����������� ������ 50 (1998), 37–45

UDK 515.122.2

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ON S-CLOSED AND EXTREMALLY DISCONNECTED FUZZY TOPOLOGICAL SPACES

Zlata Petričević

Abstract. The concepts of a filter-base and s-convergence and θ-convergence of a filter-base in a fuzzy setting are defined and investigated. Fuzzy filter-base is used to characterize fuzzy Sclosed and extremally disconnected spaces. Several other properties of these two types of spaces and comparison between different forms of compactness in fuzzy topology are established.

Introduction

The concept of filters in fuzzy set theory was introduced by Lowen and at the same time by Katsaras who studied in his work [11] fuzzy filters, ultra filters, clusters and the convergence of filters in fuzzy setting. In this paper we have developed the theory of filters a little further and introduced s-convergence and θ-convergence of a filter (filter-base).

We offer several characterizations of fuzzy S-closed and fuzzy extremally disconnected spaces in terms of fuzzy filter-bases and fuzzy nets. The results are parallel to ones which have been found in general topology. A systematic discussion of these properties in general topology is given in [7], [10], [16] and [17].

In the last section we study the implications between different forms of compactness and comparison with S-closedness and extremal disconnectedness in a fuzzy setting.

1. Preliminaries

Throughout the paper by (X, τ_{X}), or simply by X, we mean a fuzzy topological space (fts, shortly) of Chang [3]. A fuzzy singleton with support x and value α (0 < α ⩽ 1) will be denoted by x_{α}. Two fuzzy sets λ and μ are said to be qcoincident if there exists x ∈ X such that λ(x) + μ(x) > 1 and by q̄ we denote “is not q-coincident”. A fuzzy set λ is said to be a q-neighbourhood (q-nbd) of x_{α} if there is a fuzzy open set μ such that x_{α} q μ and μ ⩽ λ, where λ ⩽ μ if λ(x) ⩽ μ(x), for all x ∈ X. We denote ny N(x_{α}) (N_{q}(x_{α})) the neighbourhood (q-nbd) system of x_{α} [12]. For a fuzzy set λ in an fts X by Cl λ, SCl λ, Int λ, supp λ we denote the closure, semi-closure, interior and support of λ, respectively. By 0_{X} and 1_{X} we mean the constant fuzzy sets taking the values 0 and 1 on X, resp. For the defnition of fuzzy regularly open (RO(X)), regularly closed (RC(X)), semi-open (SO(X)) and semi-closed (SC(X)) sets we refer to Azad [2]. A fuzzy point x_{α} ∈ Cl_{δ} λ (x_{α} ∈ Cl_{θ} λ) if x_{α} q U implies U q λ, for each U ∈ RO(X) (x_{α} q U implies Cl U q λ for each U ∈ τ_{X} resp.). A fuzzy set λ is called δ-closed (θ-closed resp.) if λ = Cl_{δ} λ (λ = Cl_{θ} λ). It is known [15] that for any set λ, λ ⩽ SCl λ ⩽ Cl λ ⩽ Cl_{δ} λ ⩽ Cl_{θ} λ.

AMS Subject Classification: 54 A 40, 54 D 20

Keywords and phrases: s-converegence and θ- convergence of a filter-base in a fuzzy setting; semi-open set; S-closed, extremally disconnected fuzzy topological spaces.

An fts (X, τ_{X}) is said to be semi-regular if the fuzzy regularly open sets of X form a base for τ_{X} [2]. For (X, τ_{X}), X_{S} denotes the semiregularization of X, i.e. X_{S} = (X, τ_{S}) and N_{S}(x_{α}) the nbd-system of x_{α} in X_{S}. An fts X is called fuzzy almost regular (fuzzy regular, resp.) if every fuzzy regularly open (resp. fuzzy open) set λ of X can be expressed as a union of fuzzy regularly open (fuzzy open, resp.) sets U_{α} such that Cl U_{α} ⩽ λ, for all α. Fuzzy regularity implies fuzzy semi-regularity as well as fuzzy almost regularity (see [1] and [15]).

2. Nets and filters in fuzzy topology

Definition 2.1 [12] Let (D, ⩾) be a directed set. Let X be an ordinary set. Let J be the collection of all fuzzy points in X. A function S: D → J is called a fuzzy net in X. For n ∈ D, S(n) is often denoted by x_{α_{n}}^{n}, where x^{n} is the support and α_{n} is the value of the n-th member of the fuzzy set. Hence the net S is often denoted by {x_{α_{n}}^{n}, n ∈ D} and it is called α-net if α_{n} → α.

Definition 2.2 [12] A fuzzy net {x_{α_{n}}^{n}} is said to be q-coincident with λ ∈ I^{X} if for each n ∈ D, x_{α_{n}}^{n} is q-coincident with λ; it is said to be eventually q-coincident (or q-final) with λ if there is an m ∈ D such that if n ∈ D and n ⩾ m, then x_{α_{n}}^{n} is q-coincident with λ; it is said to be frequently q-coincident (or q-cofinal) with λ if for each m ∈ D, there is an n ∈ D such that n ⩾ m and x_{α_{n}}^{n} is q-coincident with λ.

Definition 2.3. [12] A fuzzy net {x_{α_{n}}^{n}} in an fts X is said to converge to a fuzzy point x_{α} if {x_{α_{n}}^{n}} is eventuallly q-coincident with each q-nbd of x_{α}.

Definition 2.4. [11] Let B be a nonempty family of fuzzy subsets of I^{X}. Then B is called a base for a fuzzy filter on X (or a fuzzy filter-base) if the following two conditions are satisfied:

(1) 0_{X} ∉ B;

(2) if λ_{1}, λ_{2} ∈ B, then λ_{1} ∧ λ_{2} ∈ B.

If B has the property

(3) λ ∈ B and λ ⩽ μ implies μ ∈ B, then B is called a fuzzy filter on X.

A maximal, with respect to set inclusion, fuzzy filter on X is called a fuzzy ultra-filter or a maximal filter (or filter-base). If B is a base for a fuzzy filter on X, the collection F_{B} = { μ ∈ I^{X} : ∃λ ∈ B with λ ⩽ μ } is the fuzzy filter generated by B. We say that a filter F_{1} is finer than a filter F_{2} if F_{2} < F_{1}, i.e. if for each λ ∈ F_{2}, there exists μ ∈ F_{1} such that μ ⩽ λ.

Bellow are listed some results on fuzzy filter bases (ffb, shortly) which one can prove in a straightforward manner.

Theorem 2.1. (1) Let F_{1}, F_{2} be any ffb on fts X. Then the family F_{1} ∨ F_{2} = { λ_{1} ∨ λ_{2} : λ_{1} ∈ F_{1}, λ_{2} ∈ F_{2} } is an ffb on X.

(2) If λ_{1} ∧λ_{2} ≠ 0 for each λ_{1} ∈ F_{1} and each λ_{2} ∈ F_{2}, then F_{1} ∧F_{2} = { λ_{1} ∧λ_{2} : λ_{1} ∈ F_{1}, λ_{2} ∈ F_{2} } is an ffb on X.

(3) A nonempty family B ⊂ I^{X} is an ffb on X iff for any finite collection {λ_{i}}_{1}^{n} from B, ⋀_{i=1}^{n} λ_{i} ≠ 0.

(4) Let B is an ffb on X and let f: X → Y be a function. Then f(B) = {f(λ) : λ ∈ B} is an ffb on Y. If f is onto and B is an ffb on Y, then f^{−1}(B) = {f^{−1}(μ) : μ ∈ B} is an ffb on X.

Definition 2.5. A fuzzy point x_{α} is said to be a cluster point of a filter-base B (i.e. F_{B}) if every q-nbd of x_{α} is q-coincident with each member of B.

Proposition 2.1. A fuzzy point x_{α} (0 < α ⩽ 1) in an fts X is a cluster point of a filter-base B iff x_{α} ∈ Cl λ, for each λ ∈ B.

Proof. It is straightforward. This proposition is equivalent to the definition 3.2 [6].

Definition 2.6. [6] A filter-base B is said to converge to x_{α} (denoted by B → x_{α}) if every q-nbd of x_{α} contains a member of B and x_{α} ∈ Cl λ, for every λ ∈ B.

Theorem 2.2. Let F be an ffb on fts X and x_{α} a fuzzy point. Then:

(1) F → x_{α} iff N_{q}(x_{α}) < F.

(2) If B is a base for the q-nbd system of x_{α}, then B → x_{α}.

(3) If x_{α} is a cluster point of an ffb F on X and U is a q-nbd of x_{α}, then G = {λ ∧ U : λ ∈ F } is finer than F and G → x_{α}.

(4) Let λ be a non-empty fuzzy set. If F → x_{α} and there exists μ ∈ F such that μ ⩽ λ, then x_{α} ∈ Cl λ.

The proof is straightforward.

Theorem 2.3. Let x_{α} be a fuzzy point in an fts X. Then

(1) x_{α} is a cluster point of an ffb F iff there exists an ffb B finer than F and B → x_{α}.

(2) If F → x_{α}, then x_{α} is a cluster point of F.

(3) If x_{α} is a cluster point of a fuzzy ultra-filter F, then F → x_{α}.

Proof. (1) B = N_{q}(x_{α}) ∨ {λ ∧ U : λ ∈ F, U ∈ N_{q}(x_{a})} is an ffb finer than F and B → x_{α}. Converesely, let B → x_{α} and let B < F. Let λ ∈ F. Since B < F, there exists μ ∈ F such that μ ⩽ λ; x_{α} ∈ Cl μ implies x_{α} ∈ Cl λ, for each λ ∈ F, i.e. x_{α} is a cluster point of F.

(2) It follows from (1).

(3) Since x_{α} is a cluster point of an ultra-filter F, that means that for each U ∈ N_{q}(x_{α}), U q λ, for each λ ∈ F, that implies U ∧ λ ≠ 0 and hence U ∈ F. Therefore F → x_{α}.

Corollary 2.1. If x_{α} is a cluster point of a filter F_{1} that is finer than F_{2}, then x_{α} is a cluster point of the filter F_{2}. If x_{α} is a limit of the filter F_{2}, then x_{α} is the limit of every filter F_{1} finer than F_{2}.

Proposition 2.2. [11] Let X, Y be fts’s and x_{α} be a fuzzy point in X. If f is a mapping from X to Y, continuous at x_{α}, then for every filter-base B, B → x_{α} implies f(B) → f(x)_{α}.

Definition 2.7. We say that a fuzzy point x_{α} ∈ SCl_{θ} λ if for each semi open fuzzy set U, x_{α} q U implies Cl U q λ.

Definition 2.8. A fuzzy filter (or a filter-base) F is said to:

(1) δ-accumulate to x_{α} if x_{α} ∈ Cl_{δ} λ, for each λ ∈ F [6].

(2) s-accumulate to x_{α} if x_{α} ∈ SCl_{θ} λ, for each λ ∈ F.

(1) θ-accumulate to x_{α} if x_{α} ∈ Cl_{θ} λ, for each λ ∈ F.

Definition 2.9. Let (X, τ_{X}) be an fts and let N_{S}^{q}(x_{α}) be the q-nbd filter in X_{S}, i.e. the filter generated by regular open fuzzy sets; S(x_{α}) be the filter generated by the family SO(x_{α}) = {Cl λ : x_{α} q λ ∈ SO(τ_{X})}, and X(x_{α}) be the filter generated by C(x_{α}) = {Cl λ : x_{α} q λ ∈ τ_{X}}. We say that a filter (or a filter-base) F

(1) δ-converges to x_{α} if N_{S}^{q}(x_{α}) < F;

(2) s-converges to x_{α} if S(x_{α}) < F;

(3) θ-converges to x_{α} if C(x_{α}) < F.

Proposition 2.3. A filter (or a filter-base) F in an fts X

(1) δ-converges to x_{α} iff every fuzzy regular open q-nbd of x_{α} contains some member of F and x_{α} ∈ Cl_{δ} λ for each λ ∈ F;

(2) s-converges to x_{α} iff for each fuzzy semi open q-nbd μ of x_{α} there is a λ ∈ F such that λ ⩽ Cl μ and x_{α} ∈ SCl_{θ} λ, for each λ ∈ F;

(3) θ-converges to x_{α} iff for every open q-nbd μ of x_{α} there is λ ∈ F such that λ ⩽ Cl μ and x_{α} ∈ Cl_{θ} λ for each λ ∈ F.

Proof. Easy.

Theorem 2.4. A filter-base B (or a filter) on X s-accumulates (θ-accumulates, resp.) iff there exists a filter F finer than B which s-converges (θ-converges, resp.).

Proof. Let x_{α} be an s-cluster point of B. Then x_{α} ∈ SCl_{θ} λ, for every λ ∈ B. Hence for every μ ∈ SO(x_{α}) and for each λ ∈ B, μ q x_{α} implies Cl μ q λ, i.e. Cl μ ∧ λ ≠ 0. Therefore B ∨ {Cl μ : μ ∈ SO(x_{α})} is a system of generators for some filter F finer than B and it s-converges to x_{α}. The converse is obvious. Similarly for a θ-cluster point.

3. S-closed and extremally disconnected fuzzy topological spaces

Definition 3.1. [3,5] A family {λ_{α} : α ∈ I} of fuzzy open subsets of a fuzzy topological space (X, τ_{X}) is called a cover if ⋁{λ_{α} : α ∈ I} = X. A fuzzy topological space is called compact (shortly FC) if every open cover has a finite subcover. An fts X is said to be fuzzy nearly (almost) compact (shortly FNC (FAC)) if every open cover contains a finite subfamily {λ_{α_{i}} : i = 1,…, n} such that X = ⋁_{i=1}^{n} Int Cl λ_{α_{i}} (X = ⋁_{i=1}^{n} Cl λ_{α_{i}}).

Definition 3.2. [15] An fts X is fuzzy δ-compact (denoted FδC) if every open cover contains a finite δ-open subcover.

Definition 3.3. [4] An fts X is S-closed if every fuzzy semi-open cover {λ_{α} : α ∈ I} contains a finite subfamily {λ_{α_{i}}}_{i=1}^{n} such that X = ⋁_{i=1}^{n} Cl λ_{α_{i}}.

Definition 3.4. [9] An fts X is extremally disconnected (denoted FED) if the closure of every fuzzy open set is open.

Using all these definitions it is easy to see that the following implications hold:

F δC ⟹ F C ⟹ F NC ⟹ F AC.

Theorem 3.1. A semi-regular fts X is FδC iff it is FC.

Proof. The proof follows easily from the fact that τ_{S} = τ_{X}.

Using results from [5] and [15] and theorem 3.1, we have:

Theorem 3.2. a) If fts X is semi-regular, then F δC ∼ F C ∼ F NC;

b) if X is fuzzy regular, then F δC ∼ F C ∼ F NC ∼ F AC (“∼” means “is equivalent”).

Theorem 3.3. Let an fts (X, τ_{X}) be S-closed. Then for every family of fuzzy open sets {λ_{α} : α ∈ I} of X with finite intersection property (FIP, shortly) it holds ⋀_{α} Cl λ_{α} ≠ 0.

Proof. Let {λ_{α}}_{α∈I} be a family of fuzzy open sets with FIP. If ⋀_{α} Cl λ_{α} = 0, then ⋁_{α}(1 − Cl λ_{α}) = 1_{X} and {1 − Cl λ_{α}}_{α} is a semi-open cover of X. Hence, there exists a finite subcollection {λ_{α_{i}}}_{i=1}^{n} such that ⋁_{i=1}^{n} Cl(1−Cl λ_{α_{i}}) = 1_{X}. Therefore ⋀_{i=1}^{n} λ_{α_{i}} ⩽ ⋀_{i=1}^{n}(1 − Cl(1 − Cl λ_{α_{i}})) = 0, a contradiction. Hence ⋀_{α} Cl λ_{α} ≠ 0.

Theorem 3.4. If an fts (X, τ_{X}) is extremally disconnected and for any family of fuzzy open sets {λ_{α}}_{α} of X with FIP it holds ⋀_{α} Cl λ_{α} ≠ 0, then X is S-closed.

Proof. Let {λ_{α}}_{α∈I} be a semi-open cover of X and suppose that X is not Sclosed. Hence for each finite family {λ_{α_{i}}}_{i=1}^{n}, ⋁_{i=1}^{n} Cl λ_{α_{i}} < 1. Therefore ⋀_{i=1}^{n}(1 − Cl λ_{α_{i}}) ≠ 0, i.e. {1 − Cl λ_{α}}_{α} is a family of fuzzy open sets with FIP. Hence ⋀_{α} Cl(1 − Cl λ_{α}) ≠ 0. Since X is FED and since Cl λ_{α} = Cl Int λ_{α}, for each λ_{α} ∈ SO(X) it follows that Cl λ_{α} is fuzzy open for each α ∈ I, i.e. 1 − Cl λ_{α} is fuzzy closed. Thus

⋁_{α} λ_{α} ⩽ ⋁_{α} Cl λ_{α} = ⋁_{α}(1 − (1 − Cl λ_{α})) = ⋁_{α}(1 − Cl(1 − Cl λ_{α})) < 1_{X},

a contradiction (since {λ_{α}}_{α} is a cover of X).

Proposition 3.1. An fts X is S-closed iff any cover by regular closed sets has a finite subcover.

Proof. It is trivial, since the closure of a semi-open set is regular closed, therefore semi-open. Even more, it is easy to see that RC(X) = {Cl λ_{α} : λ_{α} ∈ SO(X)}.

Corollary 3.1. S-closedness is a fuzzy semiregular property, i.e. X is Sclosed iff X_{S} is S-closed.

Proof. The proof is straightforward since an fts and its semiregularization have the same fuzzy regular closed sets.

Theorem 3.5. For an fts X the following are equivalent:

(1) X is S-closed;

(2) each filter-base in X s-accumulates;

(3) every maximal filter-base on X s-converges.

Proof. (2) ⟺ (3). See theorem 2.1.

(1) ⟺ (3). Let X be S-closed and let F be a maximal filter-base on X which doesn’t s-converge. Therefore, it doesn’t s-accumulate to any point. This implies that for each point x_{α} there exists a semi-open set μ_{α} ∈ S(x_{α}) and an element λ_{α} ∈ F such that x_{α} q μ_{α} and Cl μ_{α} q̄ λ_{α}. Without loss of generality, we may assume that C = {μ_{α}}_{x_{α}∈I^{X}} is a semi-open cover of X, because if x_{α} ∉ μ_{α}, then x_{α′} = x_{1−α} ∈ μ_{α}, for each α ∈ (0, 1). Since X is S-closed, there exists a finite subcollection {μ_{α_{i}}}_{i=1}^{n} such that X = ⋁_{i=1}^{n} Cl μ_{α_{i}}. Since F is a maximal filter-base there exists λ ≠ 0, λ ∈ F such that λ ⩽ ⋀_{i=1}^{n} λ_{α_{i}} and λ_{α_{i}} q̄ Cl μ_{α_{i}}, for each i. This implies that λ q̄ Cl μ_{α_{i}}, for each i = 1,…, n, i.e. λ q̄ ⋁_{i=1}^{n} Cl μ_{α_{i}} = X, a contradiction, since λ ≠ 0.

Conversely, suppose X is not S-closed and let {λ_{α}}_{α∈I} be a semi-open cover of X such that ⋁_{i=1}^{n} Cl λ_{α_{i}} < X for every finite family {λ_{α_{i}} : i = 1,…, n}. Then there exists an x_{t} such that Cl λ_{α_{i}}(x_{t}) < t, for each 1 ⩽ i ⩽ n. Hence μ_{α_{i}}(x_{t}) = (1 − Cl λ_{α_{i}})(x_{t}) > 1 − t, i.e. μ_{α_{i}} q x_{t} and μ_{α_{i}} ∈ SO(X). Therefore ⋀_{i=1}^{n} μ_{α_{i}} ≠ 0, for each finite intersection. Hence {μ_{α}}_{α} forms a filter-base B which s-accumulates to x_{t}. Then there exists a maximal filter-base F which s-converges to x_{t}, and this implies that ⋁_{α} Cl λ_{α} ≠ X. Therefore ⋁_{α} λ_{α} ≠ X, a contradiction, since {λ_{α}}_{α} is a cover of X.

It is known that the product of S-closed spaces is not necessarily S-closed, even in general topology. In fuzzy topology for fuzzy spaces X and Y, where X is product related to Y (cf. [2, Definition 3.7; Theorem 3.10; Theorem 4.6]) we have the following result:

Theorem 3.6. Let (X, τ_{X}) and (Y, τ_{Y}) be S-closed fts’s such that X is product related to Y. Then fuzzy topological product X × Y is S-closed.

Proof. Let {λ_{α}}_{α} and {μ_{β}}_{β} be semi-open covers for X, respectively Y. Then {λ_{α} × μ_{β}}_{α,β} is fuzzy semi-open cover of X × Y. For finite n ∈ N we have ⋁_{i=1}^{n} Cl λ_{α_{i}} × ⋁_{i=1}^{n} Cl μ_{β_{i}} = ⋁_{i=1}^{n} Cl(λ_{α_{i}} × μ_{β_{i}}), i.e. X × Y = ⋁_{i=1}^{n} Cl(λ_{α_{i}} × μ_{β_{i}}). Thus X × Y is S-closed.

Definition 3.5. [14] A function f: X → Y is said to be fuzzy semi-continuous (resp. irresolute) of f^{−1}(μ) ∈ SO(X), for any fuzzy open (resp. semi-open) set μ of Y.

Theorem 3.7. [4] If f: X → Y is an irresolute surjection from an S-closed fts X to an fts Y, then Y is S-closed.

Since fuzzy semi-continuous and almost open (i.e. f^{−1}(Cl λ) < Cl f^{−1}(λ), for each λ ∈ τ_{Y}) mapping implies that f is irresolute [6] we have the following corollary:

Corollary 3.2. An image of an S-closed space under an almost open fuzzy continuous surjection is S-closed.

Definition 3.6. A property of fts is called fuzzy semi-topological if it is preserved by fuzzy semi-homeomorphism (i.e. bijection and such that the images of semi-open sets are semi-open and the inverses of semi-open sets are semi-open).

Corollary 3.3. To be an S-closed fts is a fuzzy semi-topological property.

Theorem 3.8. For an fts (X, τ_{X}) the following are equivalent:

(1) X is FED.

(2) If a filter base on X δ-converges, then it s-converges.

(3) A filter-base X s-converges iff it θ-converges.

(4) If a filter-base on X converges with respect to the topology τ_{X}, then it s-converges.

Proof. We shall first prove the following lemma.

Lemma 3.1. If an fts X is FED, then SCl λ = Cl_{θ} λ, for each λ ∈ SO(X).

Proof of Lemma. SCl λ ⩽ Cl λ ⩽ Cl_{δ} λ ⩽ Cl_{θ} λ, for each λ ∈ I^{X}. We shall prove that Cl_{θ} λ ⩽ SCl λ, for λ ∈ SO(X). Let x_{α} ∈ SCl λ, then there exists U ∈ SO(X) such that x_{α} q U and U q̄ λ, which implies that U ⩽ λ^{c}, where λ^{c} = 1 − λ. Hence Cl U ⩽ Cl(λ^{c}). Since X is FED, then Cl U is open. Therefore Cl U ⩽ Int Cl(λ^{c}) ⩽ SCl(λ^{c}) = λ^{c}. Hence Cl U q̄ λ and so x_{α} ∉ Cl_{θ} λ.

Corollary 3.4. If X is FED, then SCl λ = Cl λ = Cl_{δ} λ = Cl_{θ} λ, for each λ ∈ SO(X).

Using the results above, the proof of theorem 3.8 is now straightforward.

Remark 3.1. From the above corollary and Lemma 7 and Theorem 12 in [9] we have the following: SCl λ = Cl λ = Cl_{δ} λ = Cl_{θ} λ, for every fuzzy set λ ∈ SO(X) ∪ P O(X), where P O(X) is the set of all fuzzy pre-open sets in X. Also it is easy to prove that definition of rc-convergence [9] is equivalent to our definition of s-convergence.

Lemma 3.2. Let X be fuzzy almost regular and S-closed. Then X is FED and fuzzy nearly compact (FNC).

Proof. Suppose that X is not FED. Then there exists a fuzzy regular open set λ such that Cl λ(x) > λ(x) for some x ∈ X and Cl λ ≠ 1_{X}. Let x_{α} ∈ Cl λ and x_{α} ∉ λ. For every open q-nbd U of x_{α}, we have U ∧ λ ≠ 0. Therefore F = {U ∧ λ : U ∈ N_{q}(x_{α})} forms a filter-base in Cl λ. Since Cl λ is S-closed relative to X, then F →^{s} y_{β}, i.e. F s-converges to some point y_{β} ∈ Cl λ. If y_{β} ∉ λ, then y_{β} q λ^{c} = 1 − λ ∈ RC(X), hence 1 − λ is semi-open and 1 − λ ∈ S(y_{β}). Therefore, there exists μ ∈ F such that μ ⩽ 1 − λ, i.e. μ q̄ λ. Since F → y_{β} in the usual sense and y_{β} ∈ Cl u ⩽ 1 − λ, therefore every member of F is q-coincident with 1 − λ, what is impossible. Hence y_{β} ∈ λ. Almost regularity and λ ∈ RO(X) imply that λ ∈ S(y_{β}), i.e. there exists V ∈ RO(X) such that y_{β} q V ⩽ Cl V ⩽ λ. Since x_{α} ∉ λ, then x_{α} q Cl V^{c} = 1 − Cl V is an open q-nbd of x_{α}. Since N_{q}(x_{α}) → x_{α}, there exists a q-nbd U of x_{α} such that (U ∧ λ) ⩽ 1 − Cl V, i.e. (U ∧ λ) q̄ Cl V. But that contradicts the fact that F →^{s} y_{β}. Therefore it must be x_{α} = y_{β}, i.e. Cl λ = λ. Hence X is FED.

To prove nearly compactness, let U be a maximal filter-base. S-closedness implies that U s-converges. Theorem 3.8.(3) implies that U θ-converges. Almost regularity implies that U δ-converges. Since X is fuzzy nearly compact (see Theorem 3.9 [13] and Theorem 2.3) iff every maximal filter-base δ-converges, then this completes the proof.

Corollary 3.5. Let X be an almost regular fts. Then X is S-closed iff X is FNC and FED iff X_{S} is regular, compact and FED.

Theorem 3.9. Let fts (X, τ_{X}) be a fuzzy regular space. Then the following are equivalent:

(1) X is compact and FED;

(2) X is S-closed;

(3) X is FAC and FED.

Proof. (1) ⟹ (2). See Theorem 3.4.

(2) ⟹ (3). That S-closed implies FAC, follows directly from the definitions. Using lemma 3.2 we have the proof of (2) ⟹ (3).

(3) ⟹ (1). The proof is straightforward.

Corollary 3.6. (cf. [4, Theorem 3.7; Corollary 3.8]) Let X be fuzzy regular and extremally disconnected. Then the following is valid:

F C ∼ F δC ∼ S-closed ∼ F AC ∼ F NC.

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(received 12.01.1998.)

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