Semi-Hausdorff Fuzzy Filters

The notion of fuzzy filters was studied by Vicente and Aranguren (1988), Lowen (1979), and Ramakrishnan and Nayagam (2002). The notion of fuzzily compactness was introduced and studied by Ramakrishnan and Nayagam (2002). In this paper, an equivalent condition of fuzzily compactness is studied and a new notion of semi-Hausdorffness on fuzzy filters, which cannot be defined in crisp theory of filters, is introduced and studied.


Introduction
The concept of fuzzy sets was introduced by Zadeh [1].The theory of fuzzy filters was studied in [2,3].The notion of fuzzy topological spaces is introduced in [4] and studied in [5,6].The notion of quasicoincident was introduced in [7], and the notion of disjointness was studied in [8] for defining separation axioms.The notion of fuzzily compact sets was introduced and studied in [9].In this paper, an equivalent condition of fuzzy compactness through fuzzy filter convergence is studied in Section 3. The notion of Hausdorff interval-valued fuzzy filter was introduced and studied in [3].In this paper, a new notion of semi-Hausdorffness, which cannot be defined in the usual theory of filters, is introduced and studied to some extend in Section 2.
Here we give a brief review of preliminaries.
Definition 1.3 [8].Two fuzzy sets μ and γ are said to be disjoint if μ ≤ γ c .Note 1. 4. By Definition 1.2, two fuzzy sets μ and γ are said to be not quasicoincident if μ(x) + γ(x) ≤ 1 for all x ∈ X, and hence disjointness in Definition 1.3 and not quasicoincidence are equivalent.
Definition 1.6 [9].Let (X,δ) be a fuzzy topological space.A collection δ 0 of fuzzy open sets is called a fuzzily open cover for A ⊆ X if for every z ∈ A, there exists γ ∈ δ 0 such that γ(z) ≥ 1/2.
Definition 1.7 [9].A subset A ⊆ X of a fuzzy topological space (X,δ) is said to be fuzzily compact if for every fuzzily open cover σ for A, there exists a finite subcollection δ 0 of σ such that for every z ∈ A, there exists γ ∈ δ 0 with γ(z) ≥ 1/2.
Definition 1.9 [3].Let (X,I) be a fuzzy filter.Let Y ⊆ X.Then (Y ,I | Y ) is called the subfilter if no element of I vanishes on Y .
(1) A function f : X→Y is said to be a fuzzy filter continuous if f −1 (γ) ∈ I 1 for every γ ∈ I 2 .(2) A function f : X→Y is said to be a fuzzy filter open if f (μ) ∈ I 2 for every μ ∈ I 1 .
(3) An injective function f : X→Y is said to a be fuzzy filter homeomorphism if f is both fuzzy filter continuous and fuzzy filter open.
Definition 1.11 [3].Let (X α ,I α ) be an indexed family of fuzzy filters.Let X = X α .Now, the product fuzzy filter I = I α is the smallest fuzzy filter for which the projection maps p α : X→X α defined by p α ((x α )) = x α are fuzzy filter continuous.
Definition 1.14 [9].Let (X,δ) and (Y ,σ) be fuzzy topological spaces.A point x ∈ X is said to be a fuzzily limit point of A if for every fuzzy open set μ ∈ δ such that μ(x) ≥ 1/2, μ(z) ≥ 1/2 for some z ∈ A − {x}.A subset C of X is said to be a fuzzily closed set if it contains all its fuzzily limit points.
Definition 1.15 [9].A function f : X→Y is said to be nearly fuzzy continuous if f −1 (A) is fuzzily closed in (X,δ) for every fuzzily closed set A in (Y ,σ).

Semi-Hausdorff fuzzy filters
Remark 2.1.From [3, Theorem 3.2], in a Hausdorff fuzzy filter, any sequence of points of X converges filterly uniquely if it converges.
But the converse need not be true, which is seen from the following example.
Example 2.2.Let X be an uncountable set and I = {μ ∈ I X | μ c has a countable support}.Clearly, (X,I) is a fuzzy filter.Let {x n } be any sequence of points of X.Let x ∈ X be an arbitrary point.Consider μ ∈ I such that (2.1) Clearly, μ(x) > 1/2.But μ(x n ) < 1/2 for all n ∈ Z + and hence x n does not converge to x fuzzily.So no sequence converges and hence every sequence converges filterly uniquely if it converges.But (X,I) is not a Hausdorff fuzzy filter.Let x, y ∈ X such that x = y.Suppose there exists μ,γ ∈ I such that μ(x) > 1/2, γ(y) > 1/2 and μ(z) + γ(z) ≤ 1, for all z ∈ X.Since μ,γ ∈ I, μ c , and γ c have countable supports, say, {x n } n∈Z+ and {y m } m∈Z+ , respectively.Hence μ and γ have value 1 on X − {x n , y m } n,m ∈ Z + .Since X is uncountable, there exists z ∈ X − {x n , y m } n,m ∈ Z + such that μ(z) = 1 and γ(z) = 1, a contradiction to the fact that μ(z) + γ(z) ≤ 1, for all z ∈ X. Definition 2.3.A fuzzy filter (X,I) is said to be a semi-Hausdorff fuzzy filter (s.H.F filter) if every sequence of points converges fuzzy filterly to at most one point.Definition 2.4 [3].Let (X,I) be a fuzzy filter.Then (X,I) is said to be a nearly fuzzy T 1 filter (n.f.T 1 ) if for every x, y ∈ X, x = y, there exists μ,γ ∈ I such that μ(x) > 1/2, γ(y) > 1/2 and μ(y Theorem 2.5.Every s.H.F filter is an n.f.T 1 filter.
The proof of Theorem 2.5 is immediate from definitions.
The converse need not be true, which is seen from the following example.
Example 2.6.Let X be an infinite set and I = {μ ∈ I X | μ c has finite support}.Clearly, (X,I) is a fuzzy filter.To prove that (X,I) is an n.f.T Then supports of μ c and γ c are {x} and {y}, respectively.Therefore, γ(x) > 1/2, γ(y) ≤ 1/2 and μ(y) > 1/2, μ(x) ≤ 1/2.To prove that (X,I) is not an s.H.F filter, consider x n of X such that x i = x j for i = j.Now, x n converges fuzzy filterly to each point of X.Let x ∈ X and μ ∈ I such that μ(x) > 1/2.Since μ c has finite support and x n is an infinite sequence of distinct points, μ(x n ) = 1 for all but finite number of points of x n .Therefore, x n converges fuzzy filterly to x and hence to all points of X. Hence(X,I) is not an s.H.F filter.
The proof of the following theorem is immediate.Proof.Let f : (X,I 1 )→(Y ,I 2 ) be a fuzzy filter open map and (X,I 1 ) be an s.H.F filter.Suppose (Y ,I 2 ) is not an s.H.F filter, then there exists (y n ) ∈ Y such that (y n ) converges fuzzy filterly to y and y .Since f is bijective, all but finite number of n's.Hence x n →a fuzzy filterly.Similarly, x n →b fuzzy filterly, which is a contradiction to the fact that (X,I 1 ) is an s.H.F filter.Lemma 2.9.Let f : (X,I 1 )→(Y ,I 2 ) be fuzzy filter continuous and let x n converge to x fuzzy filterly.Then f (x n ) converges to f (x) fuzzy filterly.
The proof is immediate.Definition 2.10 [4].Let B be a base for a fuzzy filter.Now, the collection I B = {μ | there exists some γ ∈ B such that γ ≤ μ} is the fuzzy filter generated by B.
Definition 2.11 [4].A collection S of fuzzy sets is said to be a subbase for a fuzzy filter I in finite intersections of members of S that form a base for I.
The proof of the following lemma is immediate.

Lemma 2.12. A function is fuzzy filter continuous if and only if the inverse image of subbasic fuzzy filter open set is fuzzy filter open.
Lemma 2.13.Let (X α ,I α ) be any indexed family of fuzzy filters.Then (X α ,I α ) is fuzzy filter homeomorphic to a subspace of the product fuzzy filter , where x j = x α , j = α and x j = x β , j = β = α.Then f is well defined and 1-1.To prove that f is fuzzy filter continuous, consider a subbasic fuzzy filter open set p −1 α (μ α ) ∈ I α , where μ α is fuzzy filter open in X α , and let p α : X α →X α be the projection map.Also, which is filter open.By Lemma 2.12 f is fuzzy filter continuous.
V. L. G. Nayagam and Geetha Sivaraman 5 Now, by definition of f , (2.3) is fuzzy filter open in S as a subspace of ( X α , I α ).Therefore, f −1 : S→(X α ,I α ) is fuzzy filter continuous and (X α ,I α ) is fuzzy filter homeomorphic to a subspace of ( X α , I α ).
Theorem 2.14.Let (X α ,I α ) be an indexed family of fuzzy filters.Then ( X α , I α ) is an s.H.F filter if and only if each (X α ,I α ) is an s.H.F filter.
Proof.Let (X α ,I α ) be a family of s.H.F filters.Suppose ( X α , I α ) is not an s.H.F filter, there exists a sequence {x n } of points of X α , which converges fuzzy filterly to distinct points x and y.Since x = y, there exists an index β such that x β = y β .Since the projection map p β : X α →X β is fuzzy filter continuous in the product fuzzy filter, by Lemma 2.9, p β (x n ) converges to x β and y β fuzzy filterly with x β = y α , a contradiction to s.H.F filterness of (X β ,I β ).Hence ( X α , I α ) is an s.H.F filter.Now, we prove the converse part.Let ( X α , I α ) be an s.H.F filter.By Lemma 2.13, each X α is fuzzy filter homeomorphic to a subspace of ( X α , I α ).By Theorems 2.7 and 2.8, (X α ,I α ) is an s.H.F filter.
Note 2.15.Let (Y ,I) be a fuzzy filter.For defining the pointwise convergence filter on Y X , if we take S(x,μ where x ∈ X, μ is a fuzzy filter open set in Y (i.e., μ ∈ I) and p is a fuzzy point, analogous to the crisp theory, the collection S(x,μ), and S(p,μ) need not be a subbasis for a filter as seen from the following example.
So to generalize fuzzy pointwise convergence filter, we need the following definition.
Definition 2.21.The filter generated by the subbasis {S(x, μ)} x∈X,μ∈I is called the point wise convergence filter on Y X with respect to the strong fuzzy filter (Y ,I).
Remark 2.22.In the above pointwise convergence filter on Y X with respect to a strong fuzzy filter (Y ,I), f n → f if and only if f n (x)→ f (x) fuzzy filterly for every x ∈ X.
Proof.Assume f n → f in the above filter.To prove that f n (x)→ f (x) fuzzy filterly for every x ∈ X, consider x ∈ X and a fuzzy filter open set μ ∈ I such that μ( f (x)) > 1/2.Hence f ∈ S(x,μ).Since f n → f and f ∈ S(x,μ), there exists N such that f n ∈ S(x,μ) for all n ≥ N.
Hence μ( f n (x)) > 1/2 for all n ≥ N. Hence f n (x)→ f (x) fuzzy filterly for every x ∈ X.Conversely, suppose f n (x)→ f (x) fuzzy filterly for every x ∈ X.To prove that f n → f in the strong filter, let S(x,μ) be a subbasic open set containing f .Then μ( f (x)) > 1/2.Since f n (x)→ f (x) fuzzy filterly, μ( f n (x)) > 1/2 for all n ≥ N for some N. Hence f n ∈ S(x,μ) for all n ≥ N. Hence f n → f .Definition 2.23.A filter (X,τ) is said to be semi-Hausdorff filter (semi-T 2 filter) if and only if every sequence in X has at most one limit.
Corollary 2.24.The pointwise convergence filter on Y X is a semi-T 2 filter if (Y ,I) is an s.H.F filter.
Proof.Let (Y ,I) be an s.H.F filter.Suppose the above filter on Y X is not a semi-T 2 filter, then there exists f n ∈ Y X such that f n → f and f n →g with f = g.By the above remark for all x ∈ X, which contradicts the fact that f = g.Hence the above filter on Y X is a semi-T 2 filter.
Theorem 2.25.Let (Y ,I) be a strong fuzzy filter such that Y X is semi-T 2 filter for every indexing set X in the pointwise convergence filter with respect to (Y ,I).Then (Y ,I) is an s.H.F filter.
Proof.Suppose that (Y ,I) is not an s.H.F filter, then there exists y n ∈ Y such that y n f → x, and y n f → y such that x = y.Define f n , f x , f y : X→Y by f n (z) = y n , f x (z) = x and f y (z) = y, for all z ∈ X.Then clearly, f n , f x , and f y are elements of Y X .Now, we claim that f n → f x and f n → f y .Consider a subbasic filter open set S(t,μ) containing f x , where t ∈ X and μ ∈ I. Hence μ( f x (t)) > 1/2.So μ(x) > 1/2.Since y n f → x and μ(x) > 1/2, we have μ(y n ) > 1/2 for every n ≥ N for some N. Therefore, μ( f n (t)) = μ(y n ) > 1/2 for every n ≥ N. Hence f n ∈ S(t,μ) for every n ≥ N.So f n → f x .Similarly, f n → f y .So we get a contradiction to semi-T 2 ness of Y X .Hence (Y ,I) is an s.H.F filter.
Proof.Let Y be an s.H.F filter.Suppose A is not sequentially fuzzy filterly closed, then there exists Since x n f → x and f , g are sequentially fuzzy filterly continuous functions, we have f ( , we have a contradiction to the fact that Y is an s.H.F filter.Now, the converse follows from the previous theorem by taking X = Y × Y , and f , g are projections.
Remark 2.32.The authors acknowledged the referees for pointing out that if F is a fuzzy filter, then F * = F ∪ {φ} is a fuzzy topology, and hence results in [3] and in this paper can be proved easily by this remark and more general theorems for fuzzy topological spaces.

Fuzzy filter convergence
In this section, new notions of fuzzy filter convergence and fuzzily cluster points are introduced and some fuzzy topological properties are studied through those notions.
Definition 3.1.Let (X,δ) be a fuzzy topological space and let (X,I) be a fuzzy filter.A point x ∈ X is said to be a fuzzily cluster point of (X,I) ((X,I) accumulates x) if for every μ ∈ δ with μ(x) ≥ 1/2, there exists z ∈ X such that μ(z) + ν(z) > 1, for all ν ∈ I. Definition 3.2.Let (X,δ) be a fuzzy topological space.A fuzzy filter (X,I) is said to converge to a point x ∈ X if for every μ ∈ δ with μ(x) ≥ 1/2, there exists ν ∈ I such that ν ≤ μ.
The proof of the following note is immediate from definitions.Note 3.3.If a strong fuzzy filter (X,I) converges to a point x ∈ X, then I accumulates x.Theorem 3.4.Let (X,δ) be a fuzzy topological space and let (X,I) be a fuzzy filter.A point x ∈ X is a fuzzily cluster point of (X,I) if and only if μ(x) > 1/2, for all μ ∈ I.
(a) If f is fuzzy continuous, then I→x implies f (I)→ f (x).(b) If I→x implies f (I)→ f (x) for every fuzzy filter I on X, then f is nearly fuzzy continuous.
Proof.(a) Assume that f is fuzzy continuous and I→x.To prove that f for every fuzzy filter I on X, we have to prove that f : X→Y is nearly fuzzy continuous.Let A be a fuzzily closed set in Y .Now, we prove that which is a contradiction.So x is not a fuzzily limit point of f −1 (A) and hence f −1 (A) is fuzzily closed.Theorem 3.7.A fuzzy topological space (X,δ) is fuzzily compact if and only if each strong fuzzy filter in X has at least one fuzzily cluster point.
Proof.By [7,Theorem 8], it is enough to prove that every collection of fuzzily closed sets with finite intersection property has nonempty intersection if and only if every strong fuzzy filter in X has at least one fuzzily cluster point.
"Only if " part.Now, we assume the hypothesis.Let Ω be a collection of fuzzily closed sets that satisfies finite intersection property.Let A ∈ Ω and if z ∈ A, by definition of fuzzily closed set, then z is not a fuzzily limit point of A, and hence there exists ν ∈ δ such that ν(z) ≥ 1/2 and ν(y) < 1/2, for all y ∈ A. So we have a fuzzy closed set ν c with ν c (z) ≤ 1/2 and ν c (y) > 1/2, for all y ∈ A. So for every A ∈ Ω and for each z ∈ A, there exists a fuzzy closed set γ z,A with γ z,A (y) > 1/2, for all y ∈ A and γ z,A (z) ≤ 1/2.Now, consider S = {γ z,A | A ∈ Ω and z ∈ A}.By finite intersection property of Ω, for any finite subcollection S 0 of S, we have ∧ μ∈S0 (t) > 1/2 for some t ∈ X.So the fuzzy filter

Theorem 2 . 7 .Theorem 2 . 8 .
Let (X,I) be an s.H.F filter.Let Y ⊆ X.Then (Y ,I | Y ) is also an s.H.F filter if no element of I vanishes on Y .An s.H.F filter is invariant under every bijective fuzzy filter open map.