ (a)) → (Y, ξ (a)) is continuous.

Conversely, let φ_ψ,η : (X, τ (a)) → (Y, ξ (a)) be continuous for all a ∈ (0, 1] . Let g_B ∈ (I^Y) ^{E
₂} . If ξ_η(k) (g_B) =0, then it is obvious. If ξ_η(k) (g_B) ≠0, let ξ_η(k) (g_B) = r, then g_B ∈ (ξ (r)) _η(k) . So, $φ_{ψ}^{- 1} (g_{B}) \in τ (r)_{k}$ by continuity. Hence, $τ_{k} (φ_{ψ}^{- 1} (g_{B})) \geq r = ξ_{η (k)} (g_{B}) .$ This implies the continuity of φ_ψ,η .

Proposition 4.10.φ_ψ,η : (X, T) → (Y, U) is continuous if and only if φ_ψ,η : (X, T^a) → (Y, U^a) is continuous for all a ∈ (0, 1] .

Proof. Let φ_ψ,η : (X, T) → (Y, U) be continuous. Take g_B ∈ (I^Y) ^{E
₂} . Then either g_B ∈ U_η(k) or g_B ∉ U_η(k) . If g_B ∈ U_η(k) and $g_{B} = \bar{α}$ for some constant fuzzy soft set, then $φ_{ψ}^{- 1} (\bar{α}) = \bar{α} .$ So $\begin{matrix} U_{η (k)}^{a} (g_{B}) & = & U_{η (k)}^{a} (\bar{α}) = 1 \leq 1 \\ = & T_{k}^{a} (\bar{α}) = T_{k}^{a} (φ_{ψ}^{- 1} (\bar{α})) = T_{k}^{a} (φ_{ψ}^{- 1} (g_{B})) . \end{matrix}$

Let g_B ∈ U_η(k) and $g_{B} \neq \bar{α} .$ Then $U_{η (k)}^{a} (g_{B}) = a .$ By the continuity, $φ_{ψ}^{- 1} (g_{B}) \in T_{k} .$ So, $U_{η (k)}^{a} (g_{B}) \leq T_{k}^{a} (φ_{ψ}^{- 1} (g_{B})) .$ In case $g_{B} \notin U_{η (k)}, U_{η (k)}^{a} (g_{B}) = 0 .$ So, $U_{η (k)}^{a} (g_{B}) \leq T_{k}^{a} (φ_{ψ}^{- 1} (g_{B})) .$ Hence φ_ψ,η : (X, T^a) → (Y, U^a) is continuous for all a ∈ (0, 1] .

Conversely, let φ_ψ,η : (X, T^a) → (Y, U^a) be continuous for each a ∈ (0, 1] . Then φ_ψ,η : (X, T^a (a)) → (Y, U^a (a)) is continuous by the above proposition. Since T^a (a) = T and U^a (a) = U, then the desired result is obtained.

For each a ∈ (0, 1] , I_a-FSTOP (E, K) denote the category of a-th graded fuzzy (E, K)-soft topological spaces and continuous functions.

Proposition 4.11.(1) Define V : LFSTop (E, K) → SFSTOP (E, K) by V (X, T) = (X, T^a) and V (φ_ψ,η) = φ_ψ,η. Then V is a functor.

(2) Define W : SFSTOP (E, K) → LFSTop (E, K) by W (X, τ) = (X, τ (a)) and W (φ_ψ,η) = φ_ψ,η. Then W is a functor.

Theorem 4.1.For each a ∈ (0, 1] , LFSTop (E, K) and I_a-FSTOP (E, K) are isomorphic.

Proof. Define V : LFSTop (E, K) → I_a - FSTOP (E, K) by V (X, T) = (X, T^a) and V (φ_ψ,η) = φ_ψ,η.

Define W : I_a - FSTOP (E, K) → LFSTop (E, K) by W (X, τ) = (X, τ (a)) and W (φ_ψ,η) = φ_ψ,η. Then V and W are functors, and it is clear that W ∘ V (X, T) = (X, T) . So it is enough to show that V ∘ W (X, τ) = (X, τ) .

Take (X, τ) ∈ I_a - FSTOP (E, K) , then there exists (X, T) ∈ LFSTop (E, K) such that τ = T^a .So, $\begin{matrix} V \circ W (X, τ) & = & V (X, τ (a)) = (X, (τ (a))^{a}) \\ = & (X, (T^{a} (a))^{a}) = (X, T^{a}) = (X, τ) . \end{matrix}$

Thus V ∘ W (X, τ) = (X, τ) . This completes the proof.

Theorem 4.2. I_a-FSTOP (E, K) is a bireflective full subcategory of SFSTOP (E, K) for all a ∈ (0, 1] .

Proof. Clearly I_a-FSTOP (E, K) is a full subcategory of SFSTOP (E, K). Let us take a member (X, τ) of SFSTOP (E, K). Then (X, (τ (a)) ^a) is a member of I_a-FSTOP (E, K) and idX_idE,idK : (X, τ) → (X, (τ (a)) ^a) is a continuous map. Let (Y, ξ) be a member of I_a-FSTOP (E, K) and φ_ψ,η : (X, τ) → (Y, ξ) be a continuous map.

To complete the proof we need to check thatφ_ψ,η : (X, (τ (a)) ^a) → (Y, ξ) is a continuous map.Note that for g_B ∈ (I^Y) ^{E
₂}, ξ_η(k) (g_B) =1 or a or 0.If ξ_η(k) (g_B) =1, then g_B is constant. Thus $(τ (a))_{k}^{a} (φ_{ψ}^{- 1} (g_{B})) = 1 \geq ξ_{η (k)} (g_{B}) .$ If ξ_η(k) (g_B) =0, then $ξ_{η (k)} (g_{B}) \leq (τ (a))_{k}^{a} (φ_{ψ}^{- 1} (g_{B})) .$ If ξ_η(k) (g_B) = a, then $a = ξ_{η (k)} (g_{B}) \leq τ_{k} (φ_{ψ}^{- 1} (g_{B}))$ from that φ_ψ,η : (X, τ) → (Y, ξ) is a continuous map. Thus $a \leq τ_{k} (φ_{ψ}^{- 1} (g_{B})) .$ Hence $φ_{ψ}^{- 1} (g_{B}) \in τ (a)_{k},$ and so $(τ (a))_{k}^{a} (φ_{ψ}^{- 1} (g_{B})) \geq a = ξ_{η (k)} (g_{B}) .$ Thus, φ_ψ,η : (X, (τ (a)) ^a) → (Y, ξ) is continuous. In the light of the above discussions, we get:

Theorem 4.3. LFSTop (E, K) is a bireflective full subcategory of SFSTOP (E, K) .

4.10 Functors between FYSTOP(E, K) and FSTOP(E, K)

Lemma 4.1. [9] Let (X, ζ) be a fuzzifying (E, K)-soft topological space and let $ω (ζ)_{k} (f_{A}) = ⋀_{r \in [0, 1]} ζ_{k} (σ_{r} (f_{A})),$ for all f_A ∈ (I^X) ^E and k ∈ K . Then ω (ζ) : K → I^{(I^X)^E} is a fuzzy (E, K)-soft topology on X (where σ_r (f_A) (e) = {x ∈ X ∣ f_A (e) (x) > r}).

Theorem 4.4. [9] φ_ψ,η : (X, ω (ζ)) → (Y, ω (ζ^*)) is continuous if and only if φ_ψ,η : (X, ζ) → (Y, ζ^*) is continuous.

Theorem 4.5. [9] Let (X, τ) be a fuzzy (E, K)-soft topological space. Define β^τ : K → I^{(2^X)^E} as

$β_{k}^{τ} (F_{A}) = ⋁ {⋀_{i = 1}^{m} (⋁_{r \in [0, 1]} ⋁ {τ_{k} (f_{A}) : f_{A} \in (I^{X})^{E}, σ_{r} (f_{A}) = (F_{A})_{i}}) : F_{A} = ⊓_{i = 1}^{m} (F_{A})_{i}},$ for all k ∈ K .

Then the following properties are valid.

(1) β^τ is a base of a fuzzifying (E, K)-soft topology which is denoted by ι (τ) .

(2) ω (ι (τ)) ≥ τ .

(3) ι (ω (ζ)) = ζ .

Theorem 4.6. [9] If φ_ψ,η : (X, τ¹) → (Y, τ²) is continuous, then φ_ψ,η . (X, ι (τ¹)) → (Y, ι (τ²)) is continuous.

Proposition 4.12. [9]

(1) Define ω : FYSTOP (E, K) → FSTOP (E, K) by ω (X, ζ) = (X, ω (ζ)) and V (φ_ψ,η) = φ_ψ,η. Then ω is a functor.

(2) Define ι : FSTOP (E, K) → FYSTOP (E, K) by ι (X, τ) = (X, ι (τ)) and ι (φ_ψ,η) = φ_ψ,η. Then ιis a functor.

5 Conclusion

It is well known that topology is an important area of mathematics with many applications in the domain of computer science and physical sciences. Soft topology is a relatively new and promising domain which can lead to the development of new mathematical models and innovative approaches that will significantly contribute to the solution of complex problems in natural sciences. This motivates us to observe categorical relations between all of the (fuzzy) soft topologicalspaces.

Footnotes

Acknowledgments

The authors are thankful to the associate editor for the improvement of the paper.

References

Ahmad

and Kharal

, On fuzzy soft sets, Advances in Fuzzy Systems2009 (2009), Article ID 586507.

Aygünoğlu

and Aygün

, Some notes on soft topological spaces, Neural Comput and Applic21 (2012), 113–119.

Aygünoğlu

and Aygün

, Introduction to fuzzy soft groups, Computers and Mathematics with Applications58 (2009), 1279–1286.

Aygünoğlu

, Çetkin

and Aygün

, An introduction to fuzzy soft topological spaces, Hacettepe Journal of Mathematics and Statistics43(2) (2014), 197–208.

Chang

C.L.

, Fuzzy topological spaces, J Math Anal Appl24 (1968), 182–190.

Çağman

and Enginoğlu

, Soft set theory and uni-int decision making, European Journal of Operational Research207 (2010), 848–855.

Çetkin

and Aygün

, A note on fuzzy soft topological spaces, The 8th Congerence of the European Society for Fuzzy Logic and Technology, Atlantis Press Proceedings, EUSFLAT, Milan-Italy, 2013, pp. 56–60. 10.2991/eusflat.2013.8.

Çetkin

and Aygün

, On fuzzy soft topogenous structure, Journal of Intell and Fuzzy Systems27 (2014), 247–255.

Çetkin

and Aygün

, Lowen functors in fuzzy soft topological spaces, submitted.

10.

Herrlich

, Cartesian closed topological categories, Math Colloq Univ Cape Town9 (1974), 1–16.

11.

Jun

Y.B.

, Lee

K.J.

and Park

C.H.

, Fuzzy soft set theory applied to BCK/BCI-algebras, Computers and Mathematics with Applications59 (2010), 3180–3192.

12.

Kharal

and Ahmad

, Mappings on fuzzy soft classes, Advances in Fuzzy Systems2009 (2009), Article ID 407890.

13.

Kharal

and Ahmad

, Mappings on soft classes, to appear in New Mathematics and Natural Computation.

14.

Lowen

, Fuzzy topological spaces and fuzzy compactness, J Math Anal Appl56 (1976), 621–633.

15.

Maji

P.K.

, Biswas

and Roy

A.R.

, Fuzzy soft sets, Journal of Fuzzy Mathematics9(3) (2001), 589–602.

16.

Maji

P.K.

, Biswas

and Roy

A.R.

, Soft set theory, Comput Math Appl45 (2003), 555–562.

17.

Molodtsov

, Soft set theory-First results, Computers and Mathematics with Applications37(4/5) (1999), 19–31.

18.

Pei

and Miao

, From soft sets to information systems, Granular Computing, 2005 IEEE International Conference on (2), 2005, pp. 617–621.

19.

Roy

A.R.

and Maji

P.K.

, A fuzzy soft set theoretic approach to decision making problems, Journal of Computational and Applied Mathematics203 (2007), 412–418.

20.

Shabir

and Naz

, On soft topological spaces, Computers and Mathematics with Applications61 (2011), 1786–1799.

21.

Tanay

and Kandemir

M.B.

, Topological structure of fuzzy soft sets, Computers and Mathematics with Applications61 (2011), 412–418.

22.

Yang

C.-F.

, Fuzzy soft semigroups and fuzzy soft ideals, Computers and Mathematics with Applications61 (2011), 255–261.

23.

Zadeh

L.A.

, Fuzzy Sets, Information and Control8 (1965), 338–353.

CATS of soft topological spaces

Abstract

Keywords

1 Introduction

2 Preliminaries

2.1 Soft sets

2.2 Fuzzy soft sets

3 (Fuzzy) Soft Topological Spaces

3.1 Soft topological spaces

3.2 Fuzzy soft topological spaces

4 Categorical interrelations between soft topological spaces

4.1 The category of topological spaces TOP as a subcategory of STop

4.2 The functor J : STop(E, K)→ TOP

4.3 STop(E, K) as a subcategory of QFSTop(E, K)

4.4 The functor j :QFSTop(E, K)→ STop(E, K)

4.5 STop(E, K) as a subcategory of LFSTop(E, K)

4.6 The functor I ″ : LFSTop(E, K)→ LSTop(E, K)

4.7 QFSTop(E, K) as a subcategory of FSTOP(E, K)

4.8 The functor I′ :FSTOP(E, K)→ QFSTop(E, K)

4.9 Functors between LFSTop(E, K), SFSTOP(E, K) and I a - FSTOP (E, K)

4.10 Functors between FYSTOP(E, K) and FSTOP(E, K)

5 Conclusion

Footnotes

Acknowledgments

References

4.2 The functor $J :$ STop(E, K)→ TOP

4.6 The functor $I^{″} :$ LFSTop(E, K)→ LSTop(E, K)

4.9 Functors between LFSTop(E, K), SFSTOP(E, K) and I_a - FSTOP (E, K)