Almost Fuzzy Compactness in L-fuzzy Topological Spaces

Li Hong-YananD Cui Wei

（1.School of Mathematics and Information Science，Shandong Institute of Business and Technology，Yantai，Shandong，264005）

（2.Department of Modern Science and Technology，Shenyang Municipal Party Committee Party School，Shenyang，110036）

Communicated by Lei Feng-chun

Almost Fuzzy Compactness in L-fuzzy Topological Spaces

Li Hong-Yan1anD Cui Wei2

（1.School of Mathematics and Information Science，Shandong Institute of Business and Technology，Yantai，Shandong，264005）

（2.Department of Modern Science and Technology，Shenyang Municipal Party Committee Party School，Shenyang，110036）

Communicated by Lei Feng-chun

In this paper，the notion of almost fuzzy compactness is defined in L-fuzzy topological spaces by means of inequality，where L is a completely distributive DeMorgan algebra.Its properties are discussed and many characterizations of it are presented.

L-fuzzy topological space，L-fuzzy almost compactness，L-fuzzy compactness，almost fuzzy compactness

2010 MR subject classification：54A40，54D30，03E72

Document code：A

Article ID：1674-5647（2015）03-0267-07

1　Introduction

Almost compactness is a very important concept.Many researchers have tried successfully to generalize the compactness theory of general topology to L-topology（see［1-9］）.Recently，Shi［10］introduced new definitions of almost fuzzy compactness in L-topological spaces with the help of inequality，where L is a completely distributive DeMorgan algebra.The aim of this paper is to generalize the notion of almost compactness in［10］to L-fuzzy topological spaces，thus some properties and characterizations are researched.

2　Preliminaries

In this paper，（L，∨，∧，′）is a completely distributive DeMorgan algebra（i.e.，completely distributive lattice with order-reversing involution，see［11］）.The largest element and thesmallest element in L are denoted by?and⊥，respectively.

Definition 2.1［12］An L-fuzzy topology on a set X is a map τ：LX→L such that

For a subfamily Φ?LX，2（Φ）denotes the set of all finite subfamilies of Φ.

Definition 2.2［13］is an L-fuzzy inclusion on X，it is defined asFor simplicity，it is denoted by［A~?B］instead of~?（A，B），

3　Definitions and Properties of L-fuzzy Almost Compactness

Definition 3.1Let（X，τ）be an L-fuzzy topological space and A∈LX.For all r∈L，

is called the r-interiors of A with respect to τ.The r-closures of A with respect to τ is defined as

An L-topology T can be regarded as a map χT：LX→L defined by

In this way，（X，χT）is a special L-fuzzy topological space and

This shows that Definition 3.1 can be regarded as the generalization in L-fuzzy topological space of the interiors and closures in L-topological space.

By Definition 3.1，we have the following theorem.

Theorem 3.1Let（X，T）be an L-topological space and A∈LX.Then，for all r，s∈L，

（6）A∈τr?τ（A）≥r.

Definition 3.2Let（X，τ）be an L-fuzzy topological space.G ∈LXis called L-fuzzy almost compact，if U?LX，it follows that

In an L-topological space（X，T），G∈LXis almost fuzzy compact（see［10］）if and only if for all U?T，the inequality

is satisfied.

At the same time，（X，χT）is a special L-fuzzy topological space，and for all U?T，we have r=χT（U）=?.So，G∈LXis almost fuzzy compactness if and only if for all U?T，it follows that

Thus，the following theorem can be obtained.

Theorem 3.2Let（X，T）be an L-topological space and G∈LX.Then G is almost fuzzy compact in（X，T）if and only if G is L-fuzzy almost compact in（X，χT）.

The definition of L-fuzzy compactness（see［14］）in an L-fuzzy topological space can be described as follows：Let（X，τ）be an L-fuzzy topological space.G∈LXis called L-fuzzy compact if U?LX，then

In this way，the proposition“L-fuzzy compactness?L-fuzzy almost compactness”can be proved by Theorem 3.1.

By Definitions 2.1，2.2，3.1，3.2 and Theorem 3.1，we can prove the following theorem.

Theorem 3.3Let（X，τ）be an L-fuzzy topological space and G∈LX.Then G is L-fuzzy almost compact if and only if for all P?LX，one has

Theorem 3.4Let（X，τ）be an L-fuzzy topological space and G∈LX.Then G is L-fuzzy almost compact if and only if for all r∈L and for τ（U）≤r，U?τr，one has

Proof.（1）Necessity.Suppose that G is L-fuzzy almost compact，and for all r∈L and for τ（U）≤r，U?τr.As U?τr，by Theorem 3.1，we have τ（A）≥r for all A∈U.Then we get τ（U）≥r.Therefore，τ（U）=r.By L-fuzzy almost compactness of G，we have

（2）Sufficiency.Suppose that for all r∈L and for τ（U）≤r，U?τr.We have

For all U?LX，let r=τ（U）.Then for all A∈U，τ（A）≥r.So we get U?τrby Theorem 3.1.Thus

Therefore，G is L-fuzzy almost compact.

Theorem 3.5Let（X，τ）be an L-fuzzy topological space and G∈LX.If both G and H are L-fuzzy almost compact，then G∨H is L-fuzzy almost compact.

Proof.For all U?LX，let r=τ（U）.Since G and H are L-fuzzy almost compact，we have

and

By the facts

and

we know that

Therefore，G∨H is L-fuzzy almost compact.

Theorem 3.6Let（X，τ）be an L-fuzzy topological space and G∈LX.If G is L-fuzzy almost compact，and H ∈LXwith τ（H）=τ?（H）=?，then G∧H is L-fuzzy almost compact.

Proof.For all U?LX，let r=τ（U），W=U∪H′.Then we get W ?LX.It is easy to check

Then we obtain

from the following facts

and

Therefore，G∧H is L-fuzzy almost compact.

4　Characterizations of L-fuzzy Almost Compactness

Definition 4.1Let（X，τ）be an L-fuzzy topological space，G∈LXand ??LX.Then

（1）? is r-cover（see［9］，r-shading in［15］）of G if and only if

（2）? is r+-cover（see［9］，strong r-shading in［15］）of G if and only if

（3）? is almost r-cover of G with respect to τ if and only if

（4）? is almost r+-cover of G with respect to τ if and only if

（5）? is an r-remote family（see［15］）of G if and only if

（6）? is a strong r-remote family（see［15］）of G if and only if

By Definitions 3.2，4.1 and Theorem 3.3，we can get the following theorem.

Theorem 4.1Let（X，τ）be an L-fuzzy topological space and G∈LX.Then the following conditions are equivalent：

（1）G is L-fuzzy almost compact；

（2）For any r∈L｛?｝，each r+-cover U of G with τ（U）?r has a finite subfamily V which is an almost r+-cover of G；

（3）For any r∈L｛?｝，each r+-cover of G with τ（U）?r has a finite subfamily which is an almost r-cover of G；

（4）For any r∈P（L），each r+-cover of G with τ（U）?r has a finite subfamily which is an almost r-cover（r+-cover）of G；

（5）For any r∈P（L），each r+-cover U of G with τ（U）?r has b∈α?（r）and a finite subfamily V such that V is an almost b-cover of G；

（6）For any r∈L｛?｝，each strong r-remote family P of G with τ?（P）?r′has a finite subfamily Q such thatis a strong r-remote family of G；

（7）For any r∈L｛?｝，each strong r-remote family P of G with τ?（P）?r′has a finite subfamily Q such thatis a r-remote family of G；

（8）For any r∈M（L），each strong r-remote family P of G with τ?（P）?r′has a finite subfamily Q and b∈β?（r）such thatis a strong b-remote family of G；

（9）For any r∈M（L），each strong r-remote family P of G with τ?（P）?r′has a finite subfamily Q and b∈β?（r）such thatis a b-remote family of G.

Definition 4.2［15］Let（X，τ）be an L-fuzzy topological space，G∈LXand ??LX.We define that

It is easy to prove the following theorem.

Theorem 4.2Let（X，τ）be an L-fuzzy topological space and G∈LX.Then the following conditions are equivalent：

（1）G is L-fuzzy almost compact；

（2）For any r∈L｛⊥｝（r∈M（L）），each strong βr-cover U of G with r∈β（τ（U））has a finite subfamily V such thatis a（strong）βa-cover of G；

（3）For any r∈L｛⊥｝（r∈M（L）），each strong βr-cover U of G with r∈β（τ（U））has a finite subfamily V of U and b∈L（b∈M（L））with a∈β（b）such thatis a（strong） βb-cover of G；

（4）For any r∈L｛⊥｝（r∈M（L））and any b∈β?（r），each Qr-cover U of G with τ（A）≥r for any A∈U has a finite subfamily V such thatis a Qb-cover of G；

（5）For any r∈L｛⊥｝（r∈M（L））and any b∈β?（r），each Qr-cover U of G with τ（A）≥r for any A∈U has a finite subfamily V such thatis a（strong）βb-cover of G.

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10.13447/j.1674-5647.2015.03.09

date：Oct.28，2014.

The NSF（11471297）of China.

E-mail address：lihongyan@sdibt.edu.cn（Li H Y）.