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An extension in fuzzy topological spaces by Qutaiba Ead Hassan Departamento de Geometria y Topologia Facultad de Matematicas, Universidad Complutense de Madrid 28040-Madrid .Spain e-mail: [email protected] Abstract In this paper we de…ne a fuzzy compact covering map and prove that a fuzzy continuous closed map from an induced fuzzy paracompact in to an arbitrary induced fuzzy topological space is fuzzy covering compact.We also de…ne the fuzzy countable compact by using the open cover in the sense of Lowen and it is proved that it is good extension of countable compact, also it is shown that induced fuzzy topological space is fuzzy countable compact and fuzzy paracompact if and only if it is fuzzy compact in the sense of Lowen. Morever we investigate di¤erent spaces such as fuzzy compactness, Lindelöf space, separable space, non-archimedeam fuzzy topological space and some new properties of these concepts are obtained. Keywords: fuzzy compactness, S-paracompactness, fuzzy paracompactness, fuzzy Lindelöf space, separable space, fuzzy topological space. Mathematics Subject Classi…cation (2000): 54A40, 03E72. 1 1 Introduction An important concepts in general topology are compactness, countable compact, covering compact map, paracompactness, CII , Lindelöf space, separable space and others see [5, 14, 15]. In 1965 Zada [19] introduced the theory of fuzzy sets and in 1968 Chang [3] de…ned the fuzzy topology.Several authors have tried to extend to fuzzy set theory the main notations of general topology see [1, 2, 4, 6-13], [16-18] and others. In this paper we de…ne for fuzzy topological space a notation corresponding to compact covering map studied by E. Michael [14, 15] and prove that a fuzzy continuous closed map from an induced fuzzy paracompact in to an arbitrary induced fuzzy topological space is fuzzy covering compact, We also de…ne the fuzzy countable compact by using the …nite open cover in the sense of Lowen [9] and it is proved that it is good extension of countable compact also it is shown that induced fuzzy topological space ( X, !(¿ )) is fuzzy countable compact and fuzzy paracompact if and only if it is fuzzy compact in the sense of Lowen. In addition we investigate di¤erent spaces such as, fuzzy Lindelöf space [1], fuzzy separable space [18], fuzzy paracompactness [1], S-paracompactness, ®Lindelöf [10], non-archimedeam fuzzy topological space [12] and some properties of these concepts are obtained such as: (1) Every locally family of fuzzy subsets of fuzzy ®-Lindelöf is countable.(2) fuzzy paracompact separable topological space is Lindelöf fuzzy.(3) every subspace of CII fuzzy topological space (X,T) is separable.(4) every fuzzy compact in the sense of Lowen non-archimedean fuzzy topological space is metrizable.(5) The continuous image of closed subset of N-compact space is *-fuzzy paracompact, (S*-paracompact). 2 Basic de…nition The following de…nitions have been used to obtain the results and properties developed in this paper. De…nition 1 [10] Let ® 2 (0; 1) and ¹ be a set in a fuzzy topological space (X; T ). The set ¹ is said to be ®-paracompact (resp.®¤ -paracompact) if for every ®¡open Q¡cover of ¹ has an open re…nement of it which is both locally …nite (*-locally …nite) in ¹ and an ® ¡ Q¡cover of ¹: ¹ is called Sparacompact (S¤ -paracompact) if for every ® 2 (0; 1]; ¹ is ®-paracompact (resp.®¤ -paracompact). 2 De…nition 2 [1] Let ¹ be a fuzzy set in a fuzzy topological space (X; T ) : ¹ is said to be fuzzy paracompact (resp.*-fuzzy paracompact) if for every open cover in the sense of Lowen U of ¹ and for every " 2 (0; 1], there exists an open re…nement V of U which is both locally …nite (resp.*-locally …nite) in ¹ and cover of ¹ ¡ " in the sense of Lowen. De…nition 3 [8] Let (X; T ) be a fuzzy topological space. The set [T ] = fA ½ X : ÂA 2 T g is called the original topology of X such that ÂA is the characteristic function of A, and the crisp topological space (X, [T ]) is called original topological space of (X; T ): De…nition 4 [13] A fuzzy topological space(X; T ) is called a week induction of the topological space (X; T0 ) if [T ] = T0 and each element of T is lower semi-continuous from (X; T0 ) to [0; 1] : De…nition 5 [8] Let(X; ¿ ) be a topological space and ! (¿ ) be the set of all semicontinuous function from (X; ¿ ) to the unit interval I = [0; 1] equipped with the usual topology, then (X; ! (¿ )) is called induced fuzzy topological space by (X; ¿ ). De…nition 6 [8] A fuzzy extension of a topological property is said to be good when it is possessed by (X; ! (¿ )) if and only if, the original property is possessed by (X; ¿ ) : De…nition 7 [6] A fuzzy set A in a fuzzy topological space (X; T ) is said to be N -compact if each ®- net contained in A has at least in A a cluster point with value ® such that ® 2 (0; 1]: De…nition 8 [10] Let ® 2 (0; 1]. A set A in a fuzzy topological space (X; T ) is said to be ®-Lindelöf if every ®-open Q-cover of A has a countable subfamily which is a Q-cover of A also. We say that A is S-Lindelöf if A is ®- Lindelöf for every ® 2 (0; 1]; then (X; T ) is called ®-Lindelöf (resp. S-Lindelöf) if a set X is ®-Lindelöf (resp. S-Lindelöf). De…nition 9 [1] A fuzzy topological space (X; T ) is said to be fuzzy Lindelöf, if for each family B ½ T and for each ® 2 I such that _BB2B ¸ ®, there exists for each " 2 (0; ®] a countable subset B 0 of B such that _B ¸ ® ¡ ": B2B0 De…nition 10 [18] An fuzzy topological space (X,T) is said to be separable i¤ there exists a countable sequence of fuzzy points fpi ;i=1;2;:: gsuch that for every member ¹ 6= 0 of T there exist a pi such that pi 2 ¹: 3 De…nition 11 [9]A fuzzy set ¹ in (X; T ) is fuzzy compact in the Lowen’s sense if for all family of fuzzy open sets cover { ¹j j j 2 J g such that ¹ · _f ¹j j j 2 J g and for all " > 0 there exist a …nite subfamily { ¹j j j 2 J0 g such that ¹ ¡ " · _ { ¹j j j 2 J0 g: De…nition 12 [18] A fuzzy topological space (X; T ) is said to be CII , if there exists a countable base for T: De…nition 13 En a fuzzy topological space (X,T), ¹ is fuzzy countably compact if every countable open cover { ¹i j i = 1; 2; 3; ::::g: of ¹ and for each " > 0 there exist a …nite subcover { ¹i j i = 1; 2; 3; ::::; i0 g such that ¹ ¡ " · _ { ¹i j i = 1; 2; 3; ::; i0 g 3 Fuzzy Lindelöf Spaces In this section we study two kinds of Lindelöf space (®-Lindelöf introduced by [10] and fuzzy Lindelöf introduced by [1] ) which are good extension of Lindelöf topological space. We get some additional properties for these spaces corresponding to the same results obtained in general topology. In addition to that we give relations between these concepts and other spaces such as fuzzy paracompact, fuzzy separable, ®-paracompact, and C11 : 3.1 Theorem Every locally …nite family of non empty fuzzy subsets A of ®-Lindelöf space (X; T ) is countable. Proof. Let ® 2 (0; 1] and let A be a locally …nite family of fuzzy subsets of X. Let ¯ =minf®; 1 ¡ ®g ; then ¯ 2 (0; 1) :Take Ux 2 Q (x¯ ) such that A is quasi-coincident with Ux ( A qUx for short ) for at most …nitely many members At of A: Let Ux¤ = (Ux )1¡ ® ^ Ux : Then U = fUx¤ : x 2 Xg is Q-open-cover of 1® and has a countable Q-open-subcover U ¤ of 1® : Since every member of A is quasi-coincident with u 2 U ¤ : There forj A j· @0 (aleph zero) such that @0 is the cardinal number of the set of all integers. 3.2 Theorem Let (X; T ) be an ®-paracompact regular fuzzy topological space such that for each ®¡open Q¡cover fui gi2I of X, _ ui ½ X: If X contains a dense i2I subspace A which has ®-Lindelöf property then X is ®-Lindelöf. 4 Proof. Let U = fui gi2I be an open Q¡cover of 1® : By [10, Theorem (2.32) ] there exists a locally …nite V = fvi gi2I which is open Q¡cover of 1® such that vi < ui for every i 2 I .The set I0 = fi 2 I : Aqvi g is countable by the above theorem. Since A = _i2Io fA ^ vi g then A = _i2Io fA ^ vi g and A (x) = sup fA ^ vi g (x) < sup fvi (x)g < sup fui (x)g = 1 Thus X = A has i2Io i2Io i2Io a countable Q¡cover and X is ®¡Lindelöf: 3.3 Theorem Let f be a F-continuous surjection map between fuzzy topological spaces (X; T ) and (Y; S). If (X; T ) is fuzzy Lindelöf then (Y; S) is veri…es the same property. Proof. Let ® be a constant fuzzy set in Y. Let B ½ S such that _ B ¸ ® B2B and let " 2 (0; ®], since f is F-continuous, then U = ff ¡1 (B) : B 2 Bg is an L-cover of the constant fuzzy set ± in X such that f (± ) = ®. Since X is Lindelöf, then there exists an open countable subset U0 of U such that _u2U0 u ¸ ± ¡ "; then B0 = ff (u) : u 2 U0 g is an open countable subfamily of B such that _B2B0 B ¸ ® ¡ " There for (Y; S) is fuzzy Lindelöf. 3.4 Corollary Let f be a F- continuous surjection map between weakly induced fuzzy topological spaces (X; T ) and (Y; S). If (X; T ) is fuzzy Lindelöf and (Y; S) is regular then(Y; S) is fuzzy paracompact ( resp.*-fuzzy paracompact). Proof. By above theorem and [ 1,Theorem (4.3 ) ]. 3.5 Theorem If a fuzzy topological space (X; T ) is CII ; then it is also fuzzy Lindelöf space. Proof. Let A = fAi : i 2 Ig ½ T such that _fAi : i 2 Ig ¸ ® for each ® 2 I: Since (X; T ) is CII ; then there exists a countable subfamily io B = fBn ; n = 1; 2; ¢¢g of T such that Ai = _ Bik where io may be in…nitely. k=1 Let Bo = fBik g ; i 2 I; k = 1; 2; ¢¢; io ; Bo is countable because it is subfamily of B: Let x 2 X and since _fAi : i 2 Ig ¸ ®, there exist j 2 I such that io io Aj (x) ¸ ® and we have Aj = _ Bj k which implies that _ Bjk ¸ ® and k=1 k=1 io then Bo is open cover. Let " 2 (0; 1] then _ Bi k ¸ ® ¡ " Finally (X; T ) is k=1 fuzzy Lindelöf. 5 3.6 Theorem Let (X; T ) be a fuzzy paracompact separable topological space, then (X,T) is Lindelöf fuzzy. Proof. Let B ½ T be a family such that _ B ¸ ® for each ® 2 I. Let B2B B ¤ be a T-open re…nement of B which is locally …nite and ¤_ ¤ B ¤ ¸ ® ¡ "; B 2B " 2 (0; ®]: X has countable sequence of fuzzy points fpi ; i = 1; 2; ¢¢gsuch that for every B ¤ 6= Á there exists a pi 2 B ¤ : The family fB ¤ : B ¤ 2 B ¤ g is at most countable, otherwise since each B ¤ contains at least one pi : This implies that there would be some pn contained in uncountable many B ¤ 2 B which would contradiction locally …nite. Choose for each Bi¤ 2 B¤ an element Bi 2 B such that Bi¤ < Bi ; then, _u2U0 u ¸ ® ¡ ": There for (X; T ) is Lindelöf fuzzy. 3.7 Theorem Let (X; T ) be a weakly induced regular CII fuzzy topological space, then (X; T ) is fuzzy paracompact (*-fuzzy paracompact). Proof. By [2,:Theorem (3.2)] and [1,Theorem (2.1):d] obtain (X; [T ]) is regular and CII . Then (X; [T ]) is paracompact. From [1, Theorem (3.8)] (X; [T ]) is fuzzy paracompact (*-fuzzy paracompact). 4 Fuzzy compact covering maps In this section we de…ne a notation corresponding to compact covering map studied by [14, 15] for fuzzy topological space. This de…nition will be used to suggest new theorems as will be shown below. 4.1 De…nition A fuzzy continuous mapping f from a fuzzy topological space (X; T ) into (Y; S) is called fuzzy compact covering if for every fuzzy compact ± in Y there exist a fuzzy compact ¹ in X such that f (¹) = ±: 4.2 Theorem Let f :(X; T ) ¡! (Y; S) be a continuous onto mapping. If f is compact covering then f : (X; ! (¿ )) ¡! (Y; ! (S)) is fuzzy compact covering. Proof. For every compact B in(Y; S) there exists a compact A in (X; ¿ ) such that f(A) = B: Let ¹ be a fuzzy compact in (Y; ! (S)) such that 6 f (¹¤ ) = ¹ where ¹¤ is fuzzy subset of (X; T ) and let f f ¡1 (u) : ¹ · _ ug; U ½ ! (S) u2U be an open cover of ¹¤ :To prove that ¹¤ is fuzzy compact, that is mean to …nd a …nite subfamily U0 of U such that f f ¡1 (u) : ¹ · _ ug is subcover u2U0 ¡1 ¤ of ¹ :Then f(u)1¡r : u 2 U g is open cover of B and ff (u)1¡r : u 2 Ug is open cover of A and this implies that there exists U0 ½ U such that ff¡1 (u)1¡r : u 2 U0 g is subcover of A:There for ff ¡1 (u) : ¹ · _ ug ^ fÂf¡1 (u)1¡ r : u 2 U0 g u2U is subcover of ¹¤ :Thus f is fuzzy compact covering 4.3 Theorem Every F-continuous closed mapping f : (X; ! (¿ )) ¡! (Y; ! (S)) of a fuzzy paracompact space X onto an arbitrary induced fuzzy topological space is fuzzy compact covering. Proof. Since f : (X; ! (¿ )) ¡! (Y; ! (S)) is F-continuous and closed then f :(X; ¿ ) ¡! (Y; S) is continuous and closed [11; Lemmas (2.1), (2.2)]. From the good extension of paracompactness implies (X; ¿ ) is paracompact. There for f is compact covering [14; Corollary (1.2)] ; and by above theorem f is fuzzy compact covering. 5 Additional results in fuzzy topological space In this section we de…ne the fuzzy countable compact by used the …nite open cover in the sense of Lowen.and it is proved that the fuzzy countable compact is good extension of countable compact, also it is shown that induced fuzzy topological space ( X, !(¿ )) is fuzzy countable compact and fuzzy paracompact if and only if it is fuzzy compact in the sense of Lowen. This new concept with fuzzy compact in the sense of Lowen and the concept of CII have been used. Some new results corresponding to the same concepts in general topology have been reached. We also study some others spaces such N-compact [6], fuzzy esparable space [18], fuzzy paracompact space [1] non-archimedean fuzzy [12] and we analyze the relation between these spaces and the above concepts which give new properties and notations. 7 5.1 Theorem If (X; ! (¿ )) is fuzzy countable compact then (X; ¿ ) is countable compact. Proof. Let fBn : n = 1; 2; ¢¢g be a countable open cover of (X; ¿ ):i.e _ fBn : n = 1; 2; ¢¢g = X Then _fÂBn : n = 1; 2; ¢¢g = supfÂBn : n = 1; 2; ¢¢g = 1 and fÂBn : n = 1; 2; ¢¢g is a countable open Lowen° s cover of (X; ! (¿ )): By the assumption of the fuzzy countable compactness of (X; ! (¿ )); choose " > 0, then there exists a …nite open Lowen° s subcover ÂB1 ; ÂB2 ; ÂB3 ; :::ÂBm such that supfÂBn : n = 1; 2; ¢¢; mg ¸ 1 ¡ ". Since " is arbitrary then X = _m Bn and implies that (X,¿ ) is countable compact. n=1 5.2 Theorem The induced fuzzy topological space (X; !(¿ )) is fuzzy countable compact and fuzzy paracompact i¤ (X; !(¿ )) is fuzzy compact in the sense of Lowen. Proof. It is clear that every fuzzy compact in the sense of Lowen is fuzzy countable compact. By [1] the concept of fuzzy compactness in the sense of Lowen implies the concept of fuzzy paracompactness. Let (X; !(¿ )) be a fuzzy countable compact and fuzzy paracompact, then (X; ¿ ) is paracompact [1,Theorem (3.8)], and by above theorem (X; ¿ ) is countable compact. From [5, Theorem (5.1.20)] we have (X; ¿ ) is compact, and …nally (X; !(¿ )) is fuzzy compact by [9, Theorem (4.1)]. 5.3 Lemma Every subspace of CII fuzzy topological space (X,T) is CII : Proof. By assumption T has a countable base B = fBi g ; i = 1; 2; 3; :::; then fY ^ Bi g ½ TY ; i = 1; 2; 3; :::be a countable base for TY such that (Y; TY ) is subspace of (X,T) , then (Y; TY ) is CII 5.4 Theorem In a CII fuzzy topological space (X,T), every subspace is separable Proof. By above lemma and [18, Theorem (3.4)]. 5.5 Corollary Let (X; T ) be a CII fuzzy topological space, then the continuous image of fuzzy paracompact (*-fuzzy paracompact) is Lindelöf fuzzy. Proof. Let f : (X; T ) ¡! (Y; S) be F-continuous and let A be a fuzzy paracompact subset of X. By the above theorem A is separable. Then 8 by [Theorem (3.6)] A is Lindelöf fuzzy, and …nally f (A) is Lindelöf fuzzy [Theorem (3.3)]. 5.6 Theorem Every fuzzy compact in the sense of Lowen non-archimedean fuzzy topological space is metrizable. Proof. Let (X,T) be a fuzzy compact non-archimedean fuzzy topological space. Then (X; T ) is*-fuzzy paracompact [1, Theorem (4.2)] and has a uniform base. There for (X; [T ]) is paracompact and has a uniform base. Thus by theorem of Alexandero¤ (X; [T ]) is metrizable and …nally (X; T ) is metrizable fuzzy, see [4]. 5.7 Theorem The continuous image of closed subset of N-compact space is *-fuzzy paracompact, S*-paracompact. Proof. Let f : (X; T ) ¡! (Y; S) and let A be a fuzzy closed subset of N-compact (X,T).Then A is N-compact [6, Theorem (3.1)]. 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