Download On Some Paracompactness%type Properties of Fuzzy Topological

Mathematical Methods in Science and Mechanics
On Some Paracompactness-type Properties of Fuzzy Topological
Spaces
FRANCISCO GALLEGO LUPIÁÑEZ
Dept. Geometry & Topology, and IMI
Fac. Mathematics
Univ. Complutense of Madrid
28040 Madrid, SPAIN
e-mail: [email protected]
October 15, 2014
Abstract
are good extensions of those de…ned by A.V.
Arhangel’skiii, and obtain several results about
The aim of this paper is to study some them.
paracompactness-type properties for fuzzy topoFirst, we give some de…nitions:
logical spaces. we prove that these properties are good extensions of others de…ned by
A.V.Arkhangel’skii (and studied by S.A. Peregu- De…nition 1 Let
be a fuzzy set in a fuzzy
dov) and obtain several results about them.
topological space (X; ).
We will say that
Keywords: Fuzzy sets; Topology; paracomis fuzzy a -paracompact (respectively, apactness; fuzzy perfect maps.
paracompact) if for each open cover (in Lowen’s
AMS Mathematics Subject Classi…cation sense) U of
and for each
2 (0; 1], there
(2010): 54A40, 03E72
exits an open re…nement V of U which covers
(in Lowen’s sense) and every in…nite subfamily of V contains an in…nite pairwise disjoint
subfamily (respectively, contains a disjoint pair
1 Introduction
of elements). We will say that (X; ) is fuzzy
On 1971-1972, A.V. Arhangel’kii introduced a -paracompact (respectively, a-paracompact) if
some new paracompactness-type properties: a- each constant fuzzy set in X is fuzzy a paracompactness, a -paracompactness and b- paracompact (respectively, a-paracompact).
paracompactness. Later, S.A. Peregudov did
some research on these concepts [9,10].
De…nition 2 Let
be a fuzzy set in a fuzzy
is
In this paper, we de…ne fuzzy extensions topological space (X; ). We will say that
of these notions, prove that these properties fuzzy b-paracompact if for each open cover (in
1
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238
Mathematical Methods in Science and Mechanics
For each V 2 U , there exits GV 1 (r
; 1]
2 U such that V
GV 1 (r
; 1]: Then B =
f V ^ GV = V 2 U g
and re…nes B.
Now, for each x 2 X, there exits V 2 U such
that V (x) = 1 and GV_
(x) › r
. Thus, ( V ^
GV ) (x) r
and
( V ^ GV ) (x) r
Lowen’s sense) U of and for each 2 (0; 1],
there exits an open re…nement V of U which
covers
(in Lowen’s sense) and every subfamily of V with the …nite intersection property,
is …nite. We will say that (X; ) is fuzzy bparacompact if each constant fuzzy set in X is
fuzzy b-paracompact.
.
V 2U
For each B1
B such that B1 has the …De…nition 3 [4] Let (X; T ) be a topological
space, !(T ) = f : (X; T ) ! I j
is a lower nite intersection property, we have that also, fVj
semi-continuos mapg:The fuzzy topological space /
Vj ^ GV j 2 B1 g has the …nite intersection
property, because, if for each f V1 ^ GV1 ; ::: Vn ^
(X; !(T )) is said induced by (X; T ).
GVn g B1 is ( V1 ^ GV1 ) ^ ::: ^ ( Vn ^ GVn ) 6= 0,
De…nition 4 [4] Let P be a property of topo- there exists x 2 X such that, for each i 2
logical spaces and P be a fuzzy extension of P. f1; :::ng is ( V i ^ GVi )(x) 6= 0 (then, x 2 Vi
The concept P is a good extension of P if a and GVi (x) 6= 0): So, V1 \ ::: \ Vn 6= ?:
topological space (X; T ) veri…es P if and only if
From the hypothesis, fVj / Vj ^ GV j 2 B1 g
(X; !(T )) veri…es P :
is …nite, then B1 is …nite.
Thus, (X, ) is fuzzy b-paracompact.
De…nition 5 [7] A fuzzy topological space
((=) Let U [ ] be an open cover of (X; [ ]),
(X; ) is a weakly inducement of an ordinary
then U 0 = f U / U 2 Ug is an_
open fuzzy cover
topological space (X; T ) if:
r. For
U
2 g is of (X; );and for each r 2 I;
(a) T = [ ], where [ ] = fA X =
A
U 2U
the original topology of :
each 2 (0; r], there exits an open re…nement
(b) every 2 is a lower semi-continuos map
G of U 0 such that every subfamily with
_ the …nite
from (X; T ) into I = [0; 1]:
intersection property is …nite, and
G r ,
2
which implies that
Results
_
G2G
G
r
1
for all
1
› :
G2G
Let U = fG 1 (r
; 1] / G 2 Gg, then U
[ ]
is an open re…nement of U (because, for each
G 1 (r
; 1] 2 U there exits VG 2 U such that
G
,
and G 1 (r
; 1]
V ); and U ia
VG
_ G
an open cover of X; because
G r
1:
Theorem 6 A topological space (X; [ ]) is bparacompact if and only if (X; ) is fuzzy bparacompact.
Proof. ( =) ) _
For each r 2 I, let a family
B
such that
G
r: Let 2 (0; r], then
G2G
G2B
For each U1
U such that U1 has the …nite intersection property, we have that fGj /
Gj 1 (r
; 1] 2 U1 g has also the …nite intersection property (because, if for each fG1 1 (r
; 1]
1
1
; :::; Gn (r
; 1] g U1 is G1 (r
; 1] \::: \
the family U =fG 1 (r ; 1] = G 2 Bg [ ] is an
open cover of (X; [ ]), and from the hypothesis,
it has an open re…nement U
[ ] which is a
cover of
and every subfamily of U with
the …nite intersection property is …nite.
2
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Mathematical Methods in Science and Mechanics
Gn 1 (r ; 1] 6= ?, then for each i = 1; ::; n ,there
is VGi 2 U such that Gi
VGi and there is
x 2 X such that Gi (x) › r
for all i = 1; :::; n.
So, (G1 ^ ::: ^ Gn )(x) 6= 0). By the hypothesis,
fGj / Gj 1 (r
; 1] 2 U1 g is …nite, then U1 is
…nite.
Thus, (X; [ ]) is b-paracompact.
an in…nite pairwise disjoint subfamily (resp.
contains a disjoint pair of elements). Thus,
B1 has an in…nite pairwise disjoint subfamily
(resp. has a disjoint pair of elements), because
Vj1 \ Vj2 = ? implies that ( Vj ^ GVj1 ) ^ ( Vj ^
1
2
GVj2 ) = Vj \Vj ^ (GVj1 ^ GVj2 ) = 0:
1
2
So, (X, ) is fuzzy a -paracompact (resp. aparacompact).
((=) Let U [ ] be an open cover of (X; [ ]),
Remark 7 If (X; T ) is a topological space,
open fuzzy cover
then (X; T ) is b-paracompact if and only if then U 0 = f U / U 2 Ug is an_
(X; !(T )) is fuzzy b-paracompact. (i.e. fuzzy of (X; );and for each r 2 I;
r. For
U
b-paracompactnesss is a good extension of bU 2U
each 2 (0; r], there exists an open re…nement
paracompactness).
Proof. By the above proof (now it is also valid). G of U 0 such that every in…nite subfamily of G
contains an in…nite pairwise disjoint subfamily
(resp.
contains a disjoint pair of elements), and
_
G r
.
G2G
Theorem 8 A topological space (X; [ ]) is a paracompact (resp. a-paracompact) if and only
if (X; ) is fuzzy a -paracompact (resp .aparacompact).
Proof. ( =) ) _
For each r 2 I, let a family
B
such that
G
r: Let 2 (0; r], then
Analogously to the previous theorem, U =
fG 1 (r
; 1] / G 2 Gg re…nes U and is an
open cover of X:
For each in…nite subfamily U1 of U we have
that U1 = fGj 1 (r
; 1] = j 2 Jg for some in…nite set J. Then fGj =j 2 Jg is in…nite and
contained in G: Thus, there exits an in…nite pairwise disjoint subfamily (resp. exists a disjoint
pair of elements) of it, and also U1 has an in…nite pairwise disjoint subfamily (resp. has a disjoint pair of elements), because GVj1 ^ GVj2 = 0
implies that Gj11 (r
; 1]\ Gj21 (r
; 1] = ? (if
Gj1 (x0 ); Gj2 (x0 ) 2 (r
; 1] for some x0 2 X;
then (Gj1 ^ Gj2 )(x0 ) 6= 0; and this is contradictory).
So, (X; [ ]) is a -paracompact (resp.
aparacompact).
G2B
the family U =fG 1 (r
; 1] = G 2 Bg [ ] is
an open cover of (X; [ ]), and by the hypothesis,
it has an open re…nement U
[ ] which is a
cover of
and every in…nite subfamily of U
contains an in…nite pairwise disjoint subfamily
(resp. contains a disjoint pair of elements).
Analogously to the previous theorem, for each
V 2 U , there is GV 1 (r ; 1] 2 U such that V
GV 1 (r ; 1]:Then B = f V ^ GV = V 2 U g
; re…nes
_ B, and
( V ^ GV ) (x) r
.
V 2U
B , B1 =
n For each in…nite osubfamily B1
Remark 9 If (X; T ) is a topological space,
for some in…nite set J, then (X; T ) is a -paracompact (resp.
Vj ^ GV j = j 2 J
athen fVj / j 2 Jg U is in…nite and contains paracompact) if and only if (X; !(T )) is fuzzy a 3
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Mathematical Methods in Science and Mechanics
paracompact (resp.a-paracompact) (i.e. fuzzy a - 3 References
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ness and fuzzy Hausdor¤ness are good extensions
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…rst Remark). Then, the result follows from [9,
Th.5].
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