On several types of compactness in smooth topological spaces

sets and systems ELSEVIER

Fuzzy Sets and Systems

90 (1997) 83-88

On several types of compactness in smooth topological spaces Mustafa Demirci Department

of Mathematics,

Faculty of Sciences and Arts, Abant izzet Baysal University, 14280-Bolu, Turkey Received

June 1995;

revised April 1996

Abstract The smooth closure and smooth interior of a fuzzy set w.r.t. a smooth topology were defined by Gayyar et al. (1994), and some relations between a few types of compactness were established in the presence of strong restrictions. In this paper, by constructing new definitions of smooth closure and smooth interior which have more desirable properties than those of Gayyar et al. (1994), we prove that several hypothesis in the results of Gayyar et al. (1994) can be weakened and show that the relations which hold between various types of compactness in fuzzy topological spaces in Chang’s sense (CFTS for short) (Di Concilio and Gerla, 1984; Haydar Es, 1987) can be extended to smooth topological spaces. 0 1997 Published by Elsevier Science B.V. Keywords:

Fuzzy sets; Topology; Smooth almost compactness; Smooth near compactness

1. Introduction

In 1985, Sostak [9] defined a fuzzy topology on a set X as a mapping r :I’ -+ I satisfying some natural axioms, where 1’ denotes the family of all fuzzy subsets of X, and presented the fundamental concepts of such fuzzy topological spaces. In 1992, the same structure was rediscovered by Chattopadyay et al. [2]. They call the mapping z : Ix -+ I “a gradation of openness on x”. In the same year, Ramadan [S] gave a similar definition of a fuzzy topology in Sostak’s sense under the name of “smooth topological spaces” (s.t.s.), replacing I = [0, l] by possibly more general lattices. Degrizations of compactness and the types of compactness were introduced and studied in [6, lo]. A different approach for the compactness types was taken up in [4], however the results obtained include additional conditions since the 0165-0114/97/$17.00 a 1997 Published PII SO165-01 14(96)00121-2

by Elsevier

smooth closure and smooth interior defined there do not have such nice properties as the closure and interior operators in a CFTS [7]. In this paper, we give a new definition of smooth closure and smooth interior of a fuzzy set in a s.t.s. which have almost all the properties of the corresponding operators in a CFTS. As a consequence of these definitions we reduce the additional hypotheses in the results of [4] and generalize several properties of the compactness’s types in [3,5] for s.t.s.

2. Preliminaries Smooth topology and smooth topological concepts were introduced in [S] in terms of lattices L and L’, both of which were taken to be I = [0,11. For simplicity,

Science B.V. All rights reserved

we use only the symbol

I instead

of

M. Demirci / Fuzzy Sets and Systems 90 (1997) 83-88

84

L and L’, and Ox, lx will denote the characteristic functions of the crisp sets 8 and X, respectively. A s.t.s. [8,9] is an ordered pair (X, r), where X is a non-empty set and r : Ix -+ I is a mapping satisfying the following conditions: (01) 7(0x) = r(lx) = 1. (02) VA, B E Ix, z(AnB) > z(A) A z(B). (03) For every subfamily {Ai: i E J} E Ix, Then the mapping r : Ix + I is called a “smooth topology” (s.t. for short) on X. The number z(A) is called “the degree of openness of A” [9]. If the s.t. r on X has the following fourth property. (04) r(ZX) 5 (0, l} then z corresponds in a one to one way to a fuzzy topology in Chang’s sense [l, lo] (CFT for short). A mapping z* :I’ + Z is called a “smooth cotopology” [4] iff the following three conditions are satisfied: (Cl) z* (0,) = z* (lx) = 1. (C2) VA, B E Ix, z* (AuB) 2 z*(A) A (B). (C3) For every subfamily {Ai: i E J} c Ix, z*(f7isJAJ

2 Ai.Jr*(Ai).

If z is a s.t. on X, then the mapping z* : Ix -+ I, defined by z*(A) = $A”) where A” denotes the complement of A, is a smooth cotopology. Conversely, if z* is a smooth cotopology on X, then the mapping z: Ix + I, defined by z(A) = z* (A”), is a s.t. on X [2,8,9]. In this paper, for the s.t.s. (X, r), the notation r* will be used in the sense z*(A) = z(A’). The number z* (A) is called “the degree of closedness of A” [9]. For the s.t.s. (X, z) and a E [O, 11, the family r, = {A E Ix: z(A) 2 x} defines a CFT on X, called the “cc-level CFT” [2]. The family of all closed fuzzy sets w.r.t. z, is denoted by z,*, we obviously have r,* = {A E Ix : z*(A) Z cc} [6, lo]. For A E Ix and CIE [IO,11, the closure (resp., the interior) of A w.r.t. rb, denoted by cl,(A) (resp., int,(A)), is called the z,-closure (resp., z,-interior) of A [4,6]. In this paper, we will denote cl,(A) (resp., int,(A)) by A, (resp., A,“) for simplicity. For each c1E [0, l] and A E Ix, it is clear that A=,= ~{KEz,*:

Let (X, z) and (Y, a) be two smooth topological spaces. A functionf: X --f Y is called “smooth continuous” w.r.t. z and (T iff z(f-‘(A)) > a(A) for every A E Zy [8,9]. A function f:X -+ Y is called “weakly smooth continuous” w.r.t. z and r~iff [S] a(U) > 0 *

z(f-l(U))

> 0

for every U E I’.

Furthermore, one can easily see that smooth continuity implies weak smooth continuity. However the converse implication is not true in general [8], Letf: (X, r) + (Y, a) be a function. Then [4,8]: (i) f is smooth continuous iff z* (f-‘(A)) 2 o* (A) for every A E I’. (ii) f is weakly smooth continuous iff g*(A) > 0 = z* (f-’ (A)) > 0 for every A E I’. A function f:X -+ Y is called “smooth open (resp., smooth closed)” w.r.t. smooth topologies z and 0 on X and Y, respectively iff [4, 81 r(A) d a(f (4) (rev., z*(A)Q c* (f (A))) for each A E Ix.

3. Definitions and properties of closure and interior Definition 3.1. Let (X, z) be a s.t.s., and A E Ix. Then the z-smooth closure (resp., z-smooth interior) of A, denoted by A (resp., A”), is defined by A = r){K E Ix: r*(K) > 0, A c K} (resp., A” = U{K E Ix: z(K) > 0, K E A}).

Proposition 3.1. Let (X, z) be a s.t.s., and A, B E Ix. Then

(i) (ii) (iii) (iv)

A~B=~-A”GB”, A G B * A E B, (A”)” = (A”),

A” = ((A”))“, (v) ?i = ((A’)“)“, (vi) (A)” = (A”)“.

Proof. (i) and (ii) follow directly from Definition 3.1. (iii) From Definition 3.1. we observe that

ASK}

and /t,o = u{G E z,: G c A).

(A”)” = (u {K E Ix, z(K) > 0, K E A})’ = n{K” : K E Ix, z(K) = z* (Kc) > 0, AC c Kc}

85

A4 Demirci / Fuzzy Sets and Systems 90 (1997) 83-88

AC~U)

=n{~dx9(~)>0,

=A”. (iv), (v) and (vi) can be easily derived from (iii).

0

Remark 3.1. Let r be a CFT on the non-empty set X. Then the smooth topology and smooth cotopology z,, z$ : Ix -+ I, defined by r,(A) =

Proposition 3.3. Let (X, t) be a s.t.s. and A, B E Ix. Then

(i) G = Ox, (ii) -A 5 A,

1 if A E z, 0 if A#z

(iii) (A) = A, (iv) AuB E AuB.

and z,*(A) = z(A”)

Thus A“ = (A”)” directly follows these two inclusions. (iv) Since An B G A, An B c B and using Proposition 3.1 (i) we obtain (An B)” c A” and !J (AnB)” c B”, i.e., (AnB)” c A”nB”.

for each A E Ix, Proof. Similar to Proposition

identify the CFT z and corresponding fuzzy cotopology for it, respectively. Thus, the z,-smooth closure and r,-smooth interior of A are

3.2.

c]

Proposition 3.4. Let (X, z) be a .s.t.s., and A E Ix. Then

A= r)(KEIX:

z,*(K)>O,AGK}

=r){KEZX:KCE~,

(i) z(A) > 0 =S A = A”, (ii) z*(A)>0 =z-A=A, (iii) if there exists a /I E (0, l] such that A = A0 then A = A, (iv) if there exists a /I E (0, l] such that A = Ai then A = A”.

A&K}

and A”=U{K~IX:z,(K)>O,KcA} =U{KEI~:KE~,

KGA}.

This shows that A and A” are exactly the closure and interior of A w.r.t. r in Chang’s sense [7], respectively. Proposition 3.2. Let z be a s.t. on X. Then (i) 1; = lx, (ii) A” c A, (iii) (A”)” = A”, (iv) (A n B)” c A” n B”.

Proof. (i) Let z(A) > 0. Then A E (K E Ix: z(K) > 0, K E A}, and considering Proposition 3.l(ii) we obtain A c A”. Hence A = A” follows at once from Proposition 3.2(ii). z* (A) > 0, i.e., $A”) = z*(A) > 0. (ii) Let From (i) we have A’ = (A”)“, i.e., A = ((A”)“)‘. Therefore A = A is obtained directly from Proposition 3.1(v).

Proof. (i) and (ii) can be easily obtained from Definition 3.1. (iii) From (ii) we have (A”)” G A”. By Definition 3.1, we may write

(iii) Suppose that there exists a p E (0, 11 such that A = A,. Now, considering Definition 3.1 we observe that, for each x E X,

(A”)“=~{KEZ~:~(K)>O,

A(x) = Inf {K(x) : K E Ix, z*(K) > 0, A E K}

KEA’}

=U{KEZ~:~(K)>O,

= Inf

KGU(UEI~:Z(U)>O, zU{KEI~:~(K)>O, = A”.

USA}} KcA}

Inf {K(x): K E Ix,

as(O,11

86

M. Demirci / Fuzzy Sets and Systems 90 (1997) 83--88

= _:fi,{Inf

Therefore,

{K(x): K E Ix,

f

r*(K)~c&4GK}}

-’(f(A)) = n{f-l(u)Ezx:

= IEfi, &(x)1?

UEZY, a*(u)>o,

Acf-‘(U)}

i.e., A = nn.CO,l,Aa. Thus, using the hypothesis and Proposition 3.3(ii) it is obtained that A c A = narsco,i,Aa: c A, = A, i.e., A = A. (iv) For a fl E (0, 11, let A = A;. Then, A”=U{KEZ~:~(K)>O,

KGU{UEZ~:Z(U)>/?,

GA}.

Since z(Ai) > A {z(U): U E Ix, r(U) 3 /I, U c A} 3B>0,wehaveA~E{KEZX:z(K)>0,KcA~}, i.e., Ai E A”. Thus, considering Proposition 3.2(ii) and the hypothesis we get A = Ai c A” E A, i.e., A = A”. Proposition 3.5. Let (X, z) and (Y, CT)be two smooth topological spaces. Zf the function f: X + Y is weakly smooth continuous w.r.t. z and o, then we have (i) for every A E ZX,f(A) sf(A), (ii) for every B E ZY,f-l (B) Gf-’ (B), (iii) for every B E I’, f- ’(B”) E (f- l (B))“. Proof. Letf: X + Y be a weakly smooth continuous function w.r.t. z and (T. (i) For A E Ix, considering Definition 3.1 we may write f -’ (f(A)) o* (U) > 0, f(A) c U})

= ncf-l(U)Ez5

U~ZY,

z*(f-‘(U))>O, 2 n{K

Acf-‘(U)}

E Ix: z*(K) > 0, A E K)

= A.

KEA;}

=U{KEzX:z(K)>o,

=f-‘(n{UEzY:

2 n{f-l(U)Ezx:

u~zY,~*(u)>o,

Hence f(A) E f(A). (ii) For B E I’, we get from (i) f (f -l(B)) E f(f_‘(B)) c B * ~ (f-‘(B)sf-‘(f(f-7B)))c~ f- ’(B), i.e., f - ’(B) C f - ’(B). (iii) For B E I’, using (ii) and Proposition 3.l(iii), we observe that f-‘(B”)=f-‘((B”)“)=[f-‘(B”)]“?f-’(B”) =(f-‘(B))“=[(f-‘(B))“]“. Hence f - ’(B”) E [f - ’(B)]” follows.

0

Corollary 3.1. Let f: (X, z) + (Y, o) be a smooth continuousfunction w.r.t. the smooth topologies z and o. Then (i) for every A E Ix, f (A) c f(A), (ii) for every A E I’, f - ’(A) c f - ’(A), (iii) for every A E I’, f _ ’(A”) c [f - ’(A)]“. Proposition 3.6. Let (X, z) and (Y, a) be two smooth topological spaces and A E Ix. Zf a function f :X + Y is smooth open w.r.t. z and o then f(A”) E [f(A)]“.

On the other hand, since f is weakly smooth continuous, we have

Proof. Let f:X + Y be a smooth open function w.r.t. the smooth topologies z and 0 on X and Y, respectively. For A E Ix, considering Definition 3.1 we may write

{f_‘(U)

f(A”)

A sf -l(U)}.

E IX: u E zy, o*(U)>O,

s{f-‘(U)Ez5 z*(f-‘(U))>O,

UEZY, Acf-l(U)}.

Acf-‘(U)j

= f(U{K

E Ix: z(K) > 0, K c A})

E U{~(K)EZ~:KEZ~,

z(K)>O,f(K)sf(A)}.

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M. Demirci J Fuzzy Sets and Systems 90 (1997) 83-88

88

Corollary 4.3 (Gayyar et al. [4]). A smooth nearZy

that lx = lJieJo f-’ (Ai). NOW, using the surjectivity off and Proposition 3.5(i) we find that

compact

f(lX) = lY =f

References

u f-‘(A’) bJo

= U

:)

f(f-l(Ai)) ___

isJo

smooth regular s.t.s. is smooth compact.

Corollary 4.2. Let f: (X, z) + (Y, a) be a surjective smooth continuousfunction. Then, thefollowing hold: (i) If the s.t.s. (X, z) is smooth almost compact then so is (Y, a). (ii) If the s.t.s. (X, z) is smooth nearly compact then the s.t.s. (Y, C) is smooth almost compact.

[l] C.L. Chang, Fuzzy topological spaces, J. Math. Anal. Appl. 24 (1968) 182-190. [Z] KC. Chattopadyay, R.N. Hazra and S.K. &manta, Gradation of openness: fuzzy topology, Fuzzy Sets and Systems 49 (1992) 237-242. [3] A. Di Concilio and G. Gerla, Almost compactness in fuzzy topological spaces, Fuzzy Sets and Systems 13 (1984) 187-192. [4] M.K. El Gayyar, E.E. Kerre and A.A. Ramadan, Almost compactness and near compactness in smooth topological spaces, Fuzzy Sets and Systems 62 (1994) 193-202. [S] A. Haydar Es, Almost compactness and near compactness in fuzzy topological spaces, Fuzzy Sets and Systems 22 (1987) 289-295. [6] A. Haydar Es and Dogan Coker, On several types of degrees of fuzzy compactness in fuzzy topological spaces in Sostak’s sense, J.fuzzy Math. 3 (1995) 481491. [7] Pu Pao-Ming and Liu Ying-Ming, Fuzzy topology I. Neighborhood structure of a fuzzy point, J. Math. Anal. Appl. 76 (1980) 571-599. [8] A.A. Ramadan, Smooth topological spaces, Fuzzy Sets and Systems 48 (1992) 371-375. [9] A.P. Sostak, On a fuzzy topological structure, Rend. Circ.

et al. [4]). A smooth

[lo] A.P. Sostak, On compactness and connectedness degrees of fuzzy sets in fuzzy topological spaces, in: General Topol-

f(f-’

tAilI

0

UA,

=

isJo

i.e., 1~ = uie

Jo%.

Corollary 4.1. Let (X, z) and (Y, o) be two smooth topological spaces, and f: X + Y a surjective weakly smooth continuous function w.r.t. z and a. If the s.t.s. (X, z) is smooth nearly compact then the s.t.s. (Y, a) is smooth almost compact.

Matem. Palermo (2) Suppl. 11 (1985) 89-103.

Proposition 4.3 (Gayyar almost compact compact.

smooth

regular

s.t.s.

is smooth

ogy and its Relations to Modern Analysis and Algebra

(Heldermann, Berlin, 1988) 519-532.