Separation Axioms in Intuitionistic Fuzzy Topological Spaces

Abstract

In this paper we have studied separation axioms 𝑇_𝑖, 𝑖=0,1,2 in an intuitionistic fuzzy topological space introduced by Coker. We also show the existence of functors ℬ∶IF-Top→BF-Top and 𝒟∶BF-Top→IF-Top and observe that 𝒟 is left adjoint to ℬ.

1. Introduction

Fuzzy sets were introduced by Zadeh [1] in 1965 as follows: a fuzzy set 𝐴 in a nonempty set 𝑋 is a mapping from 𝑋 to the unit interval [0,1], and 𝐴(𝑥) is interpreted as the degree of membership of 𝑥 in 𝐴. Atanassov [2] generalized this concept and introduced intuitionistic fuzzy sets which take into account both the degrees of membership and of nonmembership subject to the condition that their sum does not exceed 1. Çoker [3] subsequently initiated a study of intuitionistic fuzzy topological spaces.

In this paper we have searched for appropriate definitions of the separation axioms 𝑇_𝑖, 𝑖=0,1,2 in intuitionistic fuzzy topological spaces.

Hausdorffness in an intuitionistic fuzzy topological space has been introduced earlier by Çoker [3], Bayhan and Çoker [4], and Lupianez [5]. In [4], the authors have given six possible definitions of Hausdorffness including that given in [3], and a comparative study has been done. In this paper we have introduced another definition which generalizes the corresponding definition in a fuzzy topological space given in [6]. Our definition is more general than those given in [3, 5], and it turns out to be equivalent to 𝐹𝑇₂(𝑣𝑖) in [4].

𝑇₁-ness in an intuitionistic fuzzy topological space has been defined earlier in [4] in six possible ways. Out of those, we have chosen 𝐹𝑇₁(𝑖𝑖) as it generalizes the most appropriate definition of 𝑇₁-ness in a fuzzy topological space (cf. definition 5.1, [7]). We have also introduced a suitable definition of 𝑇₀-ness in an intuitionistic topological space.

The appropriateness of the definitions has been established by proving several basic desirable results; for example, they satisfy hereditary, productive, and projective properties. We have also shown that the functor ℬ∶ IF-Top → BF-Top preserves these separation properties.

2. Preliminaries

Throughout 𝑋 denotes a nonempty set, 𝐼 denotes the unit interval [0,1], and 𝐼₀ and 𝐼₁ denote the intervals (0,1] and [0,1), respectively. A fuzzy set in 𝑋 is a function from 𝑋 to 𝐼. The collection of all fuzzy sets in 𝑋 is denoted by 𝐼^𝑋. For any 𝐴∈𝐼^𝑋, 𝐴^′ denotes the fuzzy complement of 𝐴, and the constant fuzzy set in 𝑋, taking value 𝛼∈𝐼, is denoted by 𝛼. A crisp subset of 𝑋 will be identified with its characteristic function. If 𝑌⊆𝑋, then 𝐴∈𝐼^𝑌 will be identified with the fuzzy set in 𝑋 which takes the same value as 𝐴 if 𝑥∈𝑌 and zero if 𝑥∉𝑌.

Definition 1 (Atanassov [2]). Let 𝑋 be a nonempty set. An intuitionistic fuzzy set (IFS, in short)  𝐴 is an ordered pair (𝜇_𝐴,𝜈_𝐴) of fuzzy sets in 𝑋. Here (𝜇_𝐴,𝜈_𝐴)(𝑥) = (𝜇_𝐴(𝑥),𝜈_𝐴(𝑥)) and 𝜇_𝐴(𝑥), 𝜈_𝐴(𝑥), respectively, denote the degree of membership and the degree of nonmembership of 𝑥∈𝑋 to the set 𝐴 and 0≤𝜇_𝐴(𝑥)+𝜈_𝐴(𝑥)≤1 for each 𝑥∈𝑋.
We identify an ordinary fuzzy set 𝐴∈𝐼^𝑋 with the intuitionistic fuzzy set (𝐴,𝐴^′).

Definition 2 (Atanassov [2]). Let 𝑋 be a nonempty set and 𝐴,𝐵 be given by (𝜇_𝐴,𝜈_𝐴) and (𝜇_𝐵,𝜈_𝐵), respectively,(a)𝐴⊆𝐵if𝜇_𝐴(𝑥)≤𝜇_𝐵(𝑥)and𝜈_𝐴(𝑥)≥𝜈_𝐵(𝑥)forall𝑥∈𝑋, (b)𝐴=𝐵if𝐴⊆𝐵and𝐵⊆𝐴, (c)𝐴=(𝜈_𝐴,𝜇_𝐴), (d)𝐴∩𝐵=(𝜇_𝐴∩𝜇_𝐵,𝜈_𝐴∪𝜈_𝐵), (e)𝐴∪𝐵=(𝜇_𝐴∪𝜇_𝐵,𝜈_𝐴∩𝜈_𝐵).

Definition 3 (Çoker [3]). Let {𝐴_𝑖∶𝑖∈𝐽} be an arbitrary family of IFSs in 𝑋. Then (a)∩𝐴_𝑖=(∩𝜇_𝐴𝑖,∪𝜈_𝐴𝑖), (b)∪𝐴_𝑖=(∪𝜇_𝐴𝑖,∩𝜈_𝐴𝑖), (c)0_∼=(0,1), 1_∼=(1,0).

Definition 4 (Çoker [3]). Let 𝑋 and 𝑌 be two nonempty sets and 𝑓∶𝑋→𝑌 be a function. If 𝐴 and 𝐵 be IFSs in 𝑋 and 𝑌, respectively, then(a)𝑓(𝐴)=(𝑓(𝜇_𝐴),(1−𝑓(1−𝜈_𝐴))), (b)𝑓⁻¹(𝐵)=(𝑓⁻¹(𝜇_𝐵),𝑓⁻¹(𝜈_𝐵)). It is easy to verify that 𝑓⁻¹(∩𝐴_𝑖)=∩𝑓⁻¹(𝐴_𝑖) and 𝑓⁻¹(∪𝐴_𝑖)=∪𝑓⁻¹(𝐴_𝑖).

Definition 5 (Wong [8]). A fuzzy point 𝑥_𝑟 in 𝑋 is a fuzzy set in 𝑋 taking value 𝑟∈(0,1) at 𝑥 and zero elsewhere, and 𝑥 and 𝑟 are, respectively, called the support and value of 𝑥.
A fuzzy point 𝑥_𝑟 is said to belong to a fuzzy set 𝐴  (notation∶𝑥_𝑟∈𝐴) if 𝑟<𝐴(𝑥) (cf. [6]).
Two fuzzy points are said to be distinct if their supports are distinct.

Definition 6. Let 𝑋 be a nonempty set and 𝑥∈𝑋 a fixed element in 𝑋. If 𝛼∈(0,1) and 𝛽∈(0,1) are two fixed real numbers such that 𝛼+𝛽≤1, then the IFS 𝑥_(𝛼,𝛽)=(𝑥_𝛼,1−𝑥_(1−𝛽)) is called an intuitionistic fuzzy point (IFP, in short) in 𝑋, and 𝑥 is called its support. Two IFPs are said to be distinct if their supports are distinct.
Let 𝑥_(𝛼,𝛽) be an IFP in 𝑋 and 𝐴=(𝜇_𝐴,𝜈_𝐴) be an IFS in 𝑋. Then 𝑥_(𝛼,𝛽) is said to belong to 𝐴  (notation∶𝑥_(𝛼,𝛽)∈𝐴, in short) if 𝛼<𝜇_𝐴(𝑥), 𝛽>𝜈_𝐴(𝑥) (cf. [9]).

We identify a fuzzy point 𝑥_𝑟 in 𝑋 by the intuitionistic fuzzy point 𝑥_{(𝑟,(1−𝑟))} in 𝑋.

Proposition 7. An intuitionistic fuzzy set 𝐴 in 𝑋 is the union of all intuitionistic fuzzy points belonging to 𝐴.

The proof is on similar lines as in [10, Theorem  2.4] and hence is omitted.

Replacing fuzzy sets by intuitionistic fuzzy sets in Chang’s definition of a fuzzy topological space, we get the following.

Definition 8 (Çoker [3]). An intuitionistic fuzzy topology (IFT, in short) on a nonempty set 𝑋 is a family 𝜏 of IFSs in 𝑋 satisfying the following axioms:(1)0_∼,1_∼∈𝜏, (2)𝐺₁∩𝐺₂∈𝜏,forall𝐺_𝑖∈𝜏,𝑖=1,2, (3)∪𝐺_𝑖∈𝜏 for any arbitrary family {𝐺_𝑖∈𝜏∶𝑖∈𝐽}.The pair (𝑋,𝜏)  is called an intuitionistic fuzzy topological space (IFTS, in short), members of 𝜏 are called intuitionistic fuzzy open sets (IFOS, in short) in 𝑋, and their complements are called intuitionistic fuzzy closed sets (IFCS, in short).

Definition 9. Let (𝑋,𝜏) be an IFTS. A subfamily ℬ⊆𝜏 is called a base for 𝜏 if every 𝑈∈𝜏 can be written as a union of members of ℬ.

Proposition 10. Let (𝑋,𝜏) be an IFTS, and then a subfamily ℬ⊆𝜏 is a base for 𝜏 if and only if for all 𝑈∈𝜏 and intuitionistic fuzzy point 𝑥_(𝛼,𝛽)∈𝑈, ∃𝐵∈ℬ such that 𝑥_(𝛼,𝛽)∈𝐵⊆𝑈.

The proof is easy omitted.

Definition 11. Let (𝑋,𝜏) be an IFTS. Then a subfamily 𝒮⊆𝜏 is called a subbase for 𝜏 if the family of finite intersections of members of 𝒮 forms a base for 𝜏.
Given any collection 𝒮 of IFSs in 𝑋, containing 0_∼ and 1_∼, the set 𝜏 consisting of arbitrary unions of finite intersections of members of 𝒮 forms an IFT on 𝑋. This is the smallest IFT on 𝑋 containing 𝒮 and is called the IFT generated by 𝒮.

Definition 12 (S. J. Lee and E. P. Lee [10]). An IFS 𝑁 in an IFTS (𝑋,𝜏) is called an intuitionistic fuzzy neighborhood (IFN, in short) of an IFP 𝑥_(𝛼,𝛽) if ∃𝑈∈𝜏 such that 𝑥_(𝛼,𝛽)∈𝑈⊆𝑁.

Proposition 13. Let (𝑋,𝜏) be an IFTS. Then an IFS 𝐴 in 𝑋 is an IFOS if and only if 𝐴 is an IFN of each of IFP 𝑥_(𝛼,𝛽)∈𝐴.

The proof is on similar lines as in ([10], Theorem  2.6) and hence is omitted.

Definition 14 (S. J. Lee and E. P. Lee [10]). A map 𝑓∶(𝑋,𝜏)→(𝑌,𝛿) between IFTSs is called intuitionistic fuzzy continuous if 𝑓⁻¹(𝑈)∈𝜏,forall𝑈∈𝛿.

Definition 15 (Abu Safia et al. [12]). Let 𝑋 be a nonempty set and 𝜏₁, 𝜏₂ be two fuzzy topologies on 𝑋. Then (𝑋,𝜏₁,𝜏₂) is called a bifuzzy topological space (BFTS, in short).
A map 𝑓∶(𝑋,𝜏₁,𝜏₂)→(𝑌,𝛿₁,𝛿₂) between two BFTSs is said to be FP continuous if 𝑓⁻¹(𝑈_𝑖)∈𝜏_𝑖,forall𝑈_𝑖∈𝛿_𝑖,𝑖=1,2.

Definition 16 (Bayhan and Çoker [4]). Let 𝐴=(𝜇_𝐴,𝜈_𝐴) and 𝐵=(𝜇_𝐵,𝜈_𝐵) be IFSs in 𝑋 and 𝑌, respectively, and then 𝐴×𝐵 is the IFS in 𝑋×𝑌 defined as follows 𝐴×𝐵=𝜇_𝐴×𝜇_𝐵,𝜈_𝐴∗𝜈_𝐵, (1) where (𝜇_𝐴×𝜇_𝐵)(𝑥,𝑦)=min(𝜇_𝐴(𝑥),𝜇_𝐵(𝑦)), forall(𝑥,𝑦)∈𝑋×𝑌 and (𝜈_𝐴∗𝜈_𝐵)(𝑥,𝑦)=max(𝜈_𝐴(𝑥),𝜈_𝐵(𝑦)), forall(𝑥,𝑦)∈𝑋×𝑌.

This definition can be extended to an arbitrary family of IFSs as follows.

If {𝐴_𝑖=(𝜇_𝐴𝑖,𝜈_𝐴𝑖), 𝑖∈𝐽} is a family of IFSs in 𝑋_𝑖, then their product is defined as the IFS in Π𝑋_𝑖 given by Π𝐴_𝑖=Π𝜇_𝐴𝑖,Π^∗𝜈_𝐴𝑖, (2) where Π𝜇_𝐴𝑖(𝑥)= inf 𝜇_𝐴𝑖(𝑥_𝑖), forall𝑥=Π𝑥_𝑖∈𝑋 and Π^∗𝜈_𝐴𝑖(𝑥)=sup𝜈_𝐴𝑖(𝑥_𝑖),forall𝑥=Π𝑥_𝑖∈𝑋.

Definition 17 (Bayhan and Çoker [4]). Let (𝑥_𝑖,𝜏_𝑖),𝑖=1,2 be two IFTSs, and then the product IFT 𝜏₁×𝜏₂ on 𝑋₁×𝑋₂ is defined as the IFT generated by {𝑝_𝑖⁻¹(𝑈_𝑖)∶𝑈_𝑖∈𝜏_𝑖,𝑖=1,2} where 𝑝_𝑖∶𝑋₁×𝑋₂→𝑋_𝑖, 𝑖=1,2  are the projection maps, and the IFTS (𝑋₁×𝑋₂,𝜏₁×𝜏₂) is called the product IFTS.

This definition can be extended to an arbitrary family of IFTSs as follows.

Let {(𝑋_𝑖,𝜏_𝑖)∶𝑖∈𝐽} be a family of IFTSs. Then the product intuitionistic fuzzy topology 𝜏 on 𝑋=Π𝑋_𝑖 is the one having {𝑝_𝑗⁻¹(𝑈_𝑗)∶𝑈_𝑗∈𝜏_𝑗,𝑗∈𝐽} as a subbase where 𝑝_𝑗∶𝑋→𝑋_𝑗 is the 𝑗th projection map. (𝑋,𝜏) is called the product IFTS of the family {(𝑋_𝑖,𝜏_𝑖)∶𝑖∈𝐽}.

Definition 18. A fuzzy topological space (𝑋,𝜏) is called(a)𝑇₀ if forall𝑥,𝑦∈𝑋, 𝑥≠𝑦, ∃𝑈∈𝜏 such that either 𝑈(𝑥)=1, 𝑈(𝑦)=0, or 𝑈(𝑦)=1, 𝑈(𝑥)=0, (b)𝑇₁ if forall𝑥,𝑦∈𝑋, 𝑥≠𝑦, ∃𝑈,𝑉∈𝜏 such that 𝑈(𝑥)=1, 𝑈(𝑦)=0, 𝑉(𝑦)=1, and 𝑉(𝑥)=1, (c)𝑇₂   (Hausdorff) if for all pair of distinct fuzzy points 𝑥_𝑟,𝑦_𝑠 in 𝑋, ∃𝑈,𝑉∈𝜏 such that 𝑥_𝑟∈𝑈, 𝑦_𝑠∈𝑉, and 𝑈∩𝑉=0, (d)𝑞-𝑇₂   (𝑞-Hausdorff) if for any pair of distinct fuzzy points 𝑥_𝑟, and 𝑦_𝑠  ∃𝑈,𝑉∈𝜏 such that 𝑥_𝑟∈𝑈, 𝑦_𝑠∈𝜏 and 𝑈⊆𝑉^′. Here definitions (d), (c), (b), and (a) are from [5–7, 13], respectively.

Definition 19. Let (𝑋,𝜏₁,𝜏₂) be a BFTS. Then it is called(a)𝑇₀ if forall𝑥,𝑦∈𝑋,𝑥≠𝑦, ∃𝑈∈𝜏₁∪𝜏₂ such that 𝑈(𝑥)=1, 𝑈(𝑦)=0 or 𝑈(𝑥)=0, 𝑈(𝑦)=1,(b)𝑇₁ if forall𝑥,𝑦∈𝑋,𝑥≠𝑦, ∃𝑈∈𝜏₁ and 𝑉∈𝜏₂ such that 𝑈(𝑥)=1, 𝑈(𝑦)=0 and 𝑉(𝑥)=0, 𝑉(𝑦)=1, (c)𝑇₂ if for all pair of distinct fuzzy points 𝑥_𝑟,𝑦_𝑠 in 𝑋, ∃𝑈∈𝜏₁, 𝑉∈𝜏₂ such that 𝑥_𝑟∈𝑈, 𝑦_𝑠∈𝑉 and 𝑈∩𝑉=0. Here definitions (a) and (b) are from [14], and (c) is from [15].

For the categorical concepts used here, we refer the reader to [16].

3. Separation Axioms in Intuitionistic Fuzzy Topological Spaces

Definition 20. An IFTS (𝑋,𝜏) is called(a)𝑇₀ if forall𝑥,𝑦∈𝑋, 𝑥≠𝑦, ∃𝑈=(𝜇_𝑈,𝜈_𝑈), 𝑉=(𝜇_𝑉,𝜈_𝑉)∈𝜏 such that (𝜇_𝑈,𝜈_𝑈)(𝑥)=(1,0), (𝜇_𝑈,𝜈_𝑈)(𝑦)=(0,1) or (𝜇_𝑉,𝜈_𝑉)(𝑥)=(0,1), (𝜇_𝑉,𝜈_𝑉)(𝑦)=(1,0),(b)(Bayhan and Çoker [4]). 𝑇₁ if forall𝑥,𝑦∈𝑋, 𝑥≠𝑦, ∃𝑈=(𝜇_𝑈,𝜈_𝑈), 𝑉=(𝜇_𝑉,𝜈_𝑉)∈𝜏 such that (𝜇_𝑈,𝜈_𝑈)(𝑥)=(1,0), (𝜇_𝑈,𝜈_𝑈)(𝑦)=(0,1), (𝜇_𝑉,𝜈_𝑉)(𝑥)=(0,1) and (𝜇_𝑉,𝜈_𝑉)(𝑦)=(1,0),(c)𝑇₂   (Hausdorff) if for all pair of distinct intuitionistic fuzzy points 𝑥_(𝛼,𝛽),𝑦_(𝛾,𝛿) in 𝑋, ∃𝑈,𝑉∈𝜏 such that 𝑥_(𝛼,𝛽)∈𝑈, 𝑦_(𝛾,𝛿)∈𝑉 and 𝑈∩𝑉=0_∼, (d)𝑞-𝑇₂ if for every pair of distinct intuitionistic fuzzy points 𝑥_(𝛼,𝛽),𝑦_(𝛾,𝛿) in 𝑋, ∃𝑈 and 𝑉∈𝜏 such that 𝑥_(𝛼,𝛽)∈𝑈, 𝑦_(𝛾,𝛿)∈𝑉 and 𝑈⊆𝑉^′.

Example 21. Let 𝑋={𝑎,𝑏} and let 𝜏={0_∼,𝐴,𝐵,1_∼}, where 𝐴=⟨𝑥,(𝑎/1,𝑏/0),(𝑎/0,𝑏/1)⟩ and 𝐵=⟨𝑥,(𝑎/1,𝑏/0),(𝑎/0,𝑏/1)⟩, then (𝑋,𝜏) is an IFTS, and it is 𝑇₀, 𝑇₁, 𝑇₂ (Hausdorff) and 𝑞-𝑇₂.

We have 𝑇₂⇒𝑇₁⇒𝑇₀ and 𝑇₂⇒𝑞-𝑇₂, but none of the implication are reversible.

Now we associate a BFTS with an IFTS and vice versa on parallel lines as in Bayhan and Çoker [11].

Let (𝑋,𝜏) be an IFTS and 𝜏₁={𝜇_𝐴∣∃𝜈_𝐴∈𝐼^𝑋 such that (𝜇_𝐴,𝜈_𝐴)∈𝜏}, 𝜏₂={(1−𝜈_𝐴)∣∃𝜇_𝐴∈𝐼^𝑋 such that (𝜇_𝐴,𝜈_𝐴)∈𝜏}. It is easy to see that (𝑋,𝜏₁) and (𝑋,𝜏₂) are fuzzy topological spaces in Chang’s sense.

(𝑋,𝜏₁,𝜏₂) is called the bifuzzy topological space associated with the IFTS (𝑋,𝜏).

Proposition 22. Let (𝑋,𝜏₁,𝜏₂) be a BFTS and 𝜏_𝜏1,𝜏2={(𝑈,𝑉^′)∣𝑈∈𝜏₁,𝑉∈𝜏₂ and 𝑈⊆𝑉}. Then (𝑋,𝜏_𝜏1,𝜏2) is an IFTS and (𝜏_𝜏1,𝜏2)₁=𝜏₁, (𝜏_𝜏1,𝜏2)₂=𝜏₂.

Proof. Clearly members of 𝜏_𝜏1,𝜏2 are intuitionistic fuzzy sets, and 0_∼ and 1_∼ belong to it. Now let (𝑈_𝑖,𝑉^′_𝑖)∈𝜏_𝜏1,𝜏2, 𝑖=1,2 then (𝑈₁,𝑉^′₁)∩(𝑈₂,𝑉^′₂) = (𝑈₁∩𝑈₂,𝑉^′₁∪𝑉^′₂) = (𝑈₁∩𝑈₂,(𝑉₁∩𝑉₂)^′)∈𝜏_𝜏1,𝜏2. Further let {(𝑈_𝑖,𝑉^′_𝑖)∶𝑖∈𝐽, where 𝐽 is arbitrary}⊆𝜏_𝜏1,𝜏2. Then ⋃{(𝑈_𝑖,𝑉^′_𝑖)}=(⋃𝑈_𝑖,⋂𝑉^′_𝑖)=(⋃𝑈_𝑖,(⋃𝑉_𝑖)^′)∈𝜏_𝜏1,𝜏2. Thus (𝑋,𝜏_𝜏1,𝜏2) is an IFTS.
Now let 𝑈∈𝜏₁ then (𝑈,𝜙)∈𝜏_𝜏1,𝜏2. Therefore 𝜏₁⊆(𝜏_𝜏1,𝜏2)₁. Conversely let 𝑈∈(𝜏_𝜏1,𝜏2)₁ then ∃𝑉∈𝐼^𝑋 such that (𝑈,𝑉)∈𝜏_𝜏1,𝜏2⇒𝑈∈𝜏₁, so (𝜏_𝜏1,𝜏2)₁⊆𝜏₁. Thus (𝜏_𝜏1,𝜏2)₁=𝜏₁. Similarly we can show that (𝜏_𝜏1,𝜏2)₂=𝜏₂.

The IFTS (𝑋,𝜏_𝜏1,𝜏2) is called the IFTS associated with the BFTS (𝑋,𝜏₁,𝜏₂).

Proposition 23. Let (𝑋,𝜏) and (𝑌,𝛿) be two IFTSs and 𝑓∶(𝑋,𝜏)→(𝑌,𝛿) be IF-continuous. Then 𝑓∶(𝑋,𝜏₁,𝜏₂)→(𝑌,𝛿₁,𝛿₂) is FP-continuous (Here (𝑋,𝜏₁,𝜏₂) and (𝑌,𝛿₁,𝛿₂) are BFTSs associated with (𝑋,𝜏) and (𝑌,𝛿), resp.).

Proof. Let 𝑓∶(𝑋,𝜏)→(𝑌,𝛿) be IF-continuous. To show that 𝑓∶(𝑋,𝜏₁,𝜏₂)→(𝑌,𝛿₁,𝛿₂) is FP-continuous, take 𝑈∈𝛿₁ then ∃𝑉∈𝐼^𝑌 such that (𝑈,𝑉)∈𝛿. Hence since 𝑓 is IF-continuous, 𝑓⁻¹(𝑈,𝑉)∈𝜏, that is, (𝑓⁻¹(𝑈),𝑓⁻¹(𝑉))∈𝜏⇒𝑓⁻¹(𝑈)∈𝜏₁. Further take 𝑉₁∈𝛿₂ then ∃𝑈₁∈𝐼^𝑌 such that (𝑈₁,𝑉^′₁)∈𝛿⇒𝑓⁻¹(𝑈₁,𝑉^′₁)∈𝜏⇒(𝑓⁻¹(𝑈₁),𝑓⁻¹(𝑉^′₁))∈𝜏⇒(𝑓⁻¹(𝑈₁),(𝑓⁻¹(𝑉₁))^′)∈𝜏⇒𝑓⁻¹(𝑉₁)∈𝜏₂. Thus 𝑓 is FP-continuous.

Proposition 24. Let (𝑋,𝜏₁,𝜏₂) and (𝑌,𝛿₁,𝛿₂) be two BFTSs and (𝑋,𝜏_𝜏1,𝜏2) and (𝑌,𝛿_𝛿1,𝛿2) be the associated IFTSs respectively. Then 𝑓∶(𝑋,𝜏₁,𝜏₂)→(𝑌,𝛿₁,𝛿₂) is FP-continuous if and only if 𝑓∶(𝑋,𝜏_𝜏1,𝜏2)→(𝑌,𝛿_𝛿1,𝛿2) is IF-continuous.

Proof. Let 𝑓∶(𝑋,𝜏₁,𝜏₂)→(𝑌,𝛿₁,𝛿₂) be FP-continuous. To show that 𝑓∶(𝑋,𝜏_𝜏1,𝜏2)→(𝑌,𝛿_𝛿1,𝛿2) is IF continuous, take any 𝑈=(𝜇_𝑈,𝜈^′_𝑈)∈𝛿_𝛿1,𝛿2, where 𝜇_𝑈∈𝛿₁, 𝜈_𝑈∈𝛿₂ and 𝜇_𝑈⊆𝜈_𝑈. Now using FP-continuity of 𝑓, 𝑓⁻¹(𝜇_𝑈)∈𝜏₁ and 𝑓⁻¹(𝜈_𝑈)∈𝜏₂. So (𝑓⁻¹(𝜇_𝑈),(𝑓⁻¹(𝜈_𝑈))^′)∈𝜏_𝜏1,𝜏2⇒(𝑓⁻¹(𝜇_𝑈),𝑓⁻¹(𝜈^′_𝑈))∈𝜏_𝜏1,𝜏2⇒𝑓⁻¹(𝜇_𝑈,𝜈^′_𝑈)∈𝜏_𝜏1,𝜏2, that is, 𝑓⁻¹(𝑈)∈𝜏_𝜏1,𝜏2, showing that 𝑓 is IF-continuous.
The converse follows from the previous Proposition 23 in view of the fact that (𝜏_𝜏1,𝜏2)₁=𝜏₁, (𝜏_𝜏1,𝜏2)₂=𝜏₂ and (𝛿_𝛿1,𝛿2)₁=𝛿₁ and (𝛿_𝛿1,𝛿2)₂=𝛿₂.

The category of all BFTS together with FP-continuous functions will be denoted by BF-Top and the category of all IFTS together with IF-continuous function will be denoted by IF-Top.

Now we define ℬ∶ IF-Top → BF-Top as follows:

ℬ(𝑋,𝜏)=(𝑋,𝜏₁,𝜏₂), ℬ(𝑓)=𝑓,forall morphism 𝑓  and 𝒟∶ BF-Top → IF-Top as follows:

𝒟(𝑋,𝜏₁,𝜏₂)=(𝑋,𝜏_𝜏1,𝜏2), 𝒟(𝑓)=𝑓,forall morphism 𝑓.

It can be checked easily that ℬ and 𝒟 are covariant functors, and in view of Proposition 22 we have the following remark.

Remark 25. ℬ∘𝒟=𝐼𝑑_BF-Top, the identity functor.

Theorem 26. The functor 𝒟∶ BF-Top → IF-Top is left adjoint to the functor ℬ∶ IF-Top → BF-Top.

The proof is on parallel lines as in ([11], Theorem  3.10) and hence is omitted.

Proposition 27. The following statements are equivalent in an IFTS (𝑋,𝜏):(1)(𝑋,𝜏) is 𝑇₁,(2)({𝑥},{𝑥}^′) is intuitionistic fuzzy closed in (𝑋,𝜏), for all 𝑥∈𝑋.

Proof. (1)⇒(2) We show that ({𝑥}^′,{𝑥}) is intuitionistic fuzzy open in 𝑋. Choose any IFP 𝑦_(𝛼,𝛽) in ({𝑥}^′,{𝑥}) then, 𝑦≠𝑥. Hence ∃ IFOSs 𝑈,𝑉∈𝜏 such that 𝑈(𝑥)=(𝜇_𝑈(𝑥),𝜈_𝑈(𝑥))=(1,0), 𝑈(𝑦)=(𝜇_𝑈(𝑦),𝜈_𝑈(𝑥))=(0,1) and 𝑉(𝑥)=(𝜇_𝑉(𝑥),𝜈_𝑉(𝑥))=(0,1), 𝑉(𝑦)=(𝜇_𝑉(𝑦),𝜈_𝑉(𝑦))=(1,0). Now 𝑦_(𝛼,𝛽)∈𝑉⊆({𝑥}^′,{𝑥}). Therefore in view of Proposition 13, ({𝑥}^′,{𝑥}) is an IFOS.
(2)⇒(1) Choose 𝑥,𝑦∈𝑋 such that 𝑥≠𝑦. Then ({𝑥},{𝑥}^′) and ({𝑦},{𝑦}^′) are IFCSs and hence 𝑈=({𝑥}^′,{𝑥}), 𝑉=({𝑦}^′,{𝑦}) are IFOSs such that 𝑈(𝑥)=(0,1), 𝑈(𝑦)=(1,0), 𝑉(𝑦)=(0,1) and 𝑉(𝑥)=(1,0) showing that (𝑋,𝜏) is 𝑇₁.

Proposition 28. If an IFTS (𝑋,𝜏) is 𝑇₁, then its associated BFTS is 𝑇₁.

Proof. Let (𝑋,𝜏) be 𝑇₁. Then forall𝑥≠𝑦, ∃𝑈=(𝜇_𝑈,𝜈_𝑈) and 𝑉=(𝜇_𝑉,𝜈_𝑉)∈𝜏 such that (𝜇_𝑈,𝜈_𝑈)(𝑥)=(1,0), (𝜇_𝑈,𝜈_𝑈)(𝑦)=(0,1), (𝜇_𝑉,𝜈_𝑉)(𝑥)=(0,1), and (𝜇_𝑉,𝜈_𝑉)(𝑦)=(1,0). Now (𝜇_𝑈,𝜈_𝑈)(𝑥)=(1,0)⇒𝜇_𝑈(𝑥)=1, 𝜈_𝑈(𝑥)=0.
Similarly, 𝜇_𝑈(𝑦)=0,𝜈_𝑈(𝑦)=1, 𝜇_𝑉(𝑥)=0,𝜈_𝑉(𝑥)=1, and 𝜇_𝑉(𝑦)=1,𝜈_𝑉(𝑦)=0. Now 𝜇_𝑈∈𝜏₁, (1−𝜈_𝑉)∈𝜏₂ and further 𝜇_𝑈(𝑥)=1, 𝜇_𝑈(𝑦)=0, (1−𝜈_𝑉)(𝑥)=0, and (1−𝜈_𝑉)(𝑦)=1 showing that (𝑋,𝜏₁,𝜏₂) is a 𝑇₁ space.

Similarly it can be shown that if an IFTS (𝑋,𝜏) is 𝑇₀, then its associated BFTS is 𝑇₀.

Theorem 29. Let {(𝑋_𝑖,𝜏_𝑖)∶𝑖∈𝐽} be a family of 𝑇₁ IFTSs and (𝑋,𝜏) be their product IFTS. Then (𝑋,𝜏) is 𝑇₁ if and only if (𝑋_𝑖,𝜏_𝑖) is 𝑇₁, for all 𝑖∈𝐽.

Proof. Let (𝑋_𝑖,𝜏_𝑖) be 𝑇₁  forall𝑖∈𝐽. To show that (𝑋,𝜏) is 𝑇₁, choose 𝑥,𝑦∈𝑋, 𝑥≠𝑦. Let 𝑥=Π𝑥_𝑖, 𝑦=Π𝑦_𝑖 then ∃𝑗∈𝐽 such that 𝑥_𝑗≠𝑦_𝑗. Now since (𝑋_𝑗,𝜏_𝑗) is 𝑇₁, ∃𝑈_𝑗=(𝜇_𝑈𝑗,𝜈_𝑈𝑗) and 𝑉_𝑗=(𝜇_𝑉𝑗,𝜈_𝑉𝑗)∈𝜏_𝑗 such that 𝜇_𝑈𝑗(𝑥_𝑗)=1, 𝜈_𝑈𝑗(𝑥_𝑗)=0, 𝜇_𝑈𝑗(𝑦_𝑗)=0, 𝜈_𝑈𝑗(𝑦_𝑗)=1, and 𝜇_𝑉𝑗(𝑦_𝑗)=1, 𝜈_𝑉𝑗(𝑦_𝑗)=0, 𝜇_𝑉𝑗(𝑥_𝑗)=0, 𝜈_𝑉𝑗(𝑥_𝑗)=1. Now consider the basic IFOSs Π𝑈_𝑘 and Π𝑉_𝑘 where 𝑈_𝑘=𝑉_𝑘=(1,0) for 𝑘∈𝐽,𝑘≠𝑗 and 𝑈_𝑘=𝑈_𝐽 and 𝑉_𝑘=𝑉_𝐽 when 𝑘=𝑗. Then Π𝑈_𝑘(𝑥)=(inf𝜇_𝑈𝑘(𝑥_𝑘), sup𝜈_𝑈𝑘(𝑥_𝑘))=(1,0), Π𝑈_𝑘(𝑦)=(inf𝜇_𝑈𝑘(𝑦_𝑘), sup𝜈_𝑈𝑘(𝑦_𝑘))=(0,1). Similarly we can show that Π𝑉_𝑘(𝑦)=(1,0) and Π𝑉_𝑘(𝑥)=(0,1). Therefore (𝑋,𝜏) is 𝑇₁.
Conversly let (𝑋,𝜏) be 𝑇₁. To show that (𝑋_𝑗,𝜏_𝑗) is 𝑇₁, choose 𝑥_𝑗,𝑦_𝑗∈𝑋_𝑗 such that 𝑥_𝑗≠𝑦_𝑗. Now consider 𝑥=Π𝑥_𝑖, 𝑦=Π𝑦_𝑖 where 𝑥_𝑖=𝑦_𝑖, 𝑖≠𝑗 and the 𝑗th coordinate of 𝑥,𝑦 are 𝑥_𝑗 and 𝑦_𝑗, respectively. Then 𝑥≠𝑦, therefore since (𝑋,𝜏) is 𝑇₁, ∃𝑈=(𝜇_𝑈,𝜈_𝑈) and 𝑉=(𝜇_𝑉,𝜈_𝑉)∈𝜏 such that 𝑈(𝑥)=(1,0), 𝑈(𝑦)=(0,1), 𝑉(𝑥)=(0,1), and 𝑉(𝑦)=(1,0). Now consider the intuitionistic fuzzy points 𝑥_{(𝛼,1−𝛼)}∈𝑈 and 𝑦_{(𝛾,1−𝛾)}∈𝑉. Then ∃ basic IFOSs Π𝑈^𝛼_𝑖 and Π𝑉^𝛾_𝑖 in 𝑋 such that 𝑥_{(𝛼,1−𝛼)}∈Π𝑈^𝛼_𝑖⊆𝑈 and 𝑦_{(𝛾,1−𝛾)}∈Π𝑉^𝛾_𝑖⊆𝑉: 𝑥_{(𝛼,1−𝛼)}∈Π𝑈^𝛼_𝑖⊆𝑈⇒𝛼<inf_𝑖𝜇_𝑈𝛼𝑖𝑥_𝑖,1−𝛼>sup_𝑖𝜈_𝑈^𝛼_𝑖𝑥_𝑖⇒𝛼<𝜇_𝑈𝛼𝑖𝑥_𝑖,(1−𝛼)>𝜈_𝑈^𝛼_𝑖𝑥_𝑖,∀𝑖∈𝐽. (3) Similarly we can show that 𝑦_{(𝛾,1−𝛾)}∈Π𝑉^𝛾_𝑖⊆𝑉⇒𝛾<𝜇_𝑉^𝛾_𝑖𝑦_𝑖,1−𝛾>𝜈_𝑉^𝛾_𝑖𝑦_𝑖,∀𝑖∈𝐽. (4) Now 𝑉_𝑗=∪{𝑈^𝛾_𝐽∶𝛾∈(0,1)} is such that 𝜇_𝑉𝑗(𝑦_𝑗)=1, 𝜈_𝑉𝐽(𝑦_𝑗)=0. Further, since 𝑥_𝑖=𝑦_𝑖 for 𝑖≠𝑗, from (3) and (4), it follows that𝛼<𝜇_𝑈𝛼𝑖𝑦_𝑖,(1−𝛼)>𝜈_𝑈^𝛼_𝑖𝑦_𝑖(),,∀𝑖∈𝐽,𝑖≠𝑗,𝛼∈0,1𝛾<𝜇_𝑉^𝛾_𝑖𝑥_𝑖,1−𝛾>𝜈_𝑉^𝛾_𝑖𝑥_𝑖().,∀𝑖∈𝐽,𝑖≠𝑗,𝛾∈0,1 (5)(6) Therefore 𝑈𝑦=(0,1)⇒Π𝑈^𝛼_𝑖𝑦=(0,1)⇒inf_𝑖𝜇_𝑈𝛼𝑖𝑦_𝑖=0,sup_𝑖𝜈_𝑈𝛼𝑖𝑦_𝑖=1,∀𝛼∈(0,1)⇒𝜇_𝑈𝛼𝑗𝑦_𝑗=0,𝜈_𝑈^𝛼_𝑗𝑦_𝑗()((5=1,∀𝛼∈0,1inviewof))⇒sup_𝛼𝜇_𝑈𝛼𝑗𝑦_𝑗=0,inf_𝛼𝜈_𝑈^𝛼_𝑗𝑦_𝑗=1. (7)
Thus 𝜇_𝑈𝐽(𝑦_𝑗)=0 and 𝜈_𝑈𝑗(𝑦_𝑗)=1.  That is, 𝑈_𝑗(𝑦_𝑗)=(0,1). Similarly it can be shown that 𝑉_𝐽(𝑥_𝐽)=(0,1). Hence (𝑋_𝑗,𝜏_𝑗) is 𝑇₁.

The following theorem can be proved in a similar way.

Theorem 30. Let {(𝑋_𝑖,𝜏_𝑖)∶𝑖∈𝐽} be a family of 𝑇₀ IFTSs and (𝑋,𝜏) be their product IFTS. Then (𝑋,𝜏) is 𝑇₀ if and only if (𝑋_𝑖,𝜏_𝑖) is 𝑇₀, for all 𝑖∈𝐽.

Proposition 31. An IFTS (𝑋,𝜏) is Hausdorff, and then its associated BFTS (𝑋,𝜏₁,𝜏₂) is Hausdorff.

Proof. Let (𝑋,𝜏) be Hausdorff. To show that (𝑋,𝜏₁,𝜏₂) is Hausdorff, choose any two distinct fuzzy points 𝑥_𝛼, 𝑦_1−𝛿 in 𝑋. Now choose 𝛽,𝛾∈(0,1), and then 𝑥_(𝛼,𝛽) and 𝑦_(𝛾,𝛿) are distinct intuitionistic fuzzy points in 𝑋. Since (𝑋,𝜏) is Hausdorff, ∃𝑈,𝑉∈𝜏 such that 𝑥_(𝛼,𝛽)∈𝑈, 𝑦_(𝛾,𝛿)∈𝑉 and 𝑈∩𝑉=0_∼. Let 𝑈=(𝜇_𝑈,𝜈_𝑈), 𝑉=(𝜇_𝑉,𝜈_𝑉), then 𝑥_(𝛼,𝛽)∈𝑈⇒𝛼<𝜇_𝑈(𝑥),𝛽>𝜈_𝑈(𝑥)𝑦_(𝛾,𝛿)∈𝑉⇒𝛾<𝜇_𝑉𝑦,𝛿>𝜈_𝑉𝑦⟺1−𝛿<1−𝜈_𝑉𝑦, (8) and also we have 𝜇_𝑈+𝜈_𝑈⊆1 and 𝜇_𝑉+𝜈_𝑉⊆1.
Now 𝑈∩𝑉=0_∼⟺𝜇_𝑈∩𝜇_𝑉=0,𝜈_𝑈∪𝜈_𝑉=1. (9) From (8) we have 𝑥_𝛼∈𝜇_𝑈 and 𝑦_1−𝛿∈1−𝜈_𝑉. Now we show that 𝜇_𝑈∩(1−𝜈_𝑉)=0 as follows.
We have, 𝜈_𝑈(𝑥)<1⇒𝜈_𝑉(𝑥)=1  (in view of (9)), ⇒1−𝜈_𝑉(𝑥)=0⇒(𝜇_𝑈∩(1−𝜈_𝑉))(𝑥)=0.
Further, 𝜈_𝑉(𝑦)<1⇒𝜈_𝑈(𝑦)=1  (in view of (9)), ⇒𝜇_𝑈(𝑦)=0 since 𝜇_𝑈(𝑦)+𝜈_𝑈(𝑦)≤1⇒(𝜇_𝑈∩(1−𝜈_𝑉))(𝑦)=0.
Now take 𝑧∈𝑋 such that 𝑧≠𝑥,𝑦. If 𝜇_𝑈(𝑧)=0, then obviously (𝜇_𝑈∩(1−𝜈_𝑉))(𝑧)=0, and if 𝜇_𝑈(𝑧)≠0, then 𝜈_𝑈(𝑧)<1  (since 𝜇_𝑈+𝜈_𝑈⊆1)⇒𝜈_𝑉(𝑧)=1⇒1−𝜈_𝑉(𝑧)=0⇒(𝜇_𝑈∩(1−𝜈_𝑉))(𝑧)=0.
Thus we have 𝜇_𝑈∈𝜏₁, (1−𝜈_𝑉)∈𝜏₂ such that 𝑥_𝛼∈𝜇_𝑈, 𝑦_1−𝛿∈(1−𝜈_𝑉) and 𝜇_𝑈∩(1−𝜈_𝑉)=0, and hence (𝑋,𝜏₁,𝜏₂) is Hausdorff.

Definition 32. Let (𝑋,𝜏) be an IFTS and 𝑌⊆𝑋. Then (𝑌,𝜏∣𝑌) is called a subspace of (𝑋,𝜏) where 𝜏∣𝑌={𝑈∣𝑌=(𝜇_𝑈∣𝑌,𝜈_𝑈∣𝑌)∶𝑈∈𝜏}.

Proposition 33. If an IFTS (𝑋,𝜏) is 𝑇_𝑖, 𝑖=0,1,2, then its subspace,  (𝑌,𝜏∣𝑌) is also 𝑇_𝑖, 𝑖=0,1,2.

The proofs are easy and hence are omitted.

Proposition 34. The product IFTS (𝑋,𝜏) of {(𝑋_𝑗,𝜏_𝑗)∶𝑗∈𝐽} is initial with respect to the family of projections {𝑝_𝑗∶𝑋→𝑋_𝑗,𝑗∈𝐽}, that is, for any IFTS (𝑌,𝜂), a map 𝑔∶(𝑌,𝜂)→(𝑋,𝜏) is IF continuous if and only if the map 𝑝_𝑗∘𝑔∶(𝑌,𝜂)→(𝑋_𝑗,𝜏_𝑗) is IF continuous for all  𝑗∈𝐽.

Proof. Since projection maps are IF continuous and composition of IF-continuous maps are IF-continuous, so 𝑝_𝑗∘𝑔 is IF continuous forall𝑗∈𝐽.
Conversely, if 𝑝_𝑗∘𝑔 is IF-continuous forall𝑗∈𝐽, then (𝑝_𝑗∘𝑔)⁻¹(𝑈_𝑗)=𝑔⁻¹(𝑝_𝑗⁻¹(𝑈_𝑗)) is IFO in (𝑌,𝜂)  forall𝑈_𝑗∈𝜏_𝑗,𝑗∈𝐽 showing that inverse image of every subbasic IFOS in 𝑋 is IFO in 𝑌 which implies that 𝑔 is IF continuous.

Proposition 35. Let {(𝑋_𝑗,𝜏_𝑗)∶𝑗∈𝐽} be a family of IFTSs, (𝑋_𝑗,(𝜏_𝑗)₁,(𝜏_𝑗)₂) be the BFTS associated with (𝑋_𝑗,𝜏_𝑗), and (𝑋,𝜏₁,𝜏₂) be the BFTS associated with the product IFTS (𝑋,𝜏). Then 𝜏₁=Π(𝜏_𝑗)₁ and 𝜏₂=Π(𝜏_𝑗)₂.

Proof. The product space (𝑋,𝜏) is generated by {𝑝_𝑗⁻¹(𝑈_𝑗)∶𝑈_𝑗∈𝜏_𝑗,𝑗∈𝐽} where 𝑝^,_𝑗𝑠 are projection maps. Let 𝑈_𝑗=(𝜇_𝑈𝑗,𝜈_𝑈𝑗), and then 𝑝_𝑗⁻¹(𝑈_𝑗)=(𝑝_𝑗⁻¹(𝜇_𝑈𝑗),𝑝_𝑗⁻¹(𝜈_𝑈𝑗)).
Now members of 𝜏 are of the form: _{𝑙∈𝐿arbitrary(𝑘∈𝐾finite)}𝑝_𝑙⁻¹_𝑘𝑈_𝑙𝑘=_𝑙𝑘𝑝_𝑙⁻¹_𝑘𝜇_𝑈𝑙𝑘,𝜈_𝑈𝑙𝑘=_𝑙𝑘𝑝_𝑙⁻¹_𝑘𝜇_𝑈𝑙𝑘,𝑝_𝑙⁻¹_𝑘𝜈_𝑈𝑙𝑘=_𝑙𝑘𝑝_𝑙⁻¹_𝑘𝜇_𝑈𝑙𝑘,_𝑙𝑘𝑝_𝑙⁻¹_𝑘𝜈_𝑈𝑙𝑘. (10) So, members of 𝜏₁ are of the form ⋃_𝑙⋂_𝑘𝑝_𝑙⁻¹_𝑘(𝜇_𝑈𝑙𝑘), hence 𝜏₁=Π(𝜏_𝑗)₁, and members of 𝜏₂ are of the form 1−⋂_𝑙⋃_𝑘𝑝_𝑙⁻¹_𝑘(𝜈_𝑈𝑙𝑘)=⋃_𝑙⋂_𝑘𝑝_𝑙⁻¹_𝑘(1−𝜈_𝑈𝑙𝑘), hence 𝜏₂=Π(𝜏_𝑗)₂.

Theorem 36. An IFTS (𝑋,𝜏) is Hausdorff if and only if (Δ_𝑋,Δ^′_𝑋) is IFCS in (𝑋×𝑋,𝜏×𝜏).

Proof. We show that (Δ^′_𝑋,Δ_𝑋) is IFOS in 𝑋. Choose any (𝑥,𝑦)_(𝛼,𝛽)∈(Δ^′_𝑋,Δ_𝑋) then 𝑥≠𝑦 and 𝛼<Δ^′_𝑋(𝑥,𝑦), 𝛽>Δ_𝑋(𝑥,𝑦). Now 𝑥_(𝛼,𝛽) and 𝑦_(𝛼,𝛽) are distinct intuitionistic fuzzy points in 𝑋. Since (𝑋,𝜏) is Hausdorff, ∃ IFOSs 𝑈=(𝜇_𝑈,𝜈_𝑈) and 𝑉=(𝜇_𝑉,𝜈_𝑉) in 𝜏 such that 𝑥_(𝛼,𝛽)∈𝑈, 𝑦_(𝛼,𝛽)∈𝑉 and 𝑈∩𝑉=0_∼, that is, (𝜇_𝑈∩𝜇_𝑉,𝜈_𝑈∪𝜈_𝑉)=(0,1).
Now consider 𝑈×𝑉=(𝜇_𝑈×𝜇_𝑉,𝜈_𝑈∗𝜈_𝑉), and then (𝑥,𝑦)_(𝛼,𝛽)∈𝑈×𝑉⊆(Δ^′_𝑋,Δ_𝑋) as shown below:
(𝑥,𝑦)_(𝛼,𝛽)∈(𝜇_𝑈×𝜇_𝑉,𝜈_𝑈∗𝜈_𝑉) as 𝛼<(𝜇_𝑈×𝜇_𝑉)(𝑥,𝑦)= inf (𝜇_𝑈(𝑥),𝜇_𝑉(𝑦))  (since 𝛼<𝜇_𝑈(𝑥),𝛼<𝜇_𝑉(𝑦) both) and 𝛽>(𝜈_𝑈∗𝜈_𝑉)(𝑥,𝑦)= sup (𝜈_𝑈(𝑥),𝜈_𝑉(𝑦))   (since 𝛽>𝜈_𝑈(𝑥),𝛽>𝜈_𝑉(𝑦) both).
Further, 𝑈×𝑉⊆(Δ^′_𝑋,Δ_𝑋) since 𝑈×𝑉=(𝜇_𝑈×𝜇_𝑉,𝜈_𝑈∗𝜈_𝑉) and 𝜇_𝑈×𝜇_𝑉⊆Δ^′_𝑋 as (𝜇_𝑈×𝜇_𝑉)(𝑥,𝑥)=inf(𝜇_𝑈(𝑥),𝜇_𝑉(𝑥))=0,forall𝑥∈𝑋 and 𝜈_𝑈∗𝜈_𝑉⊇Δ_𝑋 as (𝜈_𝑈×𝜈_𝑉)(𝑥,𝑥)=sup(𝜈_𝑈(𝑥),𝜈_𝑉(𝑥))=1,forall𝑥∈𝑋.
Conversely, let (Δ_𝑋,Δ^′_𝑋) be IFCS in (𝑋,𝜏). Then (Δ^′_𝑋,Δ_𝑋) is IFOS in 𝑋.  To show that (𝑋,𝜏) is Hausdorff choose any two distinct intuitionistic fuzzy points 𝑥_(𝛼,𝛽), 𝑦_(𝛾,𝛿) in 𝑋 then 𝑥≠𝑦. Let max (𝛼,𝛾)  =  𝛼 and min (𝛽,𝛿)  =  𝛽. Consider the intuitionistic fuzzy point (𝑥,𝑦)_(𝛼,𝛽) in 𝑋×𝑋. Then (𝑥,𝑦)_(𝛼,𝛽)∈(Δ^′_𝑋,Δ_𝑋) and hence ∃ a basic IFOS 𝑈×𝑉 in (𝑋×𝑋,𝜏×𝜏) such that (𝑥,𝑦)_(𝛼,𝛽)∈𝑈×𝑉⊆(Δ^′_𝑋,Δ_𝑋).
Now, (𝑥,𝑦)_(𝛼,𝛽)∈𝑈×𝑉⇒𝛼<(𝜇_𝑈×𝜇_𝑉)(𝑥,𝑦)  = inf (𝜇_𝑈(𝑥),𝜇_𝑉(𝑦))≤𝜇_𝑈(𝑥), and 𝛽>(𝜈_𝑈∗𝜈_𝑉)(𝑥,𝑦)  = sup (𝜈_𝑈(𝑥),𝜈_𝑉(𝑦))≥𝜈_𝑈(𝑥) implies that 𝑥_(𝛼,𝛽)∈𝑈, similarly 𝑦_(𝛼,𝛽)∈𝑉 which implies that 𝑦_(𝛾,𝛿)∈𝑉. Now we show that 𝑈∩𝑉= 0_∼. Since 𝑈×𝑉⊆(Δ^′_𝑋,Δ_𝑋), (𝜇_𝑈×𝜇_𝑉)(𝑥,𝑥)≤Δ^′_𝑋(𝑥,𝑥)  =  0⇒ inf (𝜇_𝑈(𝑥),𝜇_𝑉(𝑥))= 0, forall𝑥∈𝑋 and (𝜈_𝑈×𝜈_𝑉)(𝑥,𝑥)≥Δ_𝑋(𝑥,𝑥)  =1⇒ sup (𝜈_𝑈(𝑥),𝜈_𝑉(𝑥))  =1, forall𝑥∈𝑋. Thus 𝑈∩𝑉 = (𝜇_𝑈∩𝜇_𝑉,𝜈_𝑈∪𝜈_𝑉)=(0,1) =0_∼.

Definition 37. A BFTS (𝑋,𝜏₁,𝜏₂) is called 𝑞-𝑇₂ if for any two distinct fuzzy points 𝑥_𝑟 and 𝑦_𝑠 there exists 𝑈∈𝜏₁, 𝑉∈𝜏₂ such that 𝑥_𝑟∈𝑈, 𝑦_𝑠∈𝑉 and 𝑈⊆𝑉^′.

Proposition 38. An IFTS (𝑋,𝜏) is 𝑞-𝑇₂, and then its associated BFTS (𝑋,𝜏₁,𝜏₂) is 𝑞-𝑇₂.

Proof. Let 𝑥_𝛼 and 𝑦_1−𝛿 be any two distinct fuzzy points in 𝑋. Since (𝑋,𝜏) is 𝑞-𝑇₂ and 𝑥_(𝛼,𝛽), 𝑦_(𝛾,𝛿) are distinct intuitionistic fuzzy points in 𝑋, there exist 𝐺₁=(𝜇_𝐺1,𝜈_𝐺1) and 𝐺₂=(𝜇_𝐺2,𝜈_𝐺2) such that 𝑥_(𝛼,𝛽)∈𝐺₁, 𝑦_(𝛾,𝛿)∈𝐺₂ and 𝜇_𝐺1⊆𝜇^′_𝐺2, 𝜈_𝐺1⊇𝜈^′_𝐺2. ⇒𝛼<𝜇_𝐺1(𝑥) and 𝛿>𝜈_𝐺2(𝑦)⇒𝑥_𝛼∈𝜇_𝐺1 and 𝑦_1−𝛿∈(1−𝜈_𝐺2). Thus for fuzzy points 𝑥_𝛼 and 𝑦_1−𝛿 in 𝑋, ∃𝜇_𝐺1∈𝜏₁ and (1−𝜈_𝐺2)∈𝜏₂. Further, since 𝜇_𝐺1+𝜈_𝐺1⊆1, we have 𝜇_𝐺1⊆(1−𝜈_𝐺1)⊆𝜈_𝐺2  (since 𝜈_𝐺1⊇𝜈^′_𝐺2). Hence (𝑋,𝜏₁,𝜏₂) is 𝑞-𝑇₂.

Theorem 39. Let {(𝑋_𝑖,𝜏_𝑖)∶𝑖∈𝐽} be a family of IFTs and (𝑋,𝜏) be their product IFTS. Then (𝑋,𝜏) is Hausdorff if and only if each coordinate space (𝑋_𝑖,𝜏_𝑖) is Hausdorff.

Proof. Let 𝑥_(𝛼,𝛽) and 𝑦_(𝛾,𝛿) be two distinct intuitionistic fuzzy points in 𝑋. Let 𝑥=Π𝑥_𝑖, 𝑦=Π𝑦_𝑖 then 𝑥≠𝑦 and hence ∃𝑘∈𝐽 such that 𝑥_𝑘≠𝑦_𝑘. Consider the distinct intuitionistic fuzzy points (𝑥_𝑘)_(𝛼,𝛽) and (𝑦_𝑘)_(𝛾,𝛿) in (𝑋_𝑘,𝜏_𝑘). Since (𝑋_𝑘,𝜏_𝑘) is Hausdorff, ∃ disjoint 𝑈_𝑘,𝑉_𝑘 in 𝜏_𝑘 such that (𝑥_𝑘)_(𝛼,𝛽)∈𝑈_𝑘, (𝑦_𝑘)_(𝛾,𝛿)∈𝑉_𝑘 and 𝑈_𝑘∩𝑉_𝑘=0_∼. Let 𝑈_𝑘=(𝜇_𝑈𝑘,𝜈_𝑈𝑘) and 𝑉_𝑘=(𝜇_𝑉𝑘,𝜈_𝑉𝑘), and then 𝛼<𝜇_𝑈𝑘(𝑥_𝑘), 𝛽>𝜈_𝑈𝑘(𝑥_𝑘) and 𝛾<𝜇_𝑉𝑘(𝑦_𝑘), 𝛿>𝜈_𝑉𝑘(𝑦_𝑘). Equivalently, we have, 𝛼<𝑝_𝑘⁻¹(𝜇_𝑈𝑘)(𝑥), 𝛽>𝑝_𝑘⁻¹(𝜈_𝑈𝑘)(𝑥), that is, 𝑥_(𝛼,𝛽)∈𝑝_𝑘⁻¹(𝑈_𝑘) and 𝛾<𝑝_𝑘⁻¹(𝜇_𝑉𝑘)(𝑦), 𝛿>𝑝_𝑘⁻¹(𝜈_𝑉𝑘)(𝑦), that is, 𝑦_(𝛾,𝛿)∈𝑝_𝑘⁻¹(𝑉_𝑘) and 𝑝_𝑘⁻¹(𝑈_𝑘)⋂𝑝_𝑘⁻¹(𝑉_𝑘)=0_∼, since 𝑈_𝑘⋂𝑉_𝑘=0_∼.

Conversely, let (𝑋,𝜏) be Hausdorff. To show that (𝑋_𝑗,𝜏_𝑗) is Hausdorff, and consider two distinct intuitionistic fuzzy points (𝑥_𝑗)_(𝛼,𝛽) and (𝑦_𝑗)_(𝛾,𝛿) in 𝑋_𝑗, then 𝑥_𝑗≠𝑦_𝑗. Now consider 𝑥=Π𝑥_𝑖 and 𝑦=Π𝑦_𝑖 in 𝑋 where 𝑥_𝑖=𝑦_𝑖 for 𝑖≠𝑗 and the 𝑗th coordinate of 𝑥,𝑦 are 𝑥_𝑗 and 𝑦_𝑗, respectively. Since (𝑋,𝜏) is Hausdorff, ∃𝑈,𝑉∈𝜏 such that 𝑥_(𝛼,𝛽)∈𝑈, 𝑦_(𝛾,𝛿)∈𝑉 and 𝑈∩𝑉=0_∼. Now consider the basic IFOSs Π𝑈_𝑖 and Π𝑉_𝑖 in 𝑋 such that 𝑥_(𝛼,𝛽)∈Π𝑈_𝑖⊆𝑈 and 𝑦_(𝛾,𝛿)∈Π𝑉_𝑖⊆𝑉. Here the IFS 𝑈_𝑖=(1,0) except for finitely many 𝑖’s say 𝑖₁,𝑖₂,...,𝑖_𝑘 and 𝑉_𝑖=(1,0) except for finitely many 𝑖’s say 𝑙₁,𝑙₂,…,𝑙_𝑘1.

Now, 𝑥_(𝛼,𝛽)∈Π𝑈_𝑖⊆𝑈⇒𝛼< inf {𝜇_𝑈𝑖1(𝑥_𝑖1),𝜇_𝑈𝑖2(𝑥_𝑖2),…,𝜇_𝑈𝑖𝑘(𝑥_𝑖𝑘)}, 𝛽>sup𝜈_𝑈𝑖1𝑥_𝑖1,𝜈_𝑈𝑖2𝑥_𝑖2,…,𝜈_𝑈𝑖𝑘𝑥_𝑖𝑘⇒𝛼<𝜇_𝑈𝑖𝑚𝑥_𝑖𝑚,𝛽>𝜈_𝑈𝑖𝑚𝑥_𝑖𝑚,∀𝑚=1,2,…,𝑘. (11) Similarly, 𝑦_(𝛾,𝛿)∈Π𝑉_𝑖⊆𝑉⇒𝛾<𝜇_𝑉𝑙𝑛𝑦_𝑙𝑛,𝛿>𝜈_𝑉𝑙𝑛𝑦_𝑙𝑛,∀𝑛=1,2,…,𝑘₁. (12) We claim that 𝑗∈{𝑖₁,𝑖₂,…,𝑖_𝑘} and also 𝑗∈{𝑙₁,𝑙₂,…,𝑙_𝑘1}. If it is not so, then 𝑥_𝑖𝑚=𝑦_𝑖𝑚 and 𝑥_𝑙𝑛=𝑦_𝑙𝑛, forall𝑚=1,2,…,𝑘 and forall𝑛=1,2,…,𝑘₁ and hence from (11) and (12) we have 𝛼<𝜇_𝑈𝑖𝑚𝑥_𝑖𝑚,𝛽>𝜈_𝑈𝑖𝑚𝑥_𝑖𝑚,∀𝑚=1,2,…,𝑘,𝛾<𝜇_𝑉𝑙𝑛𝑥_𝑙𝑛,𝛿>𝜈_𝑉𝑙𝑛𝑥_𝑙𝑛,∀𝑛=1,2,…,𝑘₁ (13) implying that Π𝜇_𝑈𝑖(𝑥)>0, Π𝜇_𝑉𝑖(𝑥)>0, therefore (Π𝜇_𝑈𝑖∩Π𝜇_𝑉𝑖)≠0⇒𝜇_𝑈∩𝜇_𝑉≠0 which is a contradiction to the fact that 𝑈∩𝑉=0_∼.

So, 𝛼<𝜇_𝑈𝑗(𝑥_𝑗), 𝛽>𝜈_𝑈𝑗(𝑥_𝑗), 𝛾<𝜇_𝑉𝑗(𝑦_𝑗) and 𝛿>𝜈_𝑉𝑗(𝑦_𝑗) in view of (11) and (12) showing that (𝑥_𝑗)_(𝛼,𝛽)∈𝑈_𝑗, (𝑦_𝑗)_(𝛾,𝛿)∈𝑉_𝑗.

Now it remains to show that 𝑈_𝑗∩𝑉_𝑗=0_∼, that is, 𝜇_𝑈𝑗∩𝜇_𝑉𝑗=0 and 𝜈_𝑈𝑗∪𝜈_𝑉𝑗=1. Let ∃𝑧_𝑗1 and 𝑧_𝑗2 in 𝑋_𝑗 such that (𝜇_𝑈𝑗∩𝜇_𝑉𝑗)(𝑧_𝑗1)>0 and (𝜈_𝑈𝑗∪𝜈_𝑉𝑗)(𝑧_𝑗2)<1. Consider 𝑧₁=Π𝑧_𝑖 in 𝑋 where 𝑧_𝑖=𝑥_𝑖(=𝑦_𝑖) for 𝑖≠𝑗 and 𝑧_𝑗=𝑧_𝑗1.

Now 𝜇_𝑈𝑗∩𝜇_𝑉𝑗𝑧_𝑗1>0⇒𝜇_𝑈𝑗𝑧_𝑗1>0,𝜇_𝑉𝑗𝑧_𝑗1>0. (14) In view of (11), (12), and (14) we have (Π𝜇_𝑈𝑖∩Π𝜇_𝑉𝑖)(𝑧₁)>0 implying that 𝑈∩𝑉≠0_∼ which is a contradiction. Hence 𝜇_𝑈𝑗∩𝜇_𝑉𝑗=0. Now let 𝑧₂=Π𝑧_𝑖 where 𝑧_𝑖=𝑥_𝑖(=𝑦_𝑖) for 𝑖≠𝑗 and 𝑧_𝑗=𝑧_𝑗2. Since (𝜈_𝑈𝑗∪𝜈_𝑉𝑗)(𝑧_𝑗2)<1,

We have 𝜈_𝑈𝑗𝑧_𝑗2<1,𝜈_𝑉𝑗𝑧_𝑗2<1. (15) Therefore from (11), (12), and (15) we have (Π𝜈_𝑈𝑗∪Π𝜈_𝑉𝑗)(𝑧₂)<1 implying that 𝑈∩𝑉≠0_∼, again a contradiction. Hence (𝑋_𝑗,𝜏_𝑗) is Hausdorff.