# Fuzzy Inverse Compactness

Authors: Halis Aygün; A. Arzu Bural; S. R. T. Kudri
Journal: International Journal of Mathematics and Mathematical Sciences
 (2008-02-04)

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## Abstract

We introduce definitions of fuzzy inverse compactness, fuzzy inverse countable compactness, and fuzzy inverse Lindelöfness on arbitraryL-fuzzy sets in L-fuzzy topological spaces. We prove that the proposed definitions are good extensions of the corresponding concepts in ordinary topology and obtain different characterizations of fuzzy inverse compactness.


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## Body

## 1. Introduction

In ordinary
topology, Matveev [1] has introduced a topological property called inverse
compactness which is weaker than compactness and stronger than countable
compactness. In [1, 2], inverse countable compactness and inverse Lindelöfness
have been defined and studied.A topological
spaceX is called
inversely compact if and only if for every open cover β of X, one can select a finite subcover γ of X which consists
of the elements of β or their
complements (but of course γ is prohibited
to contain both U and X∖U for any U∈β).InL-fuzzy
topological spaces fuzzy compactness has been introduced by Warner and McLean
[3] and extended to arbitrary L-fuzzy sets by
Kudri [4].In this paper,
we initiate fuzzy inverse compactness inL-fuzzy
topological spaces which is weaker than fuzzy compactness and introduce fuzzy inverse countable
compactness and fuzzy inverse Lindelöfness. We prove that proposed definitions
are good extensions of the corresponding notions in ordinary topology.


## 2. Preliminaries

We assume that
the reader is familiar with the usual notations and most of the concepts of
fuzzy topology and lattice theory.Throughout this
paperX and Y will be
nonempty ordinary sets and L=L(≤,∨,⋀,′) will denote a
completely distributive lattice with a smallest element 0 and a largest
element    1(0≠1) and with an
order-reversing involution a→a′(a∈L). τ and T will
denote L-fuzzy topology and ordinary topology, respectively.Definition 2.1 (see [5]).
An elementp of a lattice L is called prime
if and only if p≠1 and whenever a,b∈L with a⋀b≤p then a≤p or b≤p.
The set of all
prime elements of a latticeL will be denoted
by  pr(L).Definition 2.2 (see [5]).
An elementα of a lattice L is called
coprime (or union irreducible) if and only if α≠0 and whenever a,b∈L with α≤a∨b then α≤p or α≤b.
The set of all
coprime elements of a latticeL will be denoted
by M(L).From the
definitions we have thatp∈pr(L)⇔p'∈M(L).Definition 2.3 (see [3]).
LetL be a complete
lattice. A subset U of L is called Scott
open if and only if it is an upper set and is inaccessible by directed joins,
that is,
(a)
ifa∈U and a≤b, then b∈U;(b)
ifD is a
directed subset of L with ⋁D∈U, then there is a d∈D with d∈U.
The collection
of all Scott-open subsets of L is a topology
on L and is called
Scott topology of L. The Scott topology of completely distributive
lattice L is generated by
the sets of the form {x∈L∣x≰p}, where p∈pr(L). This means that the family {{x∈L∣x≰p}∣p∈pr(L)} forms a base
for Scott topology on L.
A functionf from (X,T) to L with its Scott
topology is Scott continuous if and only if f−1({x∈X∣x≰p})∈T for every p∈pr(L).Definition 2.4 (see [6]).
AnL-fuzzy topology
on X is a subset τ of LX satisfying the
following properties:
(i)
the
constant functions0 and 1 belong to τ;(ii)
iff,g∈τ, then f⋀g∈τ;(iii)
if{fi:i∈J}⊆τ, then ⋁i∈Jfi∈τ.
The pair (X,τ) is called an L-fuzzy
topological space. The elements of τ are called open L-fuzzy sets. An L-fuzzy set h is said to be
closed if and only if h'∈τ.Definition 2.5 (see [6]).
Let(X,τ) and (Y,τ∗) be L-fuzzy
topological spaces. A function φ:(X,τ)→(Y,τ∗) is called fuzzy
continuous if and only if φ−1(g)∈τ for every g∈τ∗. (φ−1(g)=g(φ(x))).Definition 2.6 (see [7]).
Let(X,T) be an ordinary
topological space. The set of all Scott continuous functions from (X,T) to L with its Scott
topology forms an L-fuzzy topology
on X, which will be denoted by ωL(T), that is,
(2.1)ωL(T)={f∈LX∣f:(X,T)→Lis Scott continuous}.Definition 2.7 (see [7]).
Let(X,T) be an ordinary
topological space. An L-fuzzy
topological property Pf is a ”good
extension” of a topological property P if and only if
the topological space (X,T) has P if and only if the induced L-fuzzy
topological space (X,ωL(T)) has Pf.Definition 2.8 (see [4]).
Let(X,τ) be an L-fuzzy
topological space, g∈LX and p∈L. A collection β=(fi)i∈J of open L-fuzzy sets is
called a p-level open
cover of the L-fuzzy set g if and only if (⋁i∈Jfi)(x)≰p for all x∈X with g(x)≥p'.
Ifg is the whole
space X(=1), then β is called a p-level open
cover of X.
Thusβ is called a p-level open
cover of X if and only if (⋁f∈βf)(x)≰p for all x∈X.Definition 2.9 (see [4]).
Let(X,τ) be an L-fuzzy
topological space and g∈LX. The L-fuzzy set g is said to be
fuzzy compact if and only if every p-level open
cover of g, where p∈pr(L), has a finite p-level subcover
of g.
Ifg is the whole
space X, then (X,τ) is called fuzzy
compact L-fuzzy
topological space.


## 3. Proposed Definitions

Definition 3.1.
LetX be nonempty set
and let β,γ⊆LX.  β is called a
partial inversement of γ if and only if β and γ can be indexed
with the same index set, sayJ:β={fi:i∈J},  γ={gi:i∈J} so that for
every i∈J either fi=g or fi=gi'.Definition 3.2.
Let(X,τ) be an L-fuzzy
topological space and g∈LX. g is said to be
fuzzy inversely compact if and only if every p-level open
cover of g, where p∈pr(L), has a partial inversement which contains a finite p-level cover of g.
Ifg is the whole
space X then (X,τ) is called fuzzy
inverse compact L-fuzzy
topological space.
This definition
can be restated as follows.g is fuzzy
inverse compact if and only if for every p-level open
cover β={fi:i∈J} of g, where p∈pr(L), there exists a finite p-level subcover γ={gi:i∈J} of g such that there
exist g1,g2,…,gn∈γ:(⋁i=1ngi)(x)≰p for every x∈X with g(x)≥p', where for each i=1,2,…,n, gi=fi, or gi=fi'.
Clearly, every
fuzzy compactL-fuzzy set is
inversely fuzzy compact.Definition 3.3.
Let(X,τ) be an L-fuzzy
topological space and g∈LX. g is said to be
fuzzy inversely countably compact if and only if for every countable p-level open
cover of g, where p∈pr(L), has a partial inversement which contains a finite p-level cover of g.
Ifg is the whole
space, then we say that the L-fuzzy
topological space (X,τ) is fuzzy
inversely countably compact.Definition 3.4.
Let(X,τ) be an L-fuzzy
topological space and g∈LX. g is said to be
fuzzy inversely Lindelöf if and only if every p-level open
cover of g, where p∈pr(L), has a partial inversement which contains a countable p-level cover of g.
Ifg is the whole
space, then we say that the L-fuzzy
topological space (X,τ) is fuzzy
inversely Lindelöf.From the
definitions, it can be easily verified that every fuzzy inversely compactL-fuzzy set g is fuzzy
inversely countably compact and fuzzy inversely Lindelöf.


## 4. Other Characterizations

Definition 4.1.
Letα∈L, ζ⊆LX, and g∈LX. ζ is called an α-level centered
family of g if and only if for any h1,h2,…,hn∈ζ there exists x∈X with g(x)≥α such that (h1⋀h2⋀⋯⋀hn)(x)≥α.Theorem 4.2.
Let(X,τ) be an L-fuzzy
topological space and g∈LX. g is fuzzy
compact if and only if for every α-level centered family ζ of closed L-fuzzy sets,
where α∈M(L), there exists x∈X with g(x)≥α such that (⋀h∈ζh)(x)≥α.Proof.
Necessity. Let α∈M(L) and let ζ be α-level centered
family of closed L-fuzzy sets such
that for each x∈X with g(x)≥α, there exists h∈ζ with h(x)≱α.
Then,(⋀h∈ζh)(x)≰α for all x∈X with g(x)≥α.
Hence(⋁h∈ζh′)(x)≱p for all x∈X with g(x)≥p', where p=α'.
Thus,β={h':h∈ζ} is a p-level open
cover of g that has no
finite p-level subcover
of g. In fact, if h1',h2',…,hn'∈β, then, since ζ is α-level centered,
there exists x∈X with g(x)≥α and (⋀i=1nhi)(x)≥α; hence (⋁i=1nhi′)(x)≤p.
Sufficiency. Suppose that β is a p-level open
cover of g with no finite p-level subcover
of g, where p∈pr(L). Then, ζ={f':f∈β} is a collection
of closed L-fuzzy sets.
Moreover, ζ is an α-level centered
family of g, where α=p'. In fact, if f1',f2',…,fn'∈ζ, then there exists x∈X with g(x)≥p'=α such that (⋁i=1nfi)(x)≤p; hence (⋀i=1nfi′)(x)≥α.
By the
hypothesis, there existsx∈X with g(x)≥p' with such that (⋀f∈βf′)(x)≥p′; hence (⋁f∈βf)(x)≤p which yields a
contradiction.Definition 4.3.
Letζ⊆LX, g∈LX, and α∈L.
ζ is called an α-level
independent family of g if and only if for any
finite f1,f2,…,fn,g1,g2,…,gm∈ζ  there exists x∈X with g(x)≥α such that (⋀i=1nfi⋀⋀i=1mgi′)(x)≥α.
In other words,ζ is an α-level
independent family of g if and only if for every
nonempty finite partial inversement ζ∗ of ζ, there exists x∈X with g(x)≥α and (⋀h∈ζ*h)(x)≥α.Theorem 4.4.
Let(X,τ) be an L-fuzzy
topological space and g∈LX. g is fuzzy
inversely compact if and only if for every α-level
independent family ζ of closed L-fuzzy sets,
where α∈M(L), there exists x∈X with g(x)≥α and (⋀h∈ζh)(x)≥α.Proof.
Necessity. Let α∈M(L). Suppose that ζ is an α-level
independent family of L-fuzzy sets,
such that (⋀h∈ζh)(x)≱α for every x∈X with g(x)≥α.
Then,(⋁h∈ζh′)(x)≰α′ for all x∈X with g(x)≥α.
Hence,β={h':h∈ζ} is a p-level open
cover of g, where p=α'.
By the fuzzy
inverse compactness ofg, there is a partial inversement γ of β which contains
a finite p-level subcover
of g.
Letγ={gi:i∈J} and β={hi':i∈J,  hi∈ζ}.
Then, there
existsg1,g2,…,gn∈γ such that (⋁i=1ngi)(x)≰p for all x∈X with g(x)≥p', where for each i∈{1,2,…,n},gi=hi, or gi=hi'.
Letg1=h1',g2=h2',…,gk=hk',gk+1=hk+1,gk+2=hk+2,…,gn=hn. Then g1',g2',…,gk',gk+1,…,gn∈ζ. Since ζ is an α-level
independent family of g, there exists x∈X with g(x)≥α such that
(4.1)(⋀i=1kgi′⋀⋀i=k+1ngi′)(x)=(⋀i=1ngi′)(x)≥α.
Hence, there
exists x∈X with g(x)≥p' such that (⋁i=1ngi)(x)≤p, which yields a contradiction.
Sufficiency. Suppose that β is a p-level open
cover of g with no partial
inversement which contains a finite p-level subcover
of g, where p∈pr(L). Then, ζ={f':f∈β} is a collection
of closed L-fuzzy sets.
Furthermore, ζ is an α-level
independent family of g, where α=p'. In fact, if f1',f2',…,fk',fk+1',…,fn'∈ζ,  then by the assumption, there exists x∈X with g(x)≥p' and (⋁i=1kfi∨⋁i=k+1nfi′)(x)≤p; hence (⋀i=1kfi′⋀⋀i=k+1nfi)(x)≥α.
By the hypothesis, there existsz∈X with g(z)≥α and (⋀f∈βf′)(z)≥α; hence (⋁f∈βf)(z)≤p, which yields
a contradiction.Theorem 4.5.
Let(X,τ) be an L-fuzzy
topological space and g∈LX. g is fuzzy
inversely countably compact if and only if for every countable α-level
independent family ζ of closed L-fuzzy sets,
where α∈M(L), there exists x∈X with g(x)≥α and (⋀h∈βh)(x)≥α.Proof.
This
is similar to Theorem4.4.Definition 4.6.
Letξ⊆LX and p∈pr(L). ξ is said to have
the finite union property (for short FUP) in G={x∈X:g(x)≥p'} if and only if  for any
finite f1,f2,…,fk,g1,g2,…,gn∈ξ,  there exists x∈G with (⋁i=1kfi∨⋁i=1ngi′)(x)≤p.Theorem 4.7.
Let(X,τ) be an L-fuzzy
topological space and g∈LX. g is fuzzy
inversely compact if and only if no ξ⊆τ with the FUP in
G satisfies (⋁f∈ξf)(x)≰p for all x∈G.Proof.
Necessity. Suppose that ξ⊆τ has the FUP in
G such that (⋁f∈ξf)(x)≰p for all x∈G.
Then,ζ={f':f∈ξ} is an α-level
independent family of closed L-fuzzy sets,
where α=p'. From Theorem 4.4, there exists x∈X with g(x)≥α and (⋀f∈ζf′)(x)≥α; hence (⋁f∈ξf)(x)≤p which yields a
contradiction.
Sufficiency. Suppose that g is not fuzzy
inverse compact. Then, there is an α-level
independent family ζ of closed L-fuzzy sets
such that (⋀h∈ζh)(x)≱α for all x∈X with g(x)≥α, where α∈M(L). Hence (⋁h∈ζh′)(x)≰p for all x∈X with g(x)≥p', where p=α'. Moreover, ξ={h':h∈ζ} is a family of
open L-fuzzy sets
having the FUP in G. This completes the proof.


## 5. Some Properties

The next
theorem shows that fuzzy inverse compactness is a good extension of inverse
compactness in general topology.Theorem 5.1.
Let(X,T) be an ordinary
topological space. (X,T) is inversely
compact if and only if the L-fuzzy
topological space (X,ωL(T)) is fuzzy
inversely compact.Proof.
Sufficiency. Let α∈M(L) and ζ be an α-level
independent family of closed subsets of X. Then, ζ∗={χA:A∈ζ} is a family of
closed L-fuzzy sets.
Furthermore, ζ∗ is an α-level
independent family of X. In fact, if χA1,χA2,…,χAn,χB1,…,χBm∈ζ∗,  then since ζ is independent, ⋂i=1nAi∩⋂i=1mBi′≠ϕ. Hence (⋀i=1nχAi⋀⋀i=1mχBi′)=χ⋂i=1nAi∩⋂i=1mBi′≠0. So, there is x∈X such that (⋀i=1nχAi⋀⋀i=1mχBi′)(x)=1≥α. Since (X,ωL(T)) is fuzzy
inversely compact, there exists z∈X with (⋀A∈ζχA)(z)≥α; hence (⋀A∈ζχA)(z)=(χ⋂A∈ζA)(z)=1 and therefore z∈⋂A∈ζA. Hence (X,T) is inversely
compact.
Necessity. Let p∈pr(L) and β=(fi)i∈J be a p-level open
cover of X consisting of
basic open L-fuzzy sets in (X,ωL(T)). Then
(5.1)fi(x)={ei∈L;x∈Ai∈T,0;x∉Ai
(for each i∈J) and (⋁i∈Jfi)(x)≰p for all x∈X.
Letζ={Ai⊆X:∃i∈J with ei≰p and fi∈β}.
It is clear
thatζ is an open
cover of X in (X,T). Due to inverse compactness of (X,T), there exists a partial inversement ζ∗ of ζ which contains
a finite cover of X.
Letζ∗={Bi:i∈J}. Then there exist B1,B2,…,Bn∈ζ∗ such that X=⋃i=1nBi, where Bi=Ai and Bi=Ai' (for each i∈{1,2,…,n}).
(i)
IfBi=Ai for all i∈{1,2,…,n}, then β∗={f1,f2,…,fn} is a finite
partial inversement of β. β∗ is also a p-level subcover
of X in (X,ωL(T)). Hence, (X,ωL(T)) is fuzzy
inversely compact.(ii)
Suppose
that there existsi0∈{1,2,…,n} such that Bi0=Ai0'.
Letβ∗={f1,f2,…,fi0−1,fi0',…,fn'}. Obviously, β∗ is a partial
inversement of β. Furthermore, β∗ is a p-level subcover
of X in (X,ωL(T)). In fact, let x∈X. Since X=⋃i=1nBi=Bi0∪⋃i=1,i≠i0nBi⇒x∈Bi0 or x∈⋃i=1,i≠i0nBi⇒x∈Ai0' or x∈⋃i=1,i≠i0nAi.
(1∘) If x∈Ai0′, then fi0′(x)=1≰p. Hence (fi0′(x)∨⋁i=1,i≠i0nfi)(x)≰p.
(2∘) If x∉Ai0′, then there is i∈{1,2,…,n}−{i0} such that x∈Ai. Hence fi(x)=ei≰p and therefore (fi0′∨⋁i=1,i≠i0nfi)(x)≰p. Consequently, β∗ is p-level subcover
of X in (X,ωL(T)).Theorem 5.2 (the goodness of fuzzy inverse countable compactness).
Let(X,T) be an ordinary
topological space. (X,T) is inversely
countably compact if and only if the L-fuzzy
topological space (X,ωL(T)) is fuzzy
inversely countably compact.Proof.
This is similar to the proof of the goodness of fuzzy inverse compactness.Theorem 5.3 (the goodness of fuzzy inverse lindelöfness).
Let(X,T) be an ordinary
topological space. (X,T) is an inverse
Lindelöf space if and only if the L-fuzzy
topological space (X,ωL(T)) is fuzzy
inverse Lindelöf space.Proof.
This is similar to the proof of the goodness of fuzzy inverse compactness.Theorem 5.4.
Let(X,τ) be an L-fuzzy
topological space. If g∈LX is fuzzy
inversely compact and h∈LX closed, then g⋀h is also fuzzy
inversely compact.Proof.
Letp∈pr(L) and let β=(fi)i∈J⊆LX be a p-level open
cover of g⋀h; that is, (⋁i∈Jfi)(x)≰p for all x∈X with (g⋀h)(x)≥p'. So β∗=(fi)i∈J∪{h'} is p-level family of g. In fact, if
(5.2)h(x)≥p'⇒(⋁i∈Jfi)(x)≰p∀x∈Xwith(g⋀h)(x)≥p'⇒(⋁k∈β*k)(x)=(⋁i∈Jfi)(x)∨h'(x)≰ph(x)≱p'⇒h'(x)≰p⇒(⋁k∈β*k)(x)≰p.
Since g is fuzzy
inversely compact, β∗ has a partial
inversement γ∗ which contains
a finite p-level subcover
of g.
Letγ∗={gi:i∈J}∪{h'} and β∗={fi:i∈J}∪{h'}.
Then there
existsg1,g2,…,gn∈γ∗ such that (⋁i=1ngi∨h')(x)≰p for all x∈X with g(x)≥p' where for each i∈{1,2,…,n},gi=fi or gi=fi'.
So(⋁i=1ngi)(x)≰p for all x∈X with (g⋀h)(x)≥p'. In fact x∈X with (g⋀h)(x)≥p⇒g(x)≥p'  and h(x)≥p',
(5.3)g(x)≥p'⇒(⋁i=1ngi∨h')(x)≰p⇒(⋁i=1ngi)(x)∨h'(x)≰p⇒(⋁i=1ngi)(x)≰p.Corollary 5.5.
Let(X,τ) be an L-fuzzy
topological space. If g is a fuzzy
inversely compact L-fuzzy set and h is a closed L-fuzzy set with h≤g, then h is fuzzy
inversely compact as well.Proof.
This follows directly from the previous theorem.Corollary 5.6.
Let(X,τ) be an L-fuzzy
topological space. If (X,τ) is fuzzy
inversely compact, then every closed L-fuzzy set on X is fuzzy
inversely compact.Proof.
This follows from Theorem5.4.Proposition 5.7.
Let(X,τ) be an L-fuzzy
topological space and g,h∈LX with h≤g. If g is fuzzy
inversely countably compact (fuzzy inversely Lindelöf) and h is closed, then h is fuzzy
inversely countably compact (fuzzy inversely Lindelöf).Proof.
This
is similar to Theorem5.4.Theorem 5.8.
Let(X,τ) be an L-fuzzy
topological space and g,h∈LX. If g and h are fuzzy
inversely compact then g∨h is fuzzy
inversely compact as well.Proof.
Suppose
thatg∨h is not fuzzy
inverse compact.
Then, there
exists anα-level
independent family ζ of g∨h consisting of
closed L-fuzzy sets such
that (⋀f∈ζf)(x)≥α for all x∈X with (g∨h)(x)≥α, where α∈M(L).
Sinceα∈M(L) we have (g∨h)(x)≥α⇔g(x)≥α or h(x)≥α.
Letf1,f2,…,fk,fk+1,…,fn∈ζ.  Since ζ is an α-level
independent family of g∨h, there exists x∈X with (g∨h)(x)≥α and (⋀i=1kfi⋀⋀i=k+1nfi')(x)≥α. That is, there exists x∈X with g(x)≥α or h(x)≥α such that (⋀i=1kfi⋀⋀i=k+1nfi')(x)≥α.
Assume thatg(x)≥α. Then, ζ is α-level
independent family of g consisting of
closed L-fuzzy sets.
Furthermore, (⋀f∈ζf)(x)≱α for all x∈X with g(x)≥α.
So,g is not
fuzzy inversely compact.Proposition 5.9.
Let(X,τ) be an L-fuzzy
topological space and g,h∈LX. If g and h are fuzzy
inversely countably compact (fuzzy inversely Lindelöf), then g∨h is fuzzy
inversely countably compact (fuzzy inversely Lindelöf).Proof.
This
is similar to Theorem5.8.Theorem 5.10.
Let(X,τ) and (Y,τ∗) be L-fuzzy
topological spaces and let  φ:(X,τ)→(Y,τ∗) be a fuzzy
continuous function such that φ−1(y) is finite for
every y∈Y. If g∈LX is fuzzy
inversely compact in (X,τ), then φ(g)∈LY is fuzzy
inversely compact in (Y,τ∗), where φ(g)(y)=⋁x∈φ−1(y)g(x).Proof.
Letα∈M(L) and ζ is an α-level
independent family of φ(g) consisting of
closed L-fuzzy sets in (Y,τ∗). Let ζ∗={φ−1(h):h∈ζ}.
Sinceφ is fuzzy
continuous, ζ∗ is a family of
closed L-fuzzy sets in (X,τ). Furthermore, ζ∗ is an α-level
independent family of g. In fact, if φ−1(h1),φ−1(h2),…,φ−1(hk),φ−1(hk+1),…,φ−1(hn)∈ζ∗, then there exists y∈Y with φ(g)(y)≥α and (⋀i=1khi⋀⋀i=k+1nhi')(y)≥α because ζ is an α−level
independent family of φ(g). φ(g)(y)=⋁x∈φ−1(y)g(x)≥α implies that there
exists x∈X with g(x)≥α and φ(x)=y because α∈M(L) and φ−1(y) is finite.
Hence, (⋀i=1kφ−1(hi)⋀⋀i=k+1nφ−1(hi'))(x)=(⋀i=1khi⋀⋀i=k+1nhi')(φ(x))≥α. From the fuzzy inverse compactness of g, there exists z∈X with g(z)≥α and (⋀h∈ζφ−1(h))(z)≥α. Then, (⋀h∈ζh)(φ(z))≥α. This means that φ(g) is fuzzy
inversely compact in (Y,τ∗).Theorem 5.11.
Let(X,τ) and (Y,τ∗) be L-fuzzy
topological spaces and let φ:(X,τ)→(Y,τ∗) be a fuzzy
continuous function such that φ−1(y) is finite for
every y∈Y. If g∈LX is fuzzy
inversely countable compact (fuzzy inversely Lindelöf) in (X,τ), then φ(g)∈LY is fuzzy
inversely countably compact (fuzzy inversely Lindelöf) in (Y,τ∗).Proof.
This is similar to the proof of Theorem5.10.



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