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International Journal of Mathematics and Mathematical Sciences
Volume 2008, Article ID 436570, 9 pages
http://dx.doi.org/10.1155/2008/436570
Research Article

Fuzzy Inverse Compactness

1Department of Mathematics, Faculty of Arts and Sciences, Kocaeli University, Kocaeli 41380, Turkey
2Department of Mathematics, Faculty of Education, Kocaeli University, Kocaeli 41380, Turkey
3Department 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.

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