Fuzzy Inverse Compactness

Abstract

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

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 𝑋 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).

In 𝐿-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 in 𝐿-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 𝑋 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 element 𝑝 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 lattice 𝐿 will be denoted by pr(𝐿).

Definition 2.2 (see [5]). An element 𝛼 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 lattice 𝐿 will be denoted by 𝑀(𝐿).

From the definitions we have that 𝑝∈pr(𝐿)⇔𝑝^′∈𝑀(𝐿).

Definition 2.3 (see [3]). Let 𝐿 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)if 𝑎∈𝑈 and 𝑎≤𝑏, then 𝑏∈𝑈;(b)if 𝐷 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 function 𝑓 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]). An 𝐿-fuzzy topology on 𝑋 is a subset 𝜏 of 𝐿^𝑋 satisfying the following properties:
(i)the constant functions 0 and 1 belong to 𝜏;(ii)if 𝑓,𝑔∈𝜏, then 𝑓⋀𝑔∈𝜏;(iii)if {𝑓_𝑖∶𝑖∈𝐽}⊆𝜏, then ⋁_𝑖∈𝐽𝑓_𝑖∈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 𝜔_𝐿(𝑇), that is, 𝜔_𝐿(𝑇)={𝑓∈𝐿^𝑋∣𝑓∶(𝑋,𝑇)→𝐿isScottcontinuous}. (2.1)

Definition 2.7 (see [7]). Let (𝑋,𝑇) be an ordinary topological space. An 𝐿-fuzzy topological property 𝑃_𝑓 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,𝜔_𝐿(𝑇)) has 𝑃_𝑓.

Definition 2.8 (see [4]). Let (𝑋,𝜏) be an 𝐿-fuzzy topological space, 𝑔∈𝐿^𝑋 and 𝑝∈𝐿. A collection 𝛽=(𝑓_𝑖)_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∈𝐽𝑓_𝑖)(𝑥)≰𝑝 for all 𝑥∈𝑋 with 𝑔(𝑥)≥𝑝^′.
If 𝑔 is the whole space 𝑋(=1), then 𝛽 is called a 𝑝-level open cover of 𝑋.
Thus 𝛽 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∈𝐿^𝑋. 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.
If 𝑔 is the whole space 𝑋, then (𝑋,𝜏) is called fuzzy compact 𝐿-fuzzy topological space.

3. Proposed Definitions

Definition 3.1. Let 𝑋 be nonempty set and let 𝛽,𝛾⊆𝐿^𝑋. 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={𝑓_𝑖∶𝑖∈𝐽}, 𝛾={𝑔_𝑖∶𝑖∈𝐽} so that for every 𝑖∈𝐽 either 𝑓_𝑖=𝑔 or 𝑓_𝑖=𝑔^′_𝑖.

Definition 3.2. Let (𝑋,𝜏) be an 𝐿-fuzzy topological space and 𝑔∈𝐿^𝑋. 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.
If 𝑔 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={𝑓_𝑖∶𝑖∈𝐽} of 𝑔, where 𝑝∈pr(𝐿), there exists a finite 𝑝-level subcover 𝛾={𝑔_𝑖∶𝑖∈𝐽} of 𝑔 such that there exist 𝑔₁,𝑔₂,…,𝑔_𝑛∈𝛾∶(⋁^𝑛_𝑖=1𝑔_𝑖)(𝑥)≰𝑝 for every 𝑥∈𝑋 with 𝑔(𝑥)≥𝑝^′, where for each 𝑖=1,2,…,𝑛, 𝑔_𝑖=𝑓_𝑖, or 𝑔_𝑖=𝑓^′_𝑖.
Clearly, every fuzzy compact 𝐿-fuzzy set is inversely fuzzy compact.

Definition 3.3. Let (𝑋,𝜏) be an 𝐿-fuzzy topological space and 𝑔∈𝐿^𝑋. 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.
If 𝑔 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∈𝐿^𝑋. 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.
If 𝑔 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 compact 𝐿-fuzzy set 𝑔 is fuzzy inversely countably compact and fuzzy inversely Lindelöf.

4. Other Characterizations

Definition 4.1. Let 𝛼∈𝐿, 𝜁⊆𝐿^𝑋, and 𝑔∈𝐿^𝑋. 𝜁 is called an 𝛼-level centered family of 𝑔 if and only if for any ℎ₁,ℎ₂,…,ℎ_𝑛∈𝜁 there exists 𝑥∈𝑋 with 𝑔(𝑥)≥𝛼 such that (ℎ₁⋀ℎ₂⋀⋯⋀ℎ_𝑛)(𝑥)≥𝛼.

Theorem 4.2. Let (𝑋,𝜏) be an 𝐿-fuzzy topological space and 𝑔∈𝐿^𝑋. 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)≥𝛼̸.
Then, (⋀_ℎ∈𝜁ℎ)(𝑥)≰𝛼 for all 𝑥∈𝑋 with 𝑔(𝑥)≥𝛼.
Hence (⋁_ℎ∈𝜁ℎ^′)(𝑥)≥𝑝̸ 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 ℎ^′₁,ℎ^′₂,…,ℎ^′_𝑛∈𝛽, then, since 𝜁 is 𝛼-level centered, there exists 𝑥∈𝑋 with 𝑔(𝑥)≥𝛼 and (⋀^𝑛_𝑖=1ℎ_𝑖)(𝑥)≥𝛼; hence (⋁^𝑛_𝑖=1ℎ^′_𝑖)(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 𝑓^′₁,𝑓^′₂,…,𝑓^′_𝑛∈𝜁, then there exists 𝑥∈𝑋 with 𝑔(𝑥)≥𝑝^′=𝛼 such that (⋁^𝑛_𝑖=1𝑓_𝑖)(𝑥)≤𝑝; hence (⋀^𝑛_𝑖=1𝑓^′_𝑖)(𝑥)≥𝛼.
By the hypothesis, there exists 𝑥∈𝑋 with 𝑔(𝑥)≥𝑝^′ with such that (⋀_𝑓∈𝛽𝑓^′)(𝑥)≥𝑝^′; hence (⋁_𝑓∈𝛽𝑓)(𝑥)≤𝑝 which yields a contradiction.

Definition 4.3. Let 𝜁⊆𝐿^𝑋, 𝑔∈𝐿^𝑋, and 𝛼∈𝐿.
𝜁 is called an 𝛼-level independent family of 𝑔 if and only if for any finite 𝑓₁,𝑓₂,…,𝑓_𝑛,𝑔₁,𝑔₂,…,𝑔_𝑚∈𝜁 there exists 𝑥∈𝑋 with 𝑔(𝑥)≥𝛼 such that (⋀^𝑛_𝑖=1𝑓_𝑖⋀⋀^𝑚_𝑖=1𝑔^′_𝑖)(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∈𝐿^𝑋. 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 of 𝑔, there is a partial inversement 𝛾 of 𝛽 which contains a finite 𝑝-level subcover of 𝑔.
Let 𝛾={𝑔_𝑖∶𝑖∈𝐽} and 𝛽={ℎ^′_𝑖∶𝑖∈𝐽, ℎ_𝑖∈𝜁}.
Then, there exists 𝑔₁,𝑔₂,…,𝑔_𝑛∈𝛾 such that (⋁^𝑛_𝑖=1𝑔_𝑖)(𝑥)≰𝑝 for all 𝑥∈𝑋 with 𝑔(𝑥)≥𝑝^′, where for each 𝑖∈{1,2,…,𝑛},𝑔_𝑖=ℎ_𝑖, or 𝑔_𝑖=ℎ^′_𝑖.
Let 𝑔₁=ℎ^′₁,𝑔₂=ℎ^′₂,…,𝑔_𝑘=ℎ^′_𝑘,𝑔_𝑘+1=ℎ_𝑘+1,𝑔_𝑘+2=ℎ_𝑘+2,…,𝑔_𝑛=ℎ_𝑛. Then 𝑔^′₁,𝑔^′₂,…,𝑔^′_𝑘,𝑔_𝑘+1,…,𝑔_𝑛∈𝜁. Since 𝜁 is an 𝛼-level independent family of 𝑔, there exists 𝑥∈𝑋 with 𝑔(𝑥)≥𝛼 such that (^𝑘_𝑖=1𝑔^′_𝑖^𝑛_𝑖=𝑘+1𝑔^′_𝑖)(𝑥)=(^𝑛_𝑖=1𝑔^′_𝑖)(𝑥)≥𝛼. (4.1) Hence, there exists 𝑥∈𝑋 with 𝑔(𝑥)≥𝑝^′ such that (⋁^𝑛_𝑖=1𝑔_𝑖)(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 𝑓^′₁,𝑓^′₂,…,𝑓^′_𝑘,𝑓^′_𝑘+1,…,𝑓^′_𝑛∈𝜁, then by the assumption, there exists 𝑥∈𝑋 with 𝑔(𝑥)≥𝑝^′ and (⋁^𝑘_𝑖=1𝑓_𝑖∨⋁^𝑛_𝑖=𝑘+1𝑓^′_𝑖)(𝑥)≤𝑝; hence (⋀^𝑘_𝑖=1𝑓^′_𝑖⋀⋀^𝑛_𝑖=𝑘+1𝑓_𝑖)(𝑥)≥𝛼.
By the hypothesis, there exists 𝑧∈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∈𝐿^𝑋. 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 Theorem 4.4.

Definition 4.6. Let 𝜉⊆𝐿^𝑋 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,…,𝑓_𝑘,𝑔₁,𝑔₂,…,𝑔_𝑛∈𝜉, there exists 𝑥∈𝐺 with (⋁^𝑘_𝑖=1𝑓_𝑖∨⋁^𝑛_𝑖=1𝑔^′_𝑖)(𝑥)≤𝑝.

Theorem 4.7. Let (𝑋,𝜏) be an 𝐿-fuzzy topological space and 𝑔∈𝐿^𝑋. 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)≥𝛼̸ 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,𝜔_𝐿(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^∗={𝜒_𝐴∶𝐴∈𝜁} is a family of closed 𝐿-fuzzy sets. Furthermore, 𝜁^∗ is an 𝛼-level independent family of 𝑋. In fact, if 𝜒_𝐴1,𝜒_𝐴2,…,𝜒_𝐴𝑛,𝜒_𝐵1,…,𝜒_𝐵𝑚∈𝜁^∗, then since 𝜁 is independent, ⋂^𝑛_𝑖=1𝐴_𝑖∩⋂^𝑚_𝑖=1𝐵^′_𝑖≠𝜙. Hence (⋀^𝑛_𝑖=1𝜒_𝐴𝑖⋀⋀^𝑚_𝑖=1𝜒^′_𝐵𝑖)=𝜒_⋂^𝑛_{𝑖=1𝐴𝑖∩⋂}^𝑚_{𝑖=1𝐵′𝑖}≠0. So, there is 𝑥∈𝑋 such that (⋀^𝑛_𝑖=1𝜒_𝐴𝑖⋀⋀^𝑚_𝑖=1𝜒^′_𝐵𝑖)(𝑥)=1≥𝛼. Since (𝑋,𝜔_𝐿(𝑇)) is fuzzy inversely compact, there exists 𝑧∈𝑋 with (⋀_𝐴∈𝜁𝜒_𝐴)(𝑧)≥𝛼; hence (⋀_𝐴∈𝜁𝜒_𝐴)(𝑧)=(𝜒_{⋂𝐴∈𝜁𝐴})(𝑧)=1 and therefore 𝑧∈⋂_𝐴∈𝜁𝐴. Hence (𝑋,𝑇) is inversely compact.
Necessity. Let 𝑝∈pr(𝐿) and 𝛽=(𝑓_𝑖)_𝑖∈𝐽 be a 𝑝-level open cover of 𝑋 consisting of basic open 𝐿-fuzzy sets in (𝑋,𝜔_𝐿(𝑇)). Then 𝑓_𝑖𝑒(𝑥)=^𝑖∈𝐿;𝑥∈𝐴_𝑖∈𝑇,0;𝑥∉𝐴_𝑖 (5.1) (for each 𝑖∈𝐽) and (⋁_𝑖∈𝐽𝑓_𝑖)(𝑥)≰𝑝 for all 𝑥∈𝑋.
Let 𝜁={𝐴_𝑖⊆𝑋∶∃𝑖∈𝐽 with 𝑒_𝑖≰𝑝 and 𝑓_𝑖∈𝛽}.
It is clear that 𝜁 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.
Let 𝜁^∗={𝐵_𝑖∶𝑖∈𝐽}. Then there exist 𝐵₁,𝐵₂,…,𝐵_𝑛∈𝜁^∗ such that 𝑋=⋃^𝑛_𝑖=1𝐵_𝑖, where 𝐵_𝑖=𝐴_𝑖 and 𝐵_𝑖=𝐴^′_𝑖 (for each 𝑖∈{1,2,…,𝑛}).
(i)If 𝐵_𝑖=𝐴_𝑖 for all 𝑖∈{1,2,…,𝑛}, then 𝛽^∗={𝑓₁,𝑓₂,…,𝑓_𝑛} is a finite partial inversement of 𝛽. 𝛽^∗ is also a 𝑝-level subcover of 𝑋 in (𝑋,𝜔_𝐿(𝑇)). Hence, (𝑋,𝜔_𝐿(𝑇)) is fuzzy inversely compact.(ii)Suppose that there exists 𝑖₀∈{1,2,…,𝑛} such that 𝐵_𝑖0=𝐴^′_𝑖0.
Let 𝛽^∗={𝑓₁,𝑓₂,…,𝑓_𝑖0−1,𝑓^′_𝑖0,…,𝑓^′_𝑛}. Obviously, 𝛽^∗ is a partial inversement of 𝛽. Furthermore, 𝛽^∗ is a 𝑝-level subcover of 𝑋 in (𝑋,𝜔_𝐿(𝑇)). In fact, let 𝑥∈𝑋. Since 𝑋=⋃^𝑛_𝑖=1𝐵_𝑖=𝐵_𝑖0∪⋃^𝑛_{𝑖=1,𝑖≠𝑖0}𝐵_𝑖⇒𝑥∈𝐵_𝑖0 or 𝑥∈⋃^𝑛_{𝑖=1,𝑖≠𝑖0}𝐵_𝑖⇒𝑥∈𝐴^′_𝑖0 or 𝑥∈⋃^𝑛_{𝑖=1,𝑖≠𝑖0}𝐴_𝑖.
(1^∘) If 𝑥∈𝐴^′_𝑖0, then 𝑓^′_𝑖0(𝑥)=1≰𝑝. Hence (𝑓^′_𝑖0(𝑥)∨⋁^𝑛_{𝑖=1,𝑖≠𝑖0}𝑓_𝑖)(𝑥)≰𝑝.
(2^∘) If 𝑥∉𝐴^′_𝑖0, then there is 𝑖∈{1,2,…,𝑛}−{𝑖₀} such that 𝑥∈𝐴_𝑖. Hence 𝑓_𝑖(𝑥)=𝑒_𝑖≰𝑝 and therefore (𝑓^′_𝑖0∨⋁^𝑛_{𝑖=1,𝑖≠𝑖0}𝑓_𝑖)(𝑥)≰𝑝. Consequently, 𝛽^∗ is 𝑝-level subcover of 𝑋 in (𝑋,𝜔_𝐿(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,𝜔_𝐿(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,𝜔_𝐿(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∈𝐿^𝑋 is fuzzy inversely compact and ℎ∈𝐿^𝑋 closed, then 𝑔⋀ℎ is also fuzzy inversely compact.

Proof. Let 𝑝∈pr(𝐿) and let 𝛽=(𝑓_𝑖)_𝑖∈𝐽⊆𝐿^𝑋 be a 𝑝-level open cover of 𝑔⋀ℎ; that is, (⋁_𝑖∈𝐽𝑓_𝑖)(𝑥)≰𝑝 for all 𝑥∈𝑋 with (𝑔⋀ℎ)(𝑥)≥𝑝^′. So 𝛽^∗=(𝑓_𝑖)_𝑖∈𝐽∪{ℎ^′} is 𝑝-level family of 𝑔. In fact, if ℎ(𝑥)≥𝑝^′⇒(_𝑖∈𝐽𝑓_𝑖)(𝑥)≰𝑝∀𝑥∈𝑋with(𝑔ℎ)(𝑥)≥𝑝^′⇒(_{𝑘∈𝛽∗}𝑘)(𝑥)=(_𝑖∈𝐽𝑓_𝑖)(𝑥)∨ℎ^′̸(𝑥)≰𝑝ℎ(𝑥)≥𝑝^′⇒ℎ^′(𝑥)≰𝑝⇒(_{𝑘∈𝛽∗}𝑘)(𝑥)≰𝑝. (5.2) Since 𝑔 is fuzzy inversely compact, 𝛽^∗ has a partial inversement 𝛾^∗ which contains a finite 𝑝-level subcover of 𝑔.
Let 𝛾^∗={𝑔_𝑖∶𝑖∈𝐽}∪{ℎ^′} and 𝛽^∗={𝑓_𝑖∶𝑖∈𝐽}∪{ℎ^′}.
Then there exists 𝑔₁,𝑔₂,…,𝑔_𝑛∈𝛾^∗ such that (⋁^𝑛_𝑖=1𝑔_𝑖∨ℎ^′)(𝑥)≰𝑝 for all 𝑥∈𝑋 with 𝑔(𝑥)≥𝑝^′ where for each 𝑖∈{1,2,…,𝑛},𝑔_𝑖=𝑓_𝑖 or 𝑔_𝑖=𝑓^′_𝑖.
So (⋁^𝑛_𝑖=1𝑔_𝑖)(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^′, 𝑔(𝑥)≥𝑝^′⇒(^𝑛_𝑖=1𝑔_𝑖∨ℎ^′)(𝑥)≰𝑝⇒(^𝑛_𝑖=1𝑔_𝑖)(𝑥)∨ℎ^′(𝑥)≰𝑝⇒(^𝑛_𝑖=1𝑔_𝑖)(𝑥)≰𝑝. (5.3)

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 Theorem 5.4.

Proposition 5.7. Let (𝑋,𝜏) be an 𝐿-fuzzy topological space and 𝑔,ℎ∈𝐿^𝑋 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 Theorem 5.4.

Theorem 5.8. Let (𝑋,𝜏) be an 𝐿-fuzzy topological space and 𝑔,ℎ∈𝐿^𝑋. If 𝑔 and ℎ are fuzzy inversely compact then 𝑔∨ℎ is fuzzy inversely compact as well.

Proof. Suppose that 𝑔∨ℎ is not fuzzy inverse compact.
Then, there exists an 𝛼-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).
Since 𝛼∈𝑀(𝐿) we have (𝑔∨ℎ)(𝑥)≥𝛼⇔𝑔(𝑥)≥𝛼 or ℎ(𝑥)≥𝛼.
Let 𝑓₁,𝑓₂,…,𝑓_𝑘,𝑓_𝑘+1,…,𝑓_𝑛∈𝜁. Since 𝜁 is an 𝛼-level independent family of 𝑔∨ℎ, there exists 𝑥∈𝑋 with (𝑔∨ℎ)(𝑥)≥𝛼 and (⋀^𝑘_𝑖=1𝑓_𝑖⋀⋀^𝑛_𝑖=𝑘+1𝑓^′_𝑖)(𝑥)≥𝛼. That is, there exists 𝑥∈𝑋 with 𝑔(𝑥)≥𝛼 or ℎ(𝑥)≥𝛼 such that (⋀^𝑘_𝑖=1𝑓_𝑖⋀⋀^𝑛_𝑖=𝑘+1𝑓^′_𝑖)(𝑥)≥𝛼.
Assume that 𝑔(𝑥)≥𝛼. Then, 𝜁 is 𝛼-level independent family of 𝑔 consisting of closed 𝐿-fuzzy sets. Furthermore, (⋀_𝑓∈𝜁𝑓)(𝑥)≥𝛼̸ 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∈𝐿^𝑋. 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 Theorem 5.8.

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∈𝐿^𝑋 is fuzzy inversely compact in (𝑋,𝜏), then 𝜑(𝑔)∈𝐿^𝑌 is fuzzy inversely compact in (𝑌,𝜏^∗), where 𝜑(𝑔)(𝑦)=⋁_{𝑥∈𝜑−1(𝑦)}𝑔(𝑥).

Proof. Let 𝛼∈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}.
Since 𝜑 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(ℎ_𝑘),𝜑⁻¹(ℎ_𝑘+1),…,𝜑⁻¹(ℎ_𝑛)∈𝜁^∗, then there exists 𝑦∈𝑌 with 𝜑(𝑔)(𝑦)≥𝛼 and (⋀^𝑘_𝑖=1ℎ_𝑖⋀⋀^𝑛_𝑖=𝑘+1ℎ^′_𝑖)(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, (⋀^𝑘_𝑖=1𝜑⁻¹(ℎ_𝑖)⋀⋀^𝑛_𝑖=𝑘+1𝜑⁻¹(ℎ^′_𝑖))(𝑥)=(⋀^𝑘_𝑖=1ℎ_𝑖⋀⋀^𝑛_𝑖=𝑘+1ℎ^′_𝑖)(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∈𝐿^𝑋 is fuzzy inversely countable compact (fuzzy inversely Lindelöf) in (𝑋,𝜏), then 𝜑(𝑔)∈𝐿^𝑌 is fuzzy inversely countably compact (fuzzy inversely Lindelöf) in (𝑌,𝜏^∗).

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