Download An extension in fuzzy topological spaces

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.
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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).
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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).
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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:
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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
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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.
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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.
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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).
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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
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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.
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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.
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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
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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)]. By [6, Theorem
(3.8)] f (A) is N-compact, and then by [1, Theorem (3.1)] and [10, Theorem
(2.7)] f (A) is *-fuzzy paracompact (S*-paracompact) respectively.
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Acknowledgment
The author is highly grateful to Dr.Francisco Gallego Lupiañez’s constructive
suggestions, instructions and guidance in the preparation of this paper.
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