Fuzzy extremally disconnected spaces

Fuzzy Sets and Systems 46 (1992) 245-250 245 North-Holland

Fuzzy extremally disconnected spaces

Banamali Ghosh Department of Pure Mathematics, University of Calcutta, 35 Ballygunge Circular Road, Calcutta 700019, India

Received May 1989 Revised May 1990

Abstract: An investigation of extremally disconnected fuzzy topological spaces is done in this papcr. Such a class of spaces is characterized in various ways by using the concepts of q-neighbourhoods, semi-q-neighbourhoods, b-closure, 0-closure, semi-closure and related notions in fuzzy setting.

AMS Subject Classification: 54A40.

Keywords: Fuzzy semi-closure, b-closure and 0-closure; q-nbd and semi-q-nbd; fuzzy pre-open and semi-pre-open sets; extremally disconnected space.

The concept of extremally-disconnectedness for topological spaces has widely been studied by many workers. In [5], Herrmann has effectively implicated the concept in studying different important classes of topological spaces. Certain nice applications of the notion of extremally disconnected spaces are found in Herrmann's work. The same concept has also been used by Jankovi6 [6] in his investigations of certain types of mappings. For quite some time, mathe- maticians have been taking keen interest in extending different kinds of mappings and also many important classes of spaces, e.g. almost compact, nearly compact and S-closed spaces, to the fuzzy setting. Thus it has become imperative for further developments of these theories, to make a detailed study of extremally disconnected fuzzy spaces. In fact, this idea was used in [8] to study fuzzy S-closed spaces. In the present paper, we further develop the notion of the extremally disconnected fuzzy topological spaces by characterizing such a class of spaces in various ways.

Throughout the paper (X, T) or simply X stands for a fuzzy topological space (fts, for short) in Chang's [2] sense. A fuzzy point in X with support x ~ X and value a~ ( 0 < ~ < 1 ) is

denoted by x~. For a fuzzy set A in X, the notations clA, intA and ( l - A ) are used to respectively stand for the closure, interior and complement of A, whereas the constant fuzzy sets taking on respectively the values 0 and 1 on X are designated by 0x and Ix respectively. A fuzzy set A in X is called fuzzy regularly open (semi-open) iff A = int cl A (resp. there exists a fuzzy open set U in X such that U ~< A ~< cl U) [1]. Complements of fuzzy regularly open (semi-open) sets are called fuzzy regularly closed (resp. fuzzy semi-closed). For an fts X, we shall denote the set of all fuzzy regularly open, regularly closed, semi-open and semi- closed sets by FRO(X), FRC(X), FSO(X) and FSC(X) respectively. It is known [1] that a fuzzy regularly open (closed) set is fuzzy semi-closed (resp. semi-open).

That the converse is false can easily be seen. For example, consider the fts (X, T), where

X = { a , b } and T={0x , lx, A},

where A(a)= A(b)= 0.48; then the fuzzy set B such that B(a)= B(b)= 0.5 is fuzzy semi-open but not fuzzy regularly closed. In general, we have the following diagram:

FRO (X) ~ 0 ( X ~ F S O (X)

FRC ( X ) ~ ~ F C ~ F S C ~ (X)

(where FO(X) and FC(X) denote the families of fuzzy open and fuzzy closed sets respectively).

It is known (see [1] for counterexamples) that no other implication other than those shown above, exists, in general. Although arbitrary union of fuzzy semi-open sets is fuzzy semi-open, the intersection of even two such sets may not be fuzzy semi-open [1]. For the case of fuzzy regularly open sets, the intersection of two members of FRO(X) is again a member of it, but union of them may not be a member of it (see [1] for details). A fuzzy set A in X is said to be q-coincident with a fuzzy set B, written as

0165-0114/92/$05.00 ~ 1992--Elsevier Science Publishers B.V. All rights reserved

246 B. Ghosh / Fuzzy extremally disconnected spaces

A q B, iff there is x • X such that A(x) + B(x) > 1 [13]. Abbreviating the word 'neighbourhood' by nbd, we say that A is a q-nbd [13] (semi-q-nbd [4]) of a fuzzy point x~ iff there exists a fuzzy open (resp. semi-open) set V such that x~ q V ~< A. It is obvious that a q-nbd of x , is a semi-q-nbd of x~, though not conversely (see [4]). The fuzzy semi-closure of a fuzzy set A in X, to be denoted by scl A, is the union of all fuzzy points x~ such that every semi-q-nbd of x~ is q-coincident with A; equivalently, scl A is the intersection of all fuzzy semi-closed sets containing A [4]. Obviously, A ~< scl A ~< cl A and scl A • FSC(X), for any fuzzy set A, also A = scl A iff A • FSC(X). The union of all fuzzy semi-open sets contained in a fuzzy set A is called the fuzzy semi-interior of A, to be denoted by sint A. Clearly, for any fuzzy set A, int A ~< sint A ~< A, and also, sint A = 1 - scl(1 - A) [8]. The operator sint being dual of the operator scl, other properties of the operator sint can be had from those of scl. For example, A = sint A iff A e FSO(X), and for any fuzzy set A, sint(sint A) = sint A. A fuzzy point x , is said to be a fuzzy b-cluster point [3] (0-cluster point [10]) of A iff i n t c l U q A (resp. c l U q A ) for every fuzzy open q-nbd U of x, . The union of all fuzzy b-cluster points (0-cluster points) of A is denoted by [A]a (resp. [A]0). A is fuzzy b-closed (0-closed) iff A = [A]~ (resp. A = [A]o) [3, 10]. In [3] and [10, 11] we find a detailed study of the fuzzy b-closure and 0-closure operators respectively. For example, it is known [10] that for any fuzzy set A, A ~ clA <~ [A]a ~< [A]o, where the reverse implications are not true, in general. Thus a fuzzy point may be a fuzzy b-cluster point of a fuzzy set without being a fuzzy 0-cluster point of it.

Definition 1 [8]. An fts (X, T) is said to be fuzzy extremally disconnected (FED, for short) iff closure of every fuzzy open set is fuzzy open in X; equivalently, every fuzzy regularly closed set is fuzzy open.

Lemma 2 [8]. I f X is fuzzy extremally disconnected then cl V = scl V, for every V • FSO(X).

Lemma 3 [11]. In an fts X, [A]a = c l A , for any A • FSO(X).

Theorem 4 [8]. An fts X is fuzzy extremally disconnected iff any two non-q-coincident fuzzy open sets in X have non-q-coincident closures.

Theorem 5. The following are equivalent for an f t s X :

(a) X is FED. (b) The closure of every fuzzy semi-open set in

X is fuzzy open. (c) The semi-closure of every fuzzy semi-open

set in X is fuzzy open. (d) The 6-closure of every fuzzy semi-open set

in X is fuzzy open. (e) Every two non-q-coincident fuzzy semi-

open sets in X have non-q-coincident closures. (f) cl A =scl A, for every fuzzy semi-open set

A inX. (g) The semi-closure of every fuzzy semi-open

set in X is fuzzy closed. (h) int A = sint A, for every fuzzy semi-closed

set A in X. (i) The semi-interior of every fuzzy semi-

closed set in X is fuzzy open.

Proof. (a) and (b) are equivalent in view of the fact that the closure of a fuzzy semi-open set is the same as the closure of a fuzzy open set, namely its interior. ( f)@ (g) and (h)@(i ) are obvious.

(a) @ (c): Let A be any fuzzy semi-open set in X. Then by Lemma 2, scl A = cl A. Hence scl A is fuzzy open in X.

(c) @ (a): Let G be any fuzzy open set in X. It is sufficient to prove cl G = scl G. Obviously, scl G ~ cl G. Let x~ ~ scl G. Then there exists a fuzzy semi-open semi-q-nbd U of x~ such that U ~ G, which means U ~< 1 - G. Hence

scl U < scl(1 - G) = 1 - G

and thus scl U ~1 G. Since scl U is a fuzzy open q-nbd of x~, x~, ~ cl G. Hence it follows that cl G = scl G.

(b) ¢:~ (d): Follows from Lemma 3. (a) ~ (f): Follows from Lemma 2. ( g ) ~ ( f ) : For any fuzzy set A in X, since

A ~< scl A ~< cl A we have cl A = cl(scl A). If A is fuzzy semi-open, by (g), sclA is fuzzy closed, and so scl A = cl A.

( i ) ~ ( h ) : For any fuzzy set A in X, int A ~< sint A <~ A, and hence int A = int(sint A).

B. Ghosh / Fuzzy extremally disconnected spaces 247

If A is fuzzy semi-closed, sint A is fuzzy open (by (i)), and then int A = sint A.

(g) ¢~, (i): Follows from the fact that sint A = 1 - scl(1 - A) and scl A = 1 - sint(1 - A), for any fuzzy set A.

( a ) f f ( e ) : Let A and B be fuzzy semi-open sets in X such that A ¢1 B. Then int A ~1 int B, and hence cl int A ¢1 cl int B (by Theorem 4). Conse- quently, cl A ¢1 cl B.

(e) ~ (b): If A • FSO(X), then A ¢1 (1 - cl A) and 1 - cl A • FSO(X). Thus by (e),

clA ¢1 cl(1 - clA) -- 1 - int clA,

and hence cl A ~< int cl A. Thus cl A = int cl A which means that cl A is fuzzy open in S.

( f ) ~ ( e ) : If A and B are non-q-coincident fuzzy semi-open sets in X, then sclA and scl B are fuzzy semi-open and so scl A ~ scl B. By (f), cl A ¢~ cl B.

In [7], a fuzzy set A in X is defined to be fuzzy pre-open iff A <~ int cl A. Let us now define as follows.

Lemma 8. For any A • FSPO(X), cl A = [A]~.

Proof. To prove this, it is sufficient to prove that [A]6 <~ cl A. Now, if x~ ~ clA then there exists a fuzzy open q-nbd U of so, such that U ¢1A and thus U ~1 cl A. But then U q int cl A, which means int ci U ¢1 cl(int cl A), and therefore int cl U ¢1A. So x~ ~ [A]6. Hence we have proved [A],~ <~ cl A.

Theorem 9. The following are equivalent for an fts (X, T).

(a) X is FED. (b) The closure of every fuzzy semi-pre-open

set in X is fuzzy open. (c) The 6-closure of every fuzzy semi-pre-

open set in S is fuzzy open. (d) The &closure of every fuzzy pre-open set

in X is fuzzy open. (e) The O-closure of every fuzzy pre-open set

in X is fuzzy open. (f) The closure of every fuzzy pre-open set in

X is fuzzy open.

Definition 6, A fuzzy set A in X is said to be fuzzy semi-pre-open iff A ~< cl(int cl A).

We shall use the notations FPO(X) and FSPO(X) to represent the sets of all fuzzy pre-open sets and fuzzy semi-pre-open sets in X respectively. Obviously, FO(X) c FPO(X) c FSPO(X), although the reverse inclusions are false.

For example, on a non-empty set X, consider the fuzzy sets A, B, C, D given by A ( x ) = J , B(x) = 0.6, C(x) = 0.2 and D(x) = 0.7, for all x • X . Then for the fts (X ,T) , where T = {0x, Ix, A , B } , we have C • F P O ( X ) , C ~ T , D • FSPO(X) and D ~ FPO(X).

Lemma 7. For any A • FPO(X), cl A = [A]~ = [A]o.

Proof. Follows from Lemma 7 and 8 by using the facts that F P O ( X ) c FSPO(X), and clA = cl(int cl A), for every A • FSPO(X).

Analogous to a lemma in [6] we now have the following.

Lemma 10. For any fuzzy set A in X, (a) int cl A <~ scl A and (b) int(scl A) = int cl A.

Proof. (a )Since sclA is fuzzy semi-closed, there exists a fuzzy closed set U in X such that intU~<sclA<~U. Then i n t U ~ < s c l A ~ c l A ~ < U and consequently, int U <~ int cl A ~< int U. Hence int cl A ~< scl A.

(b) Follows easily using (a).

Proof. It is obvious that cl A ~< [A]6 ~< [A]o, for every fuzzy set A in X. Thus it remains to show that [A]o<~clA. Now, if x ~ c l A then there exists a fuzzy open q-nbd U of x~ such that U ¢1 A and thus U ¢1 cl A. But then U q int cl A, which means c lUg l in t c lA , and so c lUglA (since A • FPO(X)). Hence x , ~ [A]o. Thus we have proved [A]o <<- cl A.

In a way similar to the Proposition 2.7 of [6] we now obtain:

Proposition 11. Let A be any fuzzy set in X. Then

(a) A • FPO(X) iff scl A = int cl A, (b) A • FPO(X) iff scl A • FRO( S) , (c) FRO(X) = FPO(X) n FSC(X).


Proof. (a) Let A e F P O ( X ) , then sclA~< scl(int cl A), and since int cl A ~ FSC(X), scl A ~< intclA. From Lemma 10(a) it follows that scl A = int cl A. The converse is obvious.

(b) Let scl A ~ FRO(X). Then scl A = int cl(scl A) and hence scl A ~< int cl(cl A) = intclA. By Lemma 10(a) it follows that sclA = int clA. By (a), A ~ FPO(X). The converse follows from (a).

(c) From A ~ FPO(X) fq FSC(X) it follows that A e FRO(X) (by (b)) and hence FRO(X) N FSC(X) c FRO(X). Again, A ~ FRO(X) implies i n t c l A = A and hence i n t c l A = s c l A = A , which means A ~ FPO(X) fq FSC(X) (by (a)). Thus FRO(X) = FPO(X) fq FSC(X).

Theorem 12. The following are equivalent for an fts X :

(a) X is FED. (b) sc lA=[A]o , for every A ~ F P O ( X ) U

FSO(X). (c) scl A = cl A, for every A ~ FSPO(X). (d) scl A = [A]o, for every A ~ FSPO(X).

Proof. ( a ) ~ ( b ) . For any fuzzy set A in X, sclA ~< [A]o. Thus it is only required to prove [A]o <~ scl A, for every A ~ FPO(X) t_J FSO(X). In fact, if x ~ , s c l A then there exists a fuzzy semi-open semi-q-nbd U of x , such that Uq A. But then there exists a fuzzy open set V in X such that V~<U~<cIV with V¢IA and hence VclclA. This means Vcl in tc lA and so cl VC] int clA. Now, if A ~ FPO(X), then A ~< int cl A and hence cl V ~ A. If A ~ FSO(X), since X is FED, cl V is fuzzy open, and thus

cl V ¢1 cl(int cl A)/> cl int A i> A,

so that cl V ¢t A. Thus in any case, x~ $ [A]o. (b) ~ (a): First let A ~ FPO(X). By Proposi-

tion 11 and Lemma 7 we have

int cl A = scl A = [A]o = cl A.

Therefore, clA is fuzzy open, and hence it follows from Theorem 9 that X is FED. Next, let A ~ FSO(X). We have

sclA ~<clA ~< [A]o = sclA

and hence scl A = cl A. Therefore it follows from Theorem 5 ((f) ::> (a)) that X is FED.

( a ) ~ ( c ) : It follows from Lemma 10 that for every fuzzy set A in X, int cl A ~< scl A ~< cl A.

Since X is FED, by Theorem 9, cl A is fuzzy open in X for every A e FSPO(X). Thus we have scl A = cl A, for every A e FSPO(X).

(c) ::> (d): Follows from Lemma 8. (d) ~ (a). Let U and V be fuzzy open such

that UcIV. Then U ~ < I - V implies sclU~< scl(1 - V) = 1 - V and hence scl U ¢1 V. Since scl U e FSO(X), scl U ¢( scl V. By Lemma 8 we obtain that cl Uq cl V, since FO(X) c FSPO(X). This shows that X is FED, by Theorem 4.

Theorem 13. ftsX.

(a) X is FED. (b) a 6 FSPO(X),

el A ¢1 cl B. (c) a ~ FSPO(X),

[A],~ ¢1 [B],~. (d) A ~ FPO(X),

[A]o ¢t [Bla. (e) A e FPO(X),

clA ~ cl B.

The following are equivalent for an

B e FSO(X) and A ~t B

B ~ FSO(X) and A ~ B

B e FSO(X) and A ~1 B

B e FSO(X) and A ~t B

Proof. (a) ~ (b): Suppose A e FSPO(X), B e FSO(X) and A ¢1 B. Then A ¢1 int B and hence cl A ¢1 int B. By Theorem 9, cl A is fuzzy open in X and hence cl A q cl int B. Since B e FSO(X), cl B = cl(int B). Thus cl A ¢1 cl B.

(b) f f (c), (c) f f (d) and (d) ~ (e) follow from Lemmas 7 and 8.

( e ) ~ ( a ) : Follows from Theorem 4, since every fuzzy open set is fuzzy pre-open and fuzzy semi-open.

Definition 14. A collection ~ of non-null fuzzy sets in an fts X is called a fuzzy filterbase in X iff every finite intersection of members of contains a member of ~ .

Herrmann [5] proved some nice characterizations of extremally disconnected space in terms of rc-convergence of filterbases. In order to extend them to the fuzzy setting we set the following definitions.

Definition 15. A fuzzy filterbase ~ in X is said to

(i) converge (b-converge, 0-converge) to a fuzzy point x~ iff for each fuzzy open q-nbd U of x , , there exist an F e ~3 such that F ~< U (resp. F~<int cl U, F~<cl U),

B. Ghosh / Fuzzy extremally disconnected spaces

(ii) rc-converge to a fuzzy point x~ iff for each fuzzy open set U with x , q cl U, there is F • such that F ~< cl U.

Lemma 16. A fuzzy set A in an fts X is fuzzy open iff it is a q-nbd of every fuzzy point with which it is q-coincident.

Proof. If A is fuzzy open then the condition is obviously satisfied. Conversely, for any fuzzy point x~ with xo~ q A, there exists by hypothesis, a fuzzy open set Ux, such that x~ q Ux~ <~ A. Let

U=t._J{Ux :xo~qA}.

Then U is fuzzy open and U ~< A. We show that A <~ U. Let x • suppA, where suppA denotes the support of A, and put A(x)-- tr . Choose m • N (--the set of all naturals) such that 1/m<~tr. For any n • N with n />m, we set o~n = 1 - o~+ ( l /n) . Then 0 < c~n ~< 1 and x~,.qA, and hence there is a fuzzy open set Ux.. such that x~,. q Ux~. <~ A, for all n/> m. This shows that

1 1 - c ~ + - + U~.(x) > 1,

n

that is

1 (x) + - .

an n

Then

O{ ~ (rtUm Uxa.)(x ) ~ U(x )"

Thus A(x)<<- U(x). Since x • suppA is arbitrary, we have A ~ < U. Hence A = U, and consequently, A is fuzzy open.

Theorem 17. For an fts (X, T), the following are equivalent.

(a) X is FED. (b) I f a fuzzy filterbase on X b-converges then

it rc-converges. (c) A fuzzy filterbase on X rc-converges iff it

O-converges. (d) If a fuzzy filterbase on X converges then it

rc-converges.

Proof. ( b ) ~ ( a ) : Let G • T and let x~ be any fuzzy point in X such that xo~ q cl G. By virtue of Lemma 16, it is sufficient to show that cl G is a q-nbd of xo~. We consider the collection ~ of all

249

fuzzy open q-nbds of x~. Obviously it is a fuzzy filterbase and b-converges to x, . By hypothesis, it rc-converges to x~. Since x~ q cl G and G • T, there exists U ~ ~ such that U ~< cl G. Thus cl G is a q-nbd of xo~.

(a) f f (b): Let ~ be any fuzzy filterbase which b-converges to a fuzzy point xo~. Let x~ q cl G, where G ~ T. Then cl G e T and consequently there is F e ~ such that F ~ < i n t c l ( c l G ) = int cl G = cl G. Hence ~ is rc-convergent to x~.

(a) ~ (c): To prove the requirement it suffices to prove that a fuzzy filterbase ~ is rc- convergent to x~ whenever it is 0 convergent to x~. Let x , q cl U, where U • T. Since X is FED, cl U • T. Then cl U is a fuzzy open q-nbd of x, . So there exists F • ~ such that F ~< cl(cl U ) = cl U. Hence ~ is rc-convergent to x~.

(c) ~ (d): Obvious. (d) ~ (a): Let G ~ T and x~ be any fuzzy point

in X such that x~ q cl G. In view of Lemma 16, we only show that cl G is a q-nbd of x~. Now, consider the collection ~ of all fuzzy open q-nbds of x~. Obviously it is a fuzzy filterbase converging to x, . Then it rc-converges to x~. Hence there exists F e ~ such that x , q F ~< cl G. Thus cl G is a q-nbd of x, .

Theorem 18. An fts (X, T) is FED iff FSO(X) c FPO(X).

Proof. Suppose X is FED, and A • FSO(X). Then there exists U • T such that U ~< A ~< cl U. Since X is FED, cl U • T. Then U ~< A ~< int cl U and hence A • F P O ( X ) . Conversely, let A • FRC(X). Then A • FSO(X), and by hypothesis, A • F P O ( X ) so that A - i n t c l A . Since A is fuzzy closed, it follows that A • T. Hence X is FED.

In order to investigate for the suitable function under which fuzzy extremally-disconnectedness is preserved, we recall the following definitions.

Definition 19. A mapping f : ( X , T)--~ (Y, T~) is said to be

(a) fuzzy semi-continuous [1] iff f - l ( U ) • FSO(X) for any U • T1,

(b) fuzzy irresolute [9] iff f - I ( U ) • F S O ( X ) for any U • FSO(Y),

(c) fuzzy almost open [12] i f f f (V) • T1 for any V • FRO(X).


From the definitions it follows that a fuzzy irresolute function is fuzzy semi-continuous but the converse is false (see [9] for coun- terexample), while the converse is true if the function is, in addition, fuzzy almost open [9]. It is also known [4] that a mapping f : (X , T)---> (Y, T1) is fuzzy semi-continuous iff for any fuzzy set A in X, f(sclA)<~clf(A). For a detailed study of the above types of functions including their different characterizations and interrela- tions we refer to [4, 9, 12].

Lemma 20. If f : (X, T)---> (Y, T~) is fuzzy almost open and fuzzy semi-continuous, then f ( A ) • FPO(Y), for every A • FPO(X).

Proof. Let A • F P O ( X ) . Since f is fuzzy semi-continuous, f(A)~<f(sclA)~<clf(A). By Proposition l l(b), sclA • FRO(X), and hence f(scl A) • FPO(Y), because f is fuzzy almost open. By Proposition l l(a) , scl(f(sclA))= int cl(f(sclA)). Thus sclf(A) ~< scl(f(sclA)) = int cl(f(scl A)) ~< clf(A). Since int cif(A) = intclf(sclA), we can conclude that f(A)<~ sclf(A) ~ in t clf(A), and consequently, f (A) • FPO(Y).

We conclude with a fuzzy analogue of a proposition in [6].

Theorem 21. Let f :(X, T)---~(Y, T1) be a fuzzy semi-continuous and fuzzy almost open surjec- tion. If X is FED then so is Y.

Proof. Let V•FSO(Y) . Since f is fuzzy semi-continuous and fuzzy almost open, f is fuzzy irresolute so that f - t ( v ) • F S O ( X ) . By Theorem 18, f - l (V) •FPO(X), and hence by Lemma 20, V •FPO(Y) . Thus FSO(Y) c FPO(Y). Hence by Theorem 18, Y is FED.

Acknowledgement

I like to express my gratitude to Dr. M.N. Mukherjee of the Department of Pure Mathe- matics, Calcutta University, for his kind help and active supervision throughout the prepara- tion of this work.

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[5] R.A. Herrmann, RC-convergence, Proc. Amer. Math. Soc. 75(2) (1979) 311-317.

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