# Measurement of Countable Compactness and Lindelöf Property inRL-Fuzzy Topological Spaces

Authors: Xiongwei Zhang; Ibtesam Alshammari; A. Ghareeb
Journal: Complexity
 (2021)

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## Abstract

Based on the concepts of pseudocomplement ofL-subsets and the implication operator where L is a completely distributive lattice with order-reversing involution, the definition of countable RL-fuzzy compactness degree and the Lindelöf property degree of an L-subset in RL-fuzzy topology are introduced and characterized. Since L-fuzzy topology in the sense of Kubiak and Šostak is a special case of RL-fuzzy topology, the degrees of RL-fuzzy compactness and the Lindelöf property are generalizations of the corresponding degrees in L-fuzzy topology.


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## Body

## 1. Introduction

Combining with fuzzy set theory, Chang [1] introduced the concept of fuzzy topology together with the definition of compactness by means of open cover in 1968. Afterwards, several researchers have tried successfully to extend the theory of compactness from the crisp topology to fuzzy setting [2–8]. The disadvantage of Chang’s fuzzy topology is that the open L-subsets were fuzzy, but the topology consisting of those open L-subsets is still a crisp subset of LX. This prompted Höhle [9] to make the first attempt to fuzzifying the openness in 1980. Later on, Höhle’s attempt developed independently by Kubiak [10] and Šostak [11] to L-fuzzy topology in 1985. In an L-fuzzy topology, open L-subsets are not crisp subset, and topology comprising those open L-subsets is an L-subset of LX. In the setting of L-fuzzy topologies, many researchers have also made plentiful investigations on the notion of compactness ([12–25]).In [26], Li and Li introduced L-topology on L-subset and discussed some of its related properties. The new kind of topology is called an RL-topology which is a generalization of L-topology. Furthermore, they introduced and characterized the concepts of RL-continuous function and RL-compactness by means of an inequality. As an extension of RL-topology, RL-fuzzy topology on an L-subset is introduced in [27]. Kubiak-Šostak’s L-fuzzy topology and RL-topology are special cases of RL-fuzzy topology. The RL-fuzzy compactness of RL-fuzzy topology is further studied. They proved that the union of two RL-fuzzy compact L-subsets is also RL-fuzzy compact, and the intersection of an RL-fuzzy compact L-subset and RL-closed subset is also RL-fuzzy compact. Moreover, they showed that RL-fuzzy compactness is an invariant under RL-fuzzy continuous functions.In this study, we introduce and characterize the degree of countableRL-fuzzy compactness and the RL-Lindelöf property of an L-subset in RL-fuzzy topology based on the concepts of pseudocomplement of L-subsets and the implication operator. Since L-fuzzy topology in the sense of Kubiak and Šostak is a special case of RL-fuzzy topology, the degrees of RL-fuzzy compactness and the RL-Lindelöf property are generalization of the corresponding degrees in L-fuzzy topology.


## 2. Preliminaries

Throughout this study,L,∨,∧,′ refers to a completely distributive lattice with order-reversing involution [2, 28], X to a nonempty set, and LX to the collection of all L-subsets on X. The greatest and the smallest elements in L and LX are 1, 0, and 1X, 0X, respectively. For μ∈LX, ν∈LY, and f:X⟶Y, we have fL⟶μy=∨μx:fx=y for each y∈Y, and fL←νx=∨μx:fL⟶μ≤ν=νfx for each x∈X. The binary operation ↦ on L defined by a↦b=∨c∈L|a∧c≤b is called the implication operator. For more properties of the implication operator, we refer the reader to [29]. An L-subset λ∈LX is said to be valuable if λ≤λ′. The family of all valuable L-subsets on X is denoted by VXL, i.e., VXL=λ∈LX:λ≤λ′. For all λ∈VXL, we define the family FXLλ by FXLλ=μ∈LX:μ≤λ. In fact, FXLλ represents the powerset of L-subset λ∈LX. If λ∈VXL and μ∈VYL, then the restriction of fL⟶ on λ, i.e., fL⟶|λ:FXLλ⟶LY such that γ∈FXLA↦fL⟶γ, is called the relative L-fuzzy mapping (briefly, RL-fuzzy mapping) from λ to μ, denoted by fL,λ⟶:λ⟶μ if fL⟶λ≤μ. The inverse image of an L-subset ν∈FYLμ under fL,λ⟶ is given by fL,λ←ν=∨γ∈FXLλ:fL⟶γ≤ν. Clearly, we have fL,λ←ν=λ∧fL←ν.Ifλ∈VXL and μ∈ℱXLλ, the pseudocomplement of μ relative to λ [26, 27], denoted by Lλμ, is defined by(1)Lλμ=λ∧μ′,ifμ≠λ,0X,ifμ=λ.The following proposition lists some properties of pseudocomplement operationLλ.Proposition 1 (see [26, 27]).
For eachλ∈VXL, μ,ν∈ℱXLλ, and μi:i∈I⊆ℱXLλ, we have(1)
Lλμ=λ⇔μ≤λ′.(2)
μ≤ν⇒Lλν≤Lλμ.(3)
Lλ∧i∈Iμi=∨i∈ILλμi.(4)
Lλ∨i∈Iμi≤∧i∈ILλμi. The equation holds provided that ∨i∈Iμi≠λ.Definition 1.
Letλ∈VXL. A function ⋞:ℱXLλ×ℱXLλ⟶L is an RL-fuzzy inclusion on X, defined as ⋞μ,γ=∧x∈X〈Lλμx∨γx, which is denoted by μ⋞γ for simplicity instead of ⋞μ,γ, i.e., μ⋞γ=∧x∈X〈Lλμx∨γx. Obviously, if λ=1X, then ⋞ is an L-fuzzy inclusion function in the sense of Šostak [17].
The following lemma gives an important property for theRL-fuzzy mapping fL,λ.Lemma 1 (see [26]).
Letλ∈VXL, μ∈VYL, and fL,λ⟶:λ⟶μ be RL-fuzzy mapping from λ to μ, and γ∈ℱXLλ. Then, for each P⊆LX, we have(2)∨y∈YfL,λ⟶γy∧∧ξ∈Pξy=∨x∈Xγx∧∧ξ∈PfL,λ←ξx.
Based on the properties of〈Lλ listed in Proposition 1, the previous equation can be rewritten as follows:(3)∧y∈YLλfL,λ⟶γy∨∨ξ∈Pξy=∧x∈XLλγx∨∨ξ∈PfL,λ←ξx.AnL-topological space [1, 2, 28] (briefly, L-ts) is a pair X,τ such that τ is a subcollection of LX which contains 0X and 1X and is closed for any suprema and finite infima. Moreover, τ is called an L-topology on X. Furthermore, elements of τ are called open L-subsets, and their complements are called closed L-subsets. A mapping f:X,τ⟶Y,δ is said to be L-continuous if and only if fL←ν∈τ for each ν∈δ.Definition 2 (see [10, 11, 30]).
AnL-fuzzy topology on the set X is the mapping τ:LX⟶L, which meets the following three conditions:(T1)τ0X=τ1X=1.(T2)∀λ,μ∈LX, τλ∧μ≥τλ∧τμ.(T3)∀λi:i∈I⊆LX, τ∨i∈Iλi≥∧i∈Iτλi.
The pairX,τ is called an L-fuzzy topological space (briefly, L-fts). Here, τλ can be regarded as the degree to which λ is an open L-subset or the degree of openness of λ. Similarly, τ∗λ=τλ′ can be regarded as the closeness degree of an L-subset λ. A mapping f:X,τ⟶Y,δ between two L-fts’s X,τ and Y,δ is said to be L-fuzzy continuous if and only if τfL←ν≥δν for each ν∈LY.
The concept ofRL-topology ϰ on an L-subset λ is introduced as follows:Definition 3 (see [26]).
Letλ∈VXL. A relative L-topology ϰ on an L-subset λ is a subcollection on ℱXLλ that satisfies the following conditions:(1)
λ∈ϰ and μ∈ϰ for all μ≤λ′.(2)
μ1∧μ2∈ϰ for each μ1, μ2∈ϰ.(3)
∨i∈Iμi∈ϰ for each μi:i∈I⊆ϰ.
The pairλ,ϰ is called a relative L-topological space on λ (briefly, RL-ts). The members of ϰ are called relative open L-subsets (briefly, RL-open subset), and an L-subset μ is called relative L-closed (briefly, RL-closed subset) if and only if 〈Lλμ∈ϰ. The family of all RL-closed subsets with respect to ϰ is denoted by 〈Lλϰ, i.e., 〈Lλϰ=ν:〈Lλν∈ϰ. Let λ∈VXL, μ∈VYL, and λ,ϰ1, μ,ϰ2 be two RL-ts’s. The relative fuzzy mapping fL,λ⟶:λ⟶μ is said to be RL-continuous if and only if fL,λ←ν∈〈Lλϰ1 for each ν∈〈Lλϰ2. It is not difficult to verify that the RL-topology on λ degenerates to L-topology if λ=1X.Theorem 1 (see [26]).
For anyRL-ts λ,ϰ, the following statements are true for 〈Lλϰ:(1)
λ∈〈Lλϰ and μ∈〈Lλϰ for all μ≤λ′.(2)
μ1∨μ2∈〈Lλϰ for each μ1, μ2∈〈Lλϰ.(3)
∧i∈Iμi∈〈Lλϰ for each μi:i∈I⊆〈Lλϰ.Definition 4 (see [27]).
Letλ∈VXL. An RL-fuzzy topology on λ is a mapping ϰ:ℱXLλ⟶L such that ϰ satisfying the following statements:(R1)ϰλ=1, ∀μ≤λ′, ϰμ=1.(R2)∀μ1, μ2∈ℱXLλ, ϰμ1∧μ2≥ϰμ1∧ϰμ2.(R3)∀μi:i∈I⊆ℱXLλ, ϰ∨i∈Iμi≥∧i∈Iϰμi.
The pairλ,ϰ is called an RL-fuzzy topological space (briefly, RL-fts) on λ. For each μ∈ℱXLλ, the values ϰμ (respectively, ϰ〈Lλμ) can be regarded as the degree of openness (respectively, closeness) of μ relative to ϰ, respectively. Moreover, if ϰμ=1 (respectively, ϰ〈Lλμ=1), then μ is called an RL-open (respectively, RL-closed) subset. It is easy to verify that when λ=1X, the RL-fuzzy topology on λ is reduced to L-fuzzy topology in the sense of Kubiak [10] and Šostak [11], that is, the RL-fuzzy topology on λ is an extension of Kubiak–Šostak’s L-fuzzy topology. Moreover, λ,ϰ is an RL-ts, and χϰ:ℱXLλ⟶L is a mapping defined by(4)χϰμ=1,ifμ∈ϰ,0,ifμ∉ϰ,Then, λ,χϰ can be seen as a special RL-fts. In this sense, λ,ϰ can also be regarded as RL-fts.Theorem 2 (see [27]).
For anyλ∈VXL and RL-fts λ,ϰ on λ. The mapping 〈Lλϰ:ℱXLλ⟶L defined by 〈Lλϰμ=ϰ〈Lλμ for each μ∈ℱXLλ satisfies the following statements:(R1)′〈Lλϰλ=1, ∀μ≤λ′, 〈Lλϰμ=1;(R2)′∀μ1, μ2∈ℱXLλ, 〈Lλϰμ1∨μ2≥〈Lλϰμ1∧〈Lλϰμ2;(R3)′∀μi:i∈I⊆ℱXLλ, 〈Lλϰ∧i∈Iμi≥∧i∈I〈Lλϰμi.±b〈Lλκ is called an RL-fuzzy cotopology (briefly, RL-cft) on λ, and the pair λ,〈Lλϰ is called an RL-fuzzy cotopological space (briefly, RL-cfts).Definition 5 (see [27]).
Letλ∈VXL, μ∈VYL, and λ,ϰ1, μ,ϰ2 are RL-fts’s on λ and μ, respectively. An RL-fuzzy mapping fL,λ:λ⟶μ is said to be RL-fuzzy continuous if and only if ϰ1〈LλfL,λ←ν≥ϰ1〈Lμν, ∀ν∈ℱYLμ. Furthermore, if λ,〈Lλϰ1 and μ,〈Lμϰ2 are corresponding RL-cfts’s of λ,ϰ1 and μ,ϰ2, respectively; then, fL,λ⟶ is called an RL-fuzzy continuous if and only if 〈Lλϰ1fL,λ←ν≥〈Lμϰ2ν, ∀ν∈ℱXLμ.Definition 6 (see [27]).
Letλ∈VXL and λ,ϰ be an RL-fts on λ. An L-subset μ∈ℱXLλ is said to be an RL-fuzzy compact with respect to ϰ if for each P⊆ℱXLλ, the following inequality holds:(5)∨γ∈PϰLλγ∨∨x∈Xμx∧∧γ∈Pγx≥∧ℛ∈2P∨x∈Xμx∧∧γ∈ℛγx.Theorem 3 (see [27]).
In the case ofλ=1X, the following conclusions hold:(1)
〈Lλμ=μ′, μ∈ℱXLλ⇔μ∈LX.(2)
RL-fuzzy compactness degenerates into L-fuzzy compactness.(3)
μ is RL-fuzzy compact if and only if μ is L-fuzzy compact.Theorem 4 (see [27]).
For anyλ∈VXL and an RL-ft ϰ on λ, we have following conclusions:(1)
Ifμ1, μ2∈ℱXLλ such that μ1 and μ2 are RL-fuzzy compacts with respect to ϰ, then μ1∨μ2 is RL-fuzzy compact with respect to ϰ.(2)
Ifμ1, μ2∈ℱXLλ such that μ1 is an RL-fuzzy compact with respect to ϰ and μ2 is an RL-closed subset, then μ1∧μ2 is an RL-fuzzy compact.Definition 7.
Letλ∈VXL, ϰ be an RL-ft on λ, and μ∈ℱXLλ. Then,(6)cϰμ=∧U⊆ℱXLλϰU↦μ⋞∨U↦∨V∈2Uμ⋞∨V=∧U⊆ℱXLλ∧γ∈Uϰγ↦∧x∈XLλμx∨∨γ∈Uγx↦∨V∈2U∧x∈XLλμx∨∨γ∈Vγx,is called the RL-fuzzy compactness degree of μ with respect to ϰ. Obviously, an L-subset μ is RL-fuzzy compact in RL-ts ϰ if and only if cχϰμ=1.
Based on Definition8 in [27], we can state the following theorem:Theorem 5.
For anyλ∈VXLlimx⟶∞, μ∈ℱXLλ, and any mapping ϰ:ℱXLλ⟶L, let(7)cϰμ=∧P⊆ℱXLλ∧ℱ∈2P∨x∈Xμx∧∧γ∈ℱγx↦∨x∈Xμx∧∧γ∈Pγx↦LλϰP=∧P⊆ℱXLλ∧ℱ∈2P∨x∈Xμx∧∧γ∈ℱγx↦∨x∈Xμx∧∧γ∈Pγx↦∨γ∈PϰLλγ.If λ,ϰ be an RL-fts, then cϰμ is called the RL-fuzzy compactness degree of μ with respect to ϰ. Clearly, μ is the RL-fuzzy compact in RL-ts ϰ iff cχϰμ=1.


## 3. Measure ofRL-Countable Compactness

Letℱ˜XLλ=U⊆ℱXLλ|Uis a countable. If λ∈VXL, λ,ϰ be an RL-ts on λ, and ϰc=U⊆ϰ|Uis a countable family. Then, μ∈ℱXLλ is countably RL-compact if and only if for each U⊆ϰc, it follows that μ⋞∨U≤∨V∈2Uμ⋞∨V. This implies that μ⋞∨U≤∨V∈2Uμ⋞∨V=1, i.e., μ∈ℱXLλ is countably RL-compact if and only if for each collection U⊆ϰc, it follows that χϰU≤μ⋞∨U≤∨V∈2Uμ⋞∨V.Definition 8.
Letλ∈VXL, ϰ be an RL-ft on λ, and μ∈ℱXLλ. Then,(8)ccϰμ=∧U⊆ℱ˜XLλϰU≤μ⋞∨U≤∨V∈2Uμ⋞∨V=∧U⊆ℱ˜XLλ∧γ∈Uϰγ↦∧x∈XLλμx∨∨γ∈Uγx↦∨V∈2U∧x∈XLλμx∨∨γ∈Vγx,is the degree of countable RL-compactness of μ with respect to ϰ. Clearly, μ is the countable RL-compact in RL-ts λ,ϰ if and only if ccχϰμ=1.Based on the properties of implication operation “↦” ([29]), we can state the following lemma:Lemma 2.
Letλ∈VXL, λ,ϰ be an RL-fts on λ, and μ∈ℱXLλ. Then, α≤ccϰμ if and only if(9)ϰU∧μ⋞∨U∧α≤∨V∈2Uμ⋞∨V,for each U⊆ℱ˜XLλ.Theorem 6.
Letλ∈VXL, λ,ϰ be an RL-fts on λ, and μ∈ℱXLλ. Then,(10)ccϰμ=∨α∈L|ϰU∧μ⋞∨U∧α≤∨V∈2Uμ⋞∨V,∀U⊆ℱ˜XLλ.
The following theorem is an intuitive result from the Definitions7 and 8.Theorem 7.
Letλ∈VXL, λ,ϰ be an RL-fts on λ, and μ∈ℱXLλ. Then, cϰμ≤ccϰμ.Theorem 8.
Letλ∈VXL, λ,ϰ be an RL-fts on λ, and μ1,μ2∈ℱXLλ. Then, ccϰμ1∧μ2≥ccϰμ1∧Lλϰμ2.Proof.
Suppose thatccϰμ1∧Lλϰμ2≥α for any α∈L and U⊆ℱ˜XLλ, then Lλϰμ2≥α and ccϰμ1≥α. Consider W=U∪Lλμ2, then W⊆ℱ˜XLλ. By Lemma 2, ϰW∧μ1⋞∨W∧α≤∨V∈2Wμ1⋞∨V. Since ϰW=ϰU∧ϰLλμ2=ϰU∧Lλϰμ2, we have(11)μ1⋞∨W=∧x∈XLλμ1x∨∨γ∈Wγx=∧x∈XLλμ1x∨Lλμ2x∨∨γ∈Uγx=∧x∈XLλμ1x∧μ2x∨∨γ∈Uγx=Lλμ1∧μ2⋞∨U,∨V∈2Wμ1⋞∨V=∨V∈2Uμ1⋞∨V∨∨V∈2Uμ1⋞∨V∨Lλμ2=∨V∈2Uμ1⋞∨V∨∨V∈2Uμ1∧μ2⋞∨V=∨V∈2Uμ1∧μ2⋞∨V.
Therefore,ϰU∧Lλϰμ2∧μ1∧μ2⋞∨V∧α≤∨V∈2Uμ1∧μ2⋞∨V. Since Lλϰμ2≥α, we have ϰU∧μ1∧μ2⋞∨U∧α≤∨V∈2Uμ1∧μ2⋞∨V. Thus, by Lemma 2, ccϰμ1∧μ2≥α, and the proof is completed.Corollary 1.
Letλ∈VXL, λ,ϰ be an RL-fts on λ, and μ∈ℱXLλ. Then, ccϰμ≥cc1X∧Lλμ.Theorem 9.
Letλ∈VXL, λ,ϰ be an RL-fts on λ, and μ1,μ2∈ℱXLλ. Then, ccϰμ1∨μ2≥ccϰμ1∧ccϰμ2.Proof.
Suppose thatccϰμ1∧ccϰμ2≥α for each α∈L and U⊆ℱ˜XLλ, then ccϰμ1≥α and ccϰμ2≥α. By using Lemma 2, we have ϰU∧μ1⋞∨U∧α≤∨V∈2Uμ1⋞∨V and ϰU∧μ2⋞∨U∧α≤∨V∈2Uμ2⋞∨V. Therefore, ϰU∧μ1⋞∨U∧μ2⋞∨U∧α≤∨V∈2Uμ1⋞∨V∧∨V∈2Uμ2⋞∨V. By(12)μ1⋞∨U∧μ2⋞∨U=∧x∈XLλμ1x∨∨γ∈Uγx∧∧x∈XLλμ2x∨∨γ∈Uγx=∧x∈XLλμ1x∨∨γ∈Uγx∧Lλμ2x∨∨γ∈Uγx=∧x∈XLλμ1x∧Lλμ2x∨∨γ∈Uγx=∧x∈XLλμ1∨μ2x∨∨γ∈Uγx=μ1∨μ2⋞∨U,∨V∈2Uμ1⋞∨V∧∨V∈2Uμ2⋞∨V=∨V∈2Uμ1⋞∨V∧μ2⋞∨V=∨V∈2Uμ1∨μ2⋞∨V,we have that ϰU∧μ1∨μ2⋞∨U∧α≤∨V∈2Uμ1∨μ2⋞∨V. By using Lemma 2, we can obtain that ccϰμ1∨μ2≥α, and thus, the proof is completed.Theorem 10.
Letλ∈VXL, μ∈VYL, λ,ϰ1, and μ,ϰ2 be RL-fts. If fL,λ⟶:λ⟶μ is RL-fuzzy continuous, then ccϰ2fL,λ⟶ν≥ccϰ1ν.Proof.
Suppose thatccϰ1ν≥α for each α∈L, and let U⊆ℱ˜YLλ and ℛ=fL,λ←U=ζ:ζ=fL,λ←γ,γ∈U. Then, ℛ⊆ℱ˜XLλ. Since ccϰ1ν≥α, we have ϰ1ℛ∧ν⋞∨ℛ∧α≤∨S∈2ℛν⋞∨S. Since fL,λ⟶ is RL-fuzzy continuous, ϰ1fL,λ←γ≥ϰ2γ for each γ∈U, i.e., ϰ1ℛ=ϰ1fL,λ←U≥ϰ2U. Based on Lemma 1, we have(13)fL,λ⟶ν⋞∨U=∧y∈YLλfL,λ⟶νy∨∨γ∈Uγy=∧x∈XLλνx∨∨η∈ℛηx=ν⋞∨ℛ,∨S∈2ℛν⋞∨S=∨S∈2ℛ∧x∈XLλνx∨∨ξ∈Sξx=∨V∈2U∧y∈YLλfL,λ⟶νy∨∨ψ∈Vψy=∨V∈2UfL,λ⟶ν⋞∨V.
Therefore, we have the following inequality:(14)ϰ2U∧fL,λ⟶ν⋞∨U∧α≤ϰ1ℛ∧ν⋞∨ℛ∧α≤∨S∈2ℛν⋞∨S=∨V∈2UfL,λ⟶ν⋞∨V.
Thus, by Lemma2, ccϰ2fL,λ⟶ν≥α, and thus completed the proof.


## 4. Lindelöf Property Degree inRL-Fts’s

Definition 9.
Letλ∈VXL, λ,ϰ be an RL-fts on λ, and μ∈ℱXLλ. Then,(15)ℒϰμ=∧U⊆ℱXLλϰU≤μ⋞∨U≤∨V∈2Uμ⋞∨V=∧U⊆ℱXLλ∧γ∈Uϰγ↦∧x∈XLλμx∨∨γ∈Uγx↦∨V∈2U∧x∈XLλμx∨∨γ∈Vγx,is the degree to which μ has the RL-Lindelöf property with respect to ϰ. Clearly, μ has the RL-Lindelöf property in RL-ts ϰ if and only if ℒχϰμ=1.Lemma 3.
Letλ∈VXL, λ,ϰ be an RL-fts on λ, and μ∈ℱXLλ. Then, α≤ℒϰμ if and only if(16)ϰU∧μ⋞∨U∧α≤∨V∈2Uμ⋞∨V,for each U⊆ℱXLλ.Theorem 11.
Letλ∈VXL, λ,ϰ be an RL-fts on λ, and μ∈ℱXLλ. Then,(17)ℒϰμ=∨α∈L|ϰU∧μ⋞∨U∧α≤∨V∈2Uμ⋞∨V,∀U⊆ℱXLλ.Theorem 12.
Letλ∈VXL, λ,ϰ be an RL-fts on λ, and μ1,μ2∈ℱXLλ. Then, ℒϰμ1∧μ2≥ℒϰμ1∧Lλϰμ2.Corollary 2.
Letλ∈VXL, λ,ϰ be an RL-fts on λ, and μ∈ℱXLλ. Then, ℒϰμ≥ℒ1X∧Lλμ.Theorem 13.
Letλ∈VXL, λ,ϰ be an RL-fts on λ, and μ1,μ2∈ℱXLλ. Then, ℒϰμ1∨μ2≥ℒϰμ1∧ℒϰμ2.Theorem 14.
Letλ∈VXL, λ,ϰ be an RL-fts on λ, and μ∈ℱXLλ. Then, ℒϰμ∧ccϰμ≤cϰμ.Proof.
Supposeℒϰμ∧ccϰμ≥α for each α∈L and U⊆ℱXLλ. Then, ℒϰμ≥α and ccϰμ≥α. Thus, ϰU∧μ⋞∨U∧α≤∨V∈2Uμ⋞∨V. For any V∈2U, V⊆ℱ˜XLλ. Therefore, ϰU≤ϰV, and by using Lemma 2, we have ϰV∧μ⋞∨V∧α≤∨W∈2Vμ⋞∨W. Moreover, we have(18)ϰU∧μ⋞∨U∧α≤ϰU∧∨V∈2Uμ⋞∨V∧α≤∨V∈2UϰV∧μ⋞∨V∧α≤∨V∈2U∨W∈2Vμ⋞∨W≤∨V∈2U∨W∈2Uμ⋞∨W≤∨W∈2Vμ⋞∨W=∨V∈2Uμ⋞∨V.then, ϰU∧α≤μ⋞∨U↦∨V∈2Uμ⋞∨V for each U⊆ℱXLλ. Also, we have ϰU↦μ⋞∨U↦∨V∈2U≥α for each U⊆ℱXLλ. Hence, cϰμ≥α. The proof is completed.
The following corollary can be considered as the multivalue extension of the principal. Ifμ has the Lindelöf property, then μ is compact if and only if it is countable compact.Corollary 3.
Letλ∈VXL, λ,ϰ be an RL-fts on λ, and μ∈ℱXLλ. Then, ℒϰμ∧ccϰμ≤ℒϰμ∧cϰμ.


## 5. Conclusion

In this paper, we introduced and characterized the countableRL-fuzzy compactness degree and Lindelöf property degree of an L-subset in RL-fuzzy topology based on pseudocomplement of L-subsets and an implication operator. Since if λ=1X, the RL-fuzzy topology on λ is reduced to L-fuzzy topology in the sense of Kubiak and Šostak; the degrees of RL-fuzzy compactness and Lindelöf property are generalizations of the corresponding degrees in L-fuzzy topology.



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*Source: 6627372-2021-02-08.xml*