IF topological vector spaces

In the present paper a notion of intuitionistic fuzzy topology on an intuitionistic fuzzy set has been developed. Also the concept of IF topological vector space has been introduced which is a combined structure of intuitionistic fuzzy vector space and intuitionistic fuzzy topology as defined by us. In this study, we generalized the action of a group on a set to intuitionistic fuzzy action. We obtained some basic results.

In this paper, by synthesizing the definition of fuzzy topologies of Chakraborty and Ahsanullah [8] and of Lowen [17], we extend it to IF setting to introduce a definition of intuitionistic fuzzy topology on intuitionistic fuzzy set. Also we introduce a notion of IF topological vector space associated with an intuitionistic fuzzy vector space [10] and intuitionistic fuzzy topology defined on this intuitionistic fuzzy vector space. Some fundamental properties of IF topological vector spaces have been investigated.

Preliminaries
Definition 2.1 ( [1]). Let X be a non-empty set. An intuitionistic fuzzy set (IFS for short) of X is defined as an object having the form A = { x, µ A (x), ν A (x) | x ∈ X}, where µ A : X → [0, 1] and ν A : X → [0, 1] denote the degree of membership (namely µ A (x)) and the degree of non-membership (namely ν A (x)) of each element x ∈ X to the set A, respectively, and 0 ≤µ A (x) + ν A (x) ≤ 1 for each x ∈ X. For the sake of simplicity we shall use the symbol A = (µ A , ν A ) for the intuitionistic fuzzy set Definition 2.2 ( [1]). Let A = (µ A , ν A ) and B = (µ B , ν B ) be intuitionistic fuzzy sets of a set X. Then (2) A = B iff A ⊆ B and B ⊆ A.
Definition 2.5 ( [5,18]). Let X be a vector space over the field K, the field of real and complex numbers, α ∈ K, A = (µ A , ν A ) and B = (µ B , ν B ) be two intuitionistic fuzzy sets of X.Then (1) the sum of A and B is defined to be the intuitionistic fuzzy set A + B = (µ A + µ B , ν A + ν B ) of X given by (2) αA is defined to be the IFS αA = (µ αA , ν αA ) of X, where Definition 2.6 ( [10]). An IFS V = (µ V , ν V ) of a vector space X over the field K is said to be intuitionistic fuzzy vector space over X if We denote the set of all intuitionistic fuzzy vector spaces over a vector space X by IF V S(X).
Lemma 2.7 ( [10]). Let V be an intuitionistic fuzzy set in a vector space X. Then, the following are equivalent: (1) V is an intuitionistic fuzzy vector space over X.
(3) For all scalars α, β and for all x, y ∈ X, we have Remark 2.8 ( [10]). Our definition of intuitionistic fuzzy vector space is equivalent to the definition of intuitionistic fuzzy subspace of [21] and [9].
Unless otherwise stated in the rest of the paper the collection of all intuitionistic fuzzy subsets of X is denoted by η X , where η = {(k, m) ∈ [0, 1] × [0, 1] : k + m ≤ 1}. Definition 2.11. [12] An intuitionistic fuzzy topology on a non-empty set X is a family τ of intuitionistic fuzzy sets in X which satisfies the following conditions: In this case (X, τ ) is called an intuitionistic fuzzy topological space. The members of τ are called the intuitionistic fuzzy open sets and the complement A C of an intuitionistic fuzzy open set in an intuitionistic fuzzy topological space (X, τ ) is called an intuitionistic fuzzy closed set.
Definition 2.12 ([12]). Let (X, τ ) and (Y, δ) be two intuitionistic fuzzy topological spaces and let f : X → Y be a function. Then f is said to be an intuitionistic fuzzy continuous if the pre-image of each intuitionistic fuzzy set of δ is an intuitionistic fuzzy set in τ.
where ∨ and ∧ are usual maximum and minimum inthe ordered set of real numbers.
Definition 2.14 ( [22]). Let X and Y be two non-empty sets and let A ∈ η X and B ∈ η Y . An intuitionistic fuzzy subset F of X × Y is said to be an intuitionistic fuzzy proper function from the intuitionistic fuzzy set A to the intuitionistic fuzzy set B if (i) F (x, y) ≤ A(x) B(y), for each (x, y) ∈ X × Y.
Definition 2.19 ([22]). The intuitionistic fuzzy proper function p A : A × B → A defined by p A ((x, y), z) = (A × B)(x, y) or (0, 1) accordingly as z = x or z = x ∀x, z ∈ X and ∀y ∈ Y is said to be the intuitionistic fuzzy projection map of A × B into A. Similarly, the intuitionistic fuzzy projection map p B : A × B → B is defined.
3 Intuitionistic fuzzy topology on intuitionistic fuzzy set Definition 3.1. An intuitionistic fuzzy set A of X is said to be constant which will be denoted by (k, m) ∼ and defined by µ A (x) = k and ν A (x) = m, (k, m) ∈ η, ∀x ∈ X.
Definition 3.2. Let A be a intuinistic fuzzy subset of X. A collection τ of intuitionistic fuzzy subsets of A satisfying is called an intuitionistic fuzzy topology or IF topology on the intuitionistic fuzzy set A. The pair (A, τ ) is called an intuitionistic fuzzy topological space. Members of τ will be called intuitionistic fuzzy open sets.
Unless otherwise mentioned by an intuitionistic fuzzy topological space we shall mean it in the sense of Definition 3.2 and (A, τ ) will denote an intuitionistic fuzzy topological space.
(iii) bijective if F is both injective and surjective.
Proof. Let x ∈ X and y be unique such that F (x, y) = A(x).
Therefore for a bijective intuitionistic fuzzy proper function F : A → B defined as in 2.14, its inverse G : B → A is defined by (iii) intuitionistic fuzzy homeomorphism if F is bijective, intuitionistic fuzzy continuous and inverse of F is also intuitionistic fuzzy continuous.
are intuitionistic fuzzy continuous proper functions, then the intuitionistic fuzzy proper function G • F : (A, τ ) → (C, τ 2 ) as defined in 3.11 is also intuitionistic fuzzy continuous.
). Since G and F are intuitionistic fuzzy continuous, for any Proof. For any z ∈ X,     Proof. The p A and p B are intuitionistic fuzzy continuous and open follows from Lemma 2.20 and 3.17. That τ × τ 1 is the smallest intuitionistic fuzzy topology in A × B with respect to which p A and p B are intuitionistic fuzzy continuous follows from the fact that if U ∈ τ , V ∈ τ 1 , then Lemma 3.22. If a be a normal element of B with respect to A, then the intuitionistic fuzzy proper function F a : Hence F a is intuitionistic fuzzy continuous.
Similarly we have, Lemma 3.23. If a be a normal element of B with respect to A, then the intuitionistic fuzzy proper function F a : if (y 1 , y 2 ) = (y 1 0 , y 2 0 ) is also intuitionistic fuzzy continuous.
Since F 1 and F 2 are intuitionistic fuzzy is intuitionistic fuzzy open set in τ 1 × τ 2 . Hence F = F 1 × F 2 is intuitionistic fuzzy continuous.
Theorem 3.25. Let (A i , τ i ) and (B i , σ i ), i = 1, 2 be intuitionistic fuzzy topological spaces and Definition 4.1 ( [17]). Given a topological space (X, τ ), the collection ω(τ ), of all fuzzy sets in X which are lower semi-continuous, as functions from X to the unit interval equipped with the usual topology, is a fuzzy topology on X. This fuzzy topology ω(τ ) is said to be the fuzzy topology generated by the topology τ.  16]). Let K be the field of real or complex numbers. Then the fuzzy usual topology on K is the fuzzy topology generated by the usual topology on K.

IF topological vector space
Throughout the section we consider V as an intuitionistic fuzzy vector space associated with a vector space X and the ground field K. We consider K to be equipped with the fuzzy usual topology ν as defined in Definition 4.2. Definition 4.3. Let X be a vector space over the field K with θ as the null vector. Let V be an intuitionistic fuzzy vector space over X, a ∈ X and k 0 ∈ K be fixed. Let us define the intuitionistic fuzzy proper functions for all x, y, z ∈ X, k, m ∈ K. Remark 4.5. Here we use the term IF topological vector space as there is a notion of intuitionistic fuzzy topological vector space in [18] where the intuitionistic fuzzy topology is in the sense of Coker and the underlying vector space is crisp vector space.
Proof. Let τ be an IF vector topology on V and k, m ∈ K. Since k ∈ K is normal element of K with respect to V , by Lemma 3.23, the intuitionistic fuzzy proper function F k : (V, τ ) → Also, by definition of IF vector topology, F : (K × V, ν × τ ) → (V, τ ) is intuitionistic fuzzy continuous. Hence by Proposition 3.14, F • F k : (V, τ ) → (V, τ ) defined by is intuitionistic fuzzy continuous. Therefore by Proposition 3.14, is intuitionistic fuzzy continuous for all k, m ∈ K. We know that the projection mapping p V : if z = x and since θ is normal of V with respect to V , by Lemma 3.22, are intuitionistic fuzzy continuous proper functions. Hence by Proposition 3.14, Since F L (k,m) is intuitionistic fuzzy continuous for all k, m ∈ K, taking k = 1, m = 1 we have F ⊕ : (V × V, τ × τ ) → (V, τ ) is intuitionistic fuzzy continuous. Hence proved. Proposition 4.7. If (V, τ ) is an IF topological vector space, then F k is an intuitionistic fuzzy homeomorphism of (V, τ ) onto itself, for all k( = 0) ∈ K.
Proof. Since (V, τ ) is an IF topological vector space, F : (K×V, ν×τ ) → (V, τ ) is intuitionistic fuzzy continuous. Also, by Lemma 3.23, the intuitionistic fuzzy proper function F k : Proposition 4.8. If (V, τ ) is an IF topological vector space and if a is a normal element of V with respect to V , then F a is an intuitionistic fuzzy homeomorphism of (V, τ ) onto itself.
Proof. If a is a normal element of V with respect to V , then F a = F ⊕ • F a is intuitionistic fuzzy continuous by continuity of F ⊕ and F a . Also, inverse of F a is F −a and hence F −a is also intuitionistic fuzzy continuous. Therefore F a is intuitionistic fuzzy homeomorphism from (V, τ ) into itself for any normal a of V with respect to V .
Let V and W be two intuitionistic fuzzy vector spaces in two vector spaces X and Y respectively and θ, θ be the null vectors of X and Y respectively.
for all x, z ∈ X, y, w ∈ Y and k, m ∈ K. (ii) F (V ) is an intuitionistic fuzzy vector space over Y .