International Journal of Mathematics and Mathematical Sciences

Volume 2008 (2008), Article ID 436570, 9 pages

http://dx.doi.org/10.1155/2008/436570

## Fuzzy Inverse Compactness

^{1}Department of Mathematics, Faculty of Arts and Sciences, Kocaeli University, Kocaeli 41380, Turkey^{2}Department of Mathematics, Faculty of Education, Kocaeli University, Kocaeli 41380, Turkey^{3}Department of Mathematics, Federal University of Parana, P.O. Box 019081, Curitiba, PR 81531-990, Brazil

Received 9 February 2007; Revised 25 November 2007; Accepted 14 January 2008

Academic Editor: Heinz Gumm

Copyright © 2008 Halis Aygün et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

We introduce definitions of fuzzy inverse compactness, fuzzy inverse countable compactness, and fuzzy inverse Lindelöfness on arbitrary -fuzzy sets in -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.

#### 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 space is called inversely compact if and only if for every open cover of , one can select a finite subcover of which consists of the elements of or their complements (but of course is prohibited to contain both and for any ).

In -fuzzy topological spaces fuzzy compactness has been introduced by Warner and McLean [3] and extended to arbitrary -fuzzy sets by Kudri [4].

In this paper, we initiate fuzzy inverse compactness in -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 paper and will be nonempty ordinary sets and will denote a completely distributive lattice with a smallest element and a largest element and with an order-reversing involution . and will denote -fuzzy topology and ordinary topology, respectively.

*Definition 2.1 (see [5]). *An element of a lattice is called prime
if and only if and whenever with then or .

The set of all
prime elements of a lattice will be denoted
by .

*Definition 2.2 (see [5]). *An element of a lattice is called
coprime (or union irreducible) if and only if and whenever with then or .

The set of all
coprime elements of a lattice will be denoted
by .

From the definitions we have that

*Definition 2.3 (see [3]). *Let be a complete
lattice. A subset of is called Scott
open if and only if it is an upper set and is inaccessible by directed joins,
that is,

(a)if and , then (b)if is a
directed subset of with , then there is a with
The collection
of all Scott-open subsets of is a topology
on and is called
Scott topology of . The Scott topology of completely distributive
lattice is generated by
the sets of the form , where . This means that the family forms a base
for Scott topology on .

A function from to with its Scott
topology is Scott continuous if and only if for every .

*Definition 2.4 (see [6]). *An -fuzzy topology
on is a subset of satisfying the
following properties:

(i)the
constant functions and belong to ;(ii)if , then ;(iii)if , then .
The pair is called an -fuzzy
topological space. The elements of are called open -fuzzy sets. An -fuzzy set is said to be
closed if and only if .

*Definition 2.5 (see [6]). *Let and be -fuzzy
topological spaces. A function is called fuzzy
continuous if and only if for every . .

*Definition 2.6 (see [7]). *Let be an ordinary
topological space. The set of all Scott continuous functions from to with its Scott
topology forms an -fuzzy topology
on , which will be denoted by , that is,

*Definition 2.7 (see [7]). *Let be an ordinary
topological space. An -fuzzy
topological property is a ”good
extension” of a topological property if and only if
the topological space has if and only if the induced -fuzzy
topological space has .

*Definition 2.8 (see [4]). *Let be an -fuzzy
topological space, and . A collection of open -fuzzy sets is
called a -level open
cover of the -fuzzy set if and only if for all with

If is the whole
space , then is called a -level open
cover of .

Thus is called a -level open
cover of if and only if for all .

*Definition 2.9 (see [4]). *Let be an -fuzzy
topological space and . The -fuzzy set is said to be
fuzzy compact if and only if every -level open
cover of , where , has a finite -level subcover
of .

If is the whole
space , then is called fuzzy
compact -fuzzy
topological space.

#### 3. Proposed Definitions

*Definition 3.1. *Let be nonempty set
and let . is called a
partial inversement of if and only if and can be indexed
with the same index set, say, so that for
every either or .

*Definition 3.2. *Let be an -fuzzy
topological space and . is said to be
fuzzy inversely compact if and only if every -level open
cover of , where , has a partial inversement which contains a finite -level cover of .

If is the whole
space then is called fuzzy
inverse compact -fuzzy
topological space.

This definition
can be restated as follows. is fuzzy
inverse compact if and only if for every -level open
cover of , where , there exists a finite -level subcover of such that there
exist for every with , where for each , , or .

Clearly, every
fuzzy compact -fuzzy set is
inversely fuzzy compact.

*Definition 3.3. *Let be an -fuzzy
topological space and . is said to be
fuzzy inversely countably compact if and only if for every countable -level open
cover of , where , has a partial inversement which contains a finite -level cover of .

If is the whole
space, then we say that the -fuzzy
topological space is fuzzy
inversely countably compact.

*Definition 3.4. *Let be an -fuzzy
topological space and . is said to be
fuzzy inversely Lindelöf if and only if every -level open
cover of , where , has a partial inversement which contains a countable -level cover of .

If is the whole
space, then we say that the -fuzzy
topological space is fuzzy
inversely Lindelöf.

From the definitions, it can be easily verified that every fuzzy inversely compact -fuzzy set is fuzzy inversely countably compact and fuzzy inversely Lindelöf.

#### 4. Other Characterizations

*Definition 4.1. *Let , , and . is called an -level centered
family of if and only if for any there exists with such that .

Theorem 4.2. *Let be an -fuzzy
topological space and . is fuzzy
compact if and only if for every -level centered family of closed -fuzzy sets,
where , there exists with such that .*

*Proof. **Necessity.* Let and let be -level centered
family of closed -fuzzy sets such
that for each with , there exists with .

Then, for all with .

Hence for all with , where .

Thus, is a -level open
cover of that has no
finite -level subcover
of . In fact, if , then, since is -level centered,
there exists with and ; hence .
*Sufficiency.* Suppose that is a -level open
cover of with no finite -level subcover
of , where . Then, is a collection
of closed -fuzzy sets.
Moreover, is an -level centered
family of , where . In fact, if , then there exists with such that ; hence .

By the
hypothesis, there exists with with such that ; hence which yields a
contradiction.

*Definition 4.3. *Let , , and .

is called an -level
independent family of if and only if for any
finite there exists with such that

In other words, is an -level
independent family of if and only if for every
nonempty finite partial inversement of , there exists with and .

Theorem 4.4. *Let be an -fuzzy
topological space and . is fuzzy
inversely compact if and only if for every -level
independent family of closed -fuzzy sets,
where , there exists with and .*

*Proof. **Necessity.* Let . Suppose that is an -level
independent family of -fuzzy sets,
such that for every with .

Then, for all with .

Hence, is a -level open
cover of , where .

By the fuzzy
inverse compactness of , there is a partial inversement of which contains
a finite -level subcover
of .

Let and , .

Then, there
exists such that for all with where for each , or .

Let . Then . Since is an -level
independent family of , there exists with such that
Hence, there
exists with such that , which yields a contradiction.
*Sufficiency.* Suppose that is a -level open
cover of with no partial
inversement which contains a finite -level subcover
of , where . Then, is a collection
of closed -fuzzy sets.
Furthermore, is an -level
independent family of , where . In fact, if then by the assumption, there exists with and ; hence

By the hypothesis, there exists with and hence which yields
a contradiction.

Theorem 4.5. *Let be an -fuzzy
topological space and . is fuzzy
inversely countably compact if and only if for every countable -level
independent family of closed -fuzzy sets,
where , there exists with and .*

*Proof. *This
is similar to Theorem 4.4.

*Definition 4.6. *Let and . is said to have
the finite union property (for short FUP) in if and only if for any
finite there exists with .

Theorem 4.7. *Let be an -fuzzy
topological space and . is fuzzy
inversely compact if and only if no with the FUP in
satisfies for all .*

*Proof. **Necessity.* Suppose that has the FUP in
such that for all .

Then, is an -level
independent family of closed -fuzzy sets,
where . From Theorem 4.4, there exists with and ; hence which yields a
contradiction.*Sufficiency.* Suppose that is not fuzzy
inverse compact. Then, there is an -level
independent family of closed -fuzzy sets
such that for all with where . Hence for all with , where . Moreover, is a family of
open -fuzzy sets
having the FUP in . 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 be an ordinary
topological space. is inversely
compact if and only if the -fuzzy
topological space is fuzzy
inversely compact.*

*Proof. **Sufficiency.* Let and be an -level
independent family of closed subsets of . Then, is a family of
closed -fuzzy sets.
Furthermore, is an -level
independent family of . In fact, if then since is independent, . Hence . So, there is such that . Since is fuzzy
inversely compact, there exists with hence and therefore . Hence is inversely
compact.*Necessity.* Let and be a -level open
cover of consisting of
basic open -fuzzy sets in . Then
(for each ) and for all .

Let with and .

It is clear
that is an open
cover of in . Due to inverse compactness of , there exists a partial inversement of which contains
a finite cover of .

Let . Then there exist such that where and (for each ).

(i)If for all , then is a finite
partial inversement of . is also a -level subcover
of in . Hence, is fuzzy
inversely compact.(ii)Suppose
that there exists such that .

Let . Obviously, is a partial
inversement of Furthermore, is a -level subcover
of in . In fact, let . Since or or

(1^{∘}) If , then . Hence

(2^{∘}) If , then there is such that . Hence and therefore Consequently, is -level subcover
of in

Theorem 5.2 (the goodness of fuzzy inverse countable compactness). *Let be an ordinary
topological space. is inversely
countably compact if and only if the -fuzzy
topological space 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 be an ordinary
topological space. is an inverse
Lindelöf space if and only if the -fuzzy
topological space is fuzzy
inverse Lindelöf space.*

*Proof. *This is similar to the proof of the goodness of fuzzy inverse compactness.

Theorem 5.4. *Let be an -fuzzy
topological space. If is fuzzy
inversely compact and closed, then is also fuzzy
inversely compact.*

*Proof. *Let and let be a -level open
cover of ; that is, for all with . So is -level family of . In fact, if
Since is fuzzy
inversely compact, has a partial
inversement which contains
a finite -level subcover
of .

Let and .

Then there
exists such that for all with where for each or .

So for all with . In fact with and ,

Corollary 5.5. *Let be an -fuzzy
topological space. If is a fuzzy
inversely compact -fuzzy set and is a closed -fuzzy set with , then is fuzzy
inversely compact as well.*

*Proof. *This follows directly from the previous theorem.

Corollary 5.6. *Let be an -fuzzy
topological space. If is fuzzy
inversely compact, then every closed -fuzzy set on is fuzzy
inversely compact.*

*Proof. *This follows from Theorem 5.4.

Proposition 5.7. *Let be an -fuzzy
topological space and with . If is fuzzy
inversely countably compact (fuzzy inversely Lindelöf) and is closed, then is fuzzy
inversely countably compact (fuzzy inversely Lindelöf).*

*Proof. *This
is similar to Theorem 5.4.

Theorem 5.8. *Let be an -fuzzy
topological space and . If and are fuzzy
inversely compact then is fuzzy
inversely compact as well.*

*Proof. *Suppose
that is not fuzzy
inverse compact.

Then, there
exists an -level
independent family of consisting of
closed -fuzzy sets such
that for all with , where .

Since we have or .

Let . Since is an -level
independent family of , there exists with and . That is, there exists with or such that .

Assume that . Then, is -level
independent family of consisting of
closed -fuzzy sets.
Furthermore, for all with .

So, is not
fuzzy inversely compact.

Proposition 5.9. *Let be an -fuzzy
topological space and . If and are fuzzy
inversely countably compact (fuzzy inversely Lindelöf), then is fuzzy
inversely countably compact (fuzzy inversely Lindelöf).*

*Proof. *This
is similar to Theorem 5.8.

Theorem 5.10. *Let and be -fuzzy
topological spaces and let be a fuzzy
continuous function such that is finite for
every . If is fuzzy
inversely compact in , then is fuzzy
inversely compact in , where *

*Proof. *Let and is an -level
independent family of consisting of
closed -fuzzy sets in . Let

Since is fuzzy
continuous, is a family of
closed -fuzzy sets in . Furthermore, is an -level
independent family of . In fact, if , then there exists with and because is an level
independent family of . implies that there
exists with and because and is finite.
Hence, . From the fuzzy inverse compactness of , there exists with and . Then, . This means that is fuzzy
inversely compact in .

Theorem 5.11. *Let and be -fuzzy
topological spaces and let be a fuzzy
continuous function such that is finite for
every . If is fuzzy
inversely countable compact (fuzzy inversely Lindelöf) in , then is fuzzy
inversely countably compact (fuzzy inversely Lindelöf) in .*

*Proof. *This is similar to the proof of Theorem 5.10.

#### References

- M. V. Matveev, “Inverse compactness,”
*Topology and Its Applications*, vol. 62, no. 2, pp. 181–191, 1995. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - V. I. Malykhin and M. V. Matveev, in
*Inverse Compactness Versus Compactness*, vol. 767 of*Annals New York Academy of Science*, pp. 153–160, New York Academy of Science, New York, NY, USA, 1995. View at MathSciNet - M. W. Warner and R. G. McLean, “On compact Hausdorff $L$-fuzzy spaces,”
*Fuzzy Sets and Systems*, vol. 56, no. 1, pp. 103–110, 1993. View at Publisher · View at Google Scholar · View at MathSciNet - S. R. T. Kudri, “Compactness in $L$-fuzzy topological spaces,”
*Fuzzy Sets and Systems*, vol. 67, no. 3, pp. 329–336, 1994. View at Publisher · View at Google Scholar · View at MathSciNet - G. Gierz, K. H. Hofmann, K. Keimel, J. D. Lawson, M. W. Mislove, and D. S. Scott, in
*A Compendium of Continuous Lattices*, Springer, Berlin, Germany, 1980. View at MathSciNet - C. L. Chang, “Fuzzy topological spaces,”
*Journal of Mathematical Analysis and Applications*, vol. 24, no. 1, pp. 182–190, 1968. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - M. W. Warner, “Fuzzy topology with respect to continuous lattices,”
*Fuzzy Sets and Systems*, vol. 35, no. 1, pp. 85–91, 1990. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet