����������� ������ 46 (1994), 17–23

UDC 515.122.8

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EXTREMAL DISCONNECTEDNESS IN FUZZY TOPOLOGICAL SPACES

Ratna Dev Sarma

Abstract. The notion of extremal disconnectedness is introduced for fuzzy topological spaces. Various properties of this notion are discussed. Interrelationship of some weaker forms of fuzzy continuity is studied in the light of extremal diconnectedness.

1. Introduction

Connectedness and its related notions are studied by various researchers for fuzzy topological space ([3], [5], [13]). In many cases, the study while remaining parallel to its counterpart in general topology shows divergence due to the fuzzy setting. For example, there are four types of connectedness, namely c_{i}-connectedness (i = 1, 2, 3, 4) defined for an arbitrary fuzzy set. But when considered globally, that is, when the fuzzy set is considered to be the whole space, all these four types coincide to one type. Because of these diversities, study of connectedness and related concepts remain to be an interesting one in the area of fuzzy topology. In this paper, we add to the existing literature one more notion, namely, extremal disconnectedness. Various properties of extremal disconnectedness is studied and the notion is shown to be compatible with the existing fuzzy topological notions such as T_{2}-axiom, fuzzy regularity, c-connectedness, zero-dimension etc. Also following localized approach, we provide a number of characterizations of fuzzy weak continuity. Alongwith, we study the interrelationship between fuzzy weak continuity and fuzzy semi-continuity in the context of extremal diconnectedness.

2. Preliminaries

Throughout the paper, X, Y, Z etc. denote ordinary sets while μ, η, σ etc. denote fuzzy sets defined on an arbitrary set. The fuzzy sets are defined with respect to the closed unit interval I = [0, 1]. The union and intersection of a family of fuzzy sets {μ_{i}} are denoted ∨μ_{i} and ∧μ_{i} respectively. The constant fuzzy sets which take each member of X to zero and one respectively are denoted by 0_{X} and 1_{X} respectively. A fuzzy point with support x and value α, 0 < α ⩽ 1, is denoted by x_{α}. While x_{α} ⩽ μ implies α ⩽ μ(x); x_{α} ε μ implies α < μ(x) and μ ⩽ η implies μ(x) ⩽ η(x) for each x. Two fuzzy sets μ and η are said to overlap, denoted by μ q η, if there exists x in X such that μ(x) + η(x) > 1. For the definition of a fuzzy topological space (an fts in brief), we refer to Chang [10]. For other definitions, results and notations used in the paper, see [1], [6], [13] or [16].

Keywords: Fuzzy point, neighbourhood, quasi-neighbourhood, quasi-coincident fuzzy sets, fuzzy continuity

3. Extremal disconnectedness

Definition 3.1. An fts X is called extremally disconnected if for every open fuzzy set μ in X, cl(μ) is a crisp clo-open fuzzy set in X.

Example 3.2. Let X be a non-empty set and T = {0_{X}, 1_{X}}∪{μ_{α}}_{α∈Λ}, where μ_{α} is any fuzzy set defined on X such that \frac{1}{2} < μ_{α}(x) < 1, for every x in X. Then the fts ⟨X, T ⟩ is extremally disconnected, although X is not c-connected.

Below we give a necessary and sufficient condition for an fts to be extremally disconnected.

Theorem 3.3. An fts X is extremally disconnected iff for every pair of nonoverlaping open fuzzy sets μ and η in X, cl(μ) and cl(η) are non-overlaping crisp fuzzy sets in X.

Proof. Let X be an extremally disconnected fts and μ and η be open fuzzy sets in X such that μ q̸ η. Suppose, if possible, cl(μ) overlaps with cl(η). Then (cl(μ))(y)+(cl(η))(y) > 1, for some y in X. Then the fuzzy point Y_{(cl(η))(y)} ⩽ cl(η) and Y_{(cl(η))(y)} q cl(μ). X being extremally disconnected, cl(μ) is open. Then cl(μ) is a Q-nbd of Y_{(cl(η))(y)} and Y_{(cl(η))(y)} ⩽ cl(η). Therefore η q cl(μ). Let η(x) + (cl(μ))(x) > 1. This implies that x_{1−η(x)} properly belongs to cl(μ), or, x_{1−η(x)+ε} ⩽ cl(μ) for some positive real number ε. Also x_{1−η(x)+ε} q η. Therefore μ q η, which is a contradiction.

Conversely, let μ be an open fuzzy set in X. Then μ and 1_{X} − cl(μ) are two open fuzzy sets such that μ q̸ 1_{X} − cl(μ). Therefore cl(μ) q̸ cl(1_{X} − cl(μ)) and cl(μ) is a crisp fuzzy set in X. Now, cl(μ) q̸ cl(1_{X} − cl(μ)) implies that cl(μ) ⩽ 1_{X} −cl(1_{X} −cl(μ)) = 1_{X} −(1_{X} −int(cl(μ))) = int(cl(μ)); so that cl(μ) = int(cl(μ)). Thus cl(μ) is a crisp clo-open fuzzy set in X. Hence X is extremally disconnected. This completes the proof.

Theorem 3.4. Every open subspace of an extremally disconnected fts is extremally disconnected.

Proof. Easy.

In [3] and [5], c-disconnectedness for general fuzzy topological spaces is introduced and studied. An fts X is c-disconnected if there exist two proper open fuzzy sets μ and η such that μ ∧ η = 0_{X} and μ ∨ η = 1_{X}. It follows that an fts X is c-disconnected iff there exists a proper crisp clo-open fuzzy set in X. The following example alongwith Example 3.2 shows that c-disconnectedness and extremal disconnectedness are two independent notions.

Example 3.5. Let A and B be non-empty disjoint sets and X = A ∪ B. Let μ_{A}^{α} and η_{B}^{β} be fuzzy sets defined by

μ_{A}^{α}(x) = \{ \begin{matrix}α, & for x ∈ A, 0 ⩽ α ⩽ 1, \\ 0, & otherwise, \end{matrix} η_{B}^{β}(y) = \{ \begin{matrix}β, & for y ∈ B, 0 ⩽ β ⩽ 1, \\ 0, & otherwise. \end{matrix}

Let T (x) = {μ_{A}^{α}∨η_{B}^{β}}_{0⩽α,β⩽1}. Then the fts ⟨X, T (x)⟩ is c-disconnected and μ_{A}^{1} and η_{B}^{1} are two non-zero open fuzzy sets in X which are providing a c-disconnectedness of 1_{X}. But this fts fails to be extremally disconnected as the closure of the open fuzzy set μ_{A}^{1/2} ∨ η_{B}^{1/2} is not crisp.

The following theorem establishes relationship between c-disconnectedness and extremal disconnectedness.

Theorem 3.6. Every subspace of an extremally disconnected Hausdorff fts is c-disconnected.

Proof. Let X be an extremally disconnected Hausdorff fts and A be a subspace of X. Let x_{α} (0 < α < 1) and y_{β} (0 < β < 1) be two fuzzy points in A with distinct supports. As X is T_{2}, considering x_{α} and y_{β} as fuzzy points in X, we get two open fuzzy sets μ and η in X such that x_{α} q η, y_{β} q η and μ ∧ η = 0_{X}. This implies that x_{α′} ε μ, y_{β′} ε η and μ ∧ η = 0_{X}, where α′ = 1 − α, β′ = 1 − β. Now, X being extremally disconnected, cl(μ) and cl(η) are crisp clo-open fuzzy sets in X, so that (cl(μ))^{a} and (cl(η))^{a}, where (cl(μ))^{a} = cl(μ)|_{A}, are crisp cloopen fuzzy sets in A. Also x_{α′} ε μ and x is in A imply that x_{α′} ε (cl(μ))^{a}. Thus (cl(μ))^{a} ≠ 0_{A}. Similarly y_{β′} ε (cl(η))^{a}. Again that μ ∧ η = 0_{X} and X is extremally disconnected together imply that cl(μ)+cl(η) ⩽ 1_{X}, in view of Theorem 3.3; whence (cl(μ))^{a} + (cl(η))^{a} ⩽ 1_{A}. Therefore (cl(μ))^{a}(y) + (cl(η))^{a}(y) ⩽ 1, that is, (cl(μ))^{a}(y) + 1 − β < (cl(μ))^{a}(y) + (cl(η))^{a}(y) ⩽ 1. Or, in other words (cl(μ))^{a}(y) < β. Thus (cl(μ))^{a} ≠ 1_{A}. Therefore (cl(μ))^{a} is a proper crisp clo-open fuzzy set in A. Consequently A is c-disconnected. This completes the proof.

Remark. In the above theorem, the condition that ”X is Hausdorff” can not be dropped. For example, the fts given in Example 3.2 is extremally disconnected. But here even X itself is not c-disconnected. It may be noticed that X is not a Hausdorff fts in this case.

Example 3.7. Let X = {x, y}, S = {μ_{1}, μ_{2}} ∪ {μ_{αβ}}_{1/2<α,β⩽1} and consider the fuzzy topology T (x) generated by S, where

μ_{1}(x) = 1, μ_{2}(x) = 0, μ_{αβ}(x) = α,
μ_{1}(y) = 0, μ_{2}(y) = 1, μ_{αβ}(y) = β.

Then ⟨X, T (x)⟩ is both extremally disconnected as well as c-disconnected. Here it can be verified that X is a T_{2}-fts .

Definition 3.8.[8] An fts X is called zero-dimensional if every crisp fuzzy point in X has a base of crisp clo-open fuzzy sets.

An fts X is called a regular fuzzy space ([1], [12]) if each open fuzzy set μ in X is a union of open fuzzy sets μ_{i} in X, such that cl(μ_{i}) ⩽ μ for each i. We have

Theorem 3.9. Every non-empty extremally disconnected regular fts is zerodimensional.

Proof. Easy.

In the rest of the paper, we discuss extremal disconnectedness in the light of some weaker forms of fuzzy continuity.

Definition 3.10.[4] A mapping f: X → Y from an fts X to an fts Y is said to be fuzzy weakly continuous (briefly f.w.c.) at a fuzzy point x_{α}, if for each neighbourhood η of f(x_{α}) in Y, there exists a neighbourhood μ of x_{α} in X such that f(μ) ⩽ cl(η).

f is said to be fuzzy weakly continuous if it is so at each fuzzy point in X.

Below we provide several characterizations of fuzzy weak continuity. Its proof is similar to that of Theorem 3.4 [4] and it is left for the reader as an exercise.

Theorem 3.11. For a mapping f: X → Y from an fts X to an fts Y, the following are equivalent:

(i) f is fuzzy weakly continuous; (ii) for each open fuzzy set η in Y, f^{−1}(η) ⩽ int(f^{−1}(cl(η))); (iii) for each closed fuzzy set σ in Y, cl(f^{−1}(int(σ))) ⩽ f^{−1}(σ); (iv) f is f.w.c. at every x_{α}, where 0 < α < 1;

(v) for each fuzzy point x_{α} in X, and for every open fuzzy set η in Y properly containing f(x_{α}), there is an open fuzzy set μ properly containing x_{α} such that f(μ) ⩽ cl(η);

(vi) for any fuzzy point x_{α} in X and for each Q-nbd η of f(x_{α}), int(f^{−1}(cl(η))) is a Q-nbd of x_{α};

(vii) for any fuzzy point x_{α} in X and for each Q-nbd η of f(x_{α}), there exists a Q-nbd μ of x_{α} such that f(μ) ⩽ cl(η);

(viii) for each fuzzy net {S_{n}} converging to any fuzzy point x_{α}, the image fuzzy net {f(S_{n})} eventually overlaps with the closure of each Q-nbd of f(x_{α});

(ix) for each open fuzzy set η in Y, cl(f^{−1}(η)) ⩽ f^{−1}(cl(η)).

Definition 3.12. A mapping f: X → Y, where A ⊂ X, is said to be a retraction of X onto A if f(x) = x iff x is in A.

Theorem 3.13. Let A ⊂ X and f: X → A be a fuzzy weakly continuous retraction of X onto A. If X is extremally disconnected Hausdorff fts, then A is a closed subspace of X.

Proof. Suppose, if possible, χ_{A} is not closed. Then there exists a fuzzy point x_{α} in X, such that x_{α} ⩽ cl(χ_{A}), but x_{α} ⩽̸ χ_{A}. Then x is not in A. Therefore f(x) ≠ x. Since X is Hausdorff, there exist disjoint Q-nbds μ and η of x_{α} and f(x_{α}) respectively. Now, X being extremally disconnected, μ ∧ cl(η) = 0_{X}. For, if y_{β} ⩽ μ ∧ cl(η), then μ(y) > 0 and (cl(η))(y) > 0. Since cl(η) is a crisp fuzzy set, (cl(η))(y) = 1. Thus y_{1} ⩽ cl(η) and μ(y) > 0. Therefore, y_{1} ⩽ cl(η) and y_{1} q μ. Hence μ q η, which is a contradiction, as μ ∧ η = 0_{X}.

Now, let σ be a Q-nbd of x_{α}. Then μ ∧ σ is a Q-nbd of x_{α}. Therefore μ ∧ σ overlaps with χ_{A} at some point z in X, as x_{α} ⩽ cl(χ_{A}). This implies that

(μ ∧ σ)(z) ≠ 0 and z is in A.

That is,

μ(z) ≠ 0, σ(z) ≠ 0 and f(z) = z.

Therefore, (cl(η))(z) = 0, σ(z) ≠ 0 and f(z) = z, using the fact that μ∧cl(η) = 0_{X}. Thus,

(cl(η))(f(z)) = 0, f(σ)(f(z)) ≠ 0.

Hence, f(σ) ⩽̸ cl(η). This contradicts the fact that f is f.w.c. at x_{α}, in view of Theorem 3.11. Thus our assumption is wrong and consequently χ_{A} is a closed fuzzy set in X. Hence A is a closed subspace of X.

Definition 3.14. (a) A fuzzy set μ is called semi-open if there exists an open fuzzy set μ_{0} such that μ_{0} ⩽ μ ⩽ cl(μ_{0}).

(b) The complement of a semi-open fuzzy set is called a semi-closed fuzzy set.

(c) Intersection of all semi-closed fuzzy sets containing a fuzzy set μ is called the semi-closure of μ and is denoted by s-cl(μ).

Azad [9] has defined a fuzzy set σ to be semi-closed if there exists an open fuzzy set η in X such that

int(1_{X} − η) ⩽ σ ⩽ 1_{X} − η.

However, this definition can easily be shown to be equivalent to the above mentioned definition.

For a fuzzy set μ, cl(μ) and s-cl(μ) need not be same. This can be verified from the following example.

Example 3.15. Let X be any non-empty set and T (x) = {0_{X}, μ_{1/3}, μ_{2/3}, 1_{X}}, where μ_{i}(x) = i for all x in X, i = 1/3, 2/3. Then for the fuzzy set μ, defined by μ(x) = \frac{1}{2} for all x in X, we have s-cl(μ) = μ and cl(μ) = μ_{2/3}.

Further, from this example it is clear that s-cl(μ) and cl(μ) need not be same even if μ is semi-open. However, if the fts is extremally disconnected, the closure and semi-closure of a semi-open fuzzy set turn out to be same. Before proving this we state the following result.

Theorem 3.16.[11] Let μ be a fuzzy set in an fts X. Then x_{α} ⩽ s-cl(μ) iff every semi-open fuzzy set overlapping with x_{α} overlaps with μ.

Theorem 3.17. In an extremally disconnected fts X, cl(μ) = s-cl(μ) for every semi-open fuzzy set μ in X.

Proof. In general, s-cl(μ) ⩽ cl(μ), as every closed fuzzy set is also semi-closed. Thus it suffices to show that cl(μ) ⩽ s-cl(μ) for every semi-open fuzzy set in X. Let x_{α} be a fuzzy point in X such that x_{α} ⩽̸ s-cl(μ). Then there exists, by Theorem 3.16, a semi-open fuzzy set η in X such that x_{α} q η and η q̸ μ. This implies that x_{α} q η and int(η) q̸ int(μ). Since X is extremally disconnected, by Theorem 3.3, we have cl(int(μ)) q̸ cl(int(η)). Also by Theorem 4.2[9] , η ⩽ cl(int(η)). Therefore, cl(int(μ)) q̸ η. Again, μ being semi-open there exists an open fuzzy set μ_{0} such that μ_{0} ⩽ μ ⩽ cl(μ_{0}). Therefore,

μ_{0} ⩽ int(μ) ⩽ μ ⩽ cl(μ) ⩽ cl(μ_{0}).

This implies that

cl(μ_{0}) ⩽ cl(int(μ)) ⩽ cl(μ) ⩽ cl(cl(μ)) ⩽ cl(cl(μ_{0})).

Hence cl(μ) = cl(int(μ)). Thus it follows that cl(μ) q̸ η. Also x_{α} q η. Therefore,

(cl(μ))(x) + η(x) ⩽ 1 and α + η(x) > 1,

so that α > (cl(μ))(x). Hence x_{α} ⩽̸ cl(μ). Consequently, cl(μ) ⩽ s-cl(μ). This proves the theorem.

The last theorem is in the spirit of Theorem 3.2 of Noiri [15]. It establishes the relationship between fuzzy semi-continuity and fuzzy weak continuity in the light of extremal disconnectedness. We first recall the definition of fuzzy semi-continuity.

Definition 3.18.[7] A mapping f: X → Y from an fts X to fts Y is said to be fuzzy semi-continuous at a fuzzy point x_{α} if for every open fuzzy set η containing f(x_{α}) there exists a semi-open fuzzy set μ containing x_{α} in X such that f(μ) ⩽ η.

f is said to be fuzzy semi-comtinuous, if it is so at every fuzzy point in X.

Theorem 3.19. A mapping f: X → Y from an fts X to an fts Y is fuzzy semi-continuous iff f^{−1}(η) is semi-open for every open fuzzy set η in Y.

Proof. Easy.

From the definition it is clear that fuzzy continuity implies fuzzy semicontinuity. The converse is, however, not true. Also, fuzzy weak continuity and fuzzy semi-continuity are two independent notions.

Example 3.20. Let X be any non-empty set and T_{1}(x) = {0_{X}, μ_{1/3}, 1_{X}}, T_{2}(x) = {0_{X}, μ_{1/2}, 1_{X}}, where μ_{j}(x) = j, j = 1/3, 1/2, for every x in X. Then the identity mapping i: ⟨X, T_{1}(x)⟩ → ⟨X, T_{2}(x)⟩ is fuzzy semi-continuous but neither fuzzy continuous nor fuzzy weakly continuous, as it is not f.w.c. at any x_{α}, where 1/3 < α ⩽ 1/2. Again the identity mapping i′: ⟨X, T_{2}(x)⟩ → ⟨X, T_{1}(x)⟩ is fuzzy weakly continuous but not fuzzy semi-continuous as it is not fuzzy semi-continuous at any x_{α}, where 0 < α ⩽ 1/3.

We are now in a position to prove the following theorem.

Theorem 3.21. If an fts X is extremally disconnected and f: X → Y is fuzzy semi-continuous, then f is fuzzy weakly continuous.

Proof. Let η be an open fuzzy set in Y. Then f^{−1}(η) is a semi-open fuzzy set in X, as f is fuzzy semi-continuous. Let x_{α} ⩽̸ f^{−1}(cl(η)). Then f(x_{α}) ⩽̸ cl(η). Therefore there exists a closed fuzzy set σ containing η such that f(x_{α}) ⩽̸ σ. Since f is fuzzy semi-continuous, f^{−1}(1_{Y} − σ) is semi-open in X. But

f^{−1}(1_{Y} − σ) = 1_{X} − f^{−1}(σ).

Thus f^{−1}(σ) is a semi-closed fuzzy set in X. Also, f^{−1}(η) ⩽ f^{−1}(σ) as η ⩽ σ and x_{α} ⩽̸ f^{−1}(σ) as f(x_{α}) ⩽̸ σ. This implies that x_{α} ⩽̸ s-cl(f^{−1}(η)). Therefore, by Theorem 3.17, x_{α} ⩽̸ cl(f^{−1}(η)), as X is extremally disconnected. Thus cl(f^{−1}(η)) ⩽ f^{−1}(cl(η)). Hence by Theorem 3.11, f is fuzzy weakly continuous.

Here the condition ”extremal disconnectedness” cannot be dropped. For example, ⟨X, T_{1}(x)⟩ is not extremally disconnected in Example 3.20. Although the identity mapping is fuzzy semi-continuous, it is not fuzzy weakly continuous.

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(received 28.07.1993.)

Mr. Ratna Dev Sarma, c/o S. Gurbux Singh, WZ–1200A, Rani Bagh, Delhi–110034, INDIA