Abstract

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

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 -subsets were fuzzy, but the topology consisting of those open -subsets is still a crisp subset of . 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 -fuzzy topology in 1985. In an -fuzzy topology, open -subsets are not crisp subset, and topology comprising those open -subsets is an -subset of . In the setting of -fuzzy topologies, many researchers have also made plentiful investigations on the notion of compactness ([12–25]).

In [26], Li and Li introduced -topology on -subset and discussed some of its related properties. The new kind of topology is called an -topology which is a generalization of -topology. Furthermore, they introduced and characterized the concepts of -continuous function and -compactness by means of an inequality. As an extension of -topology, -fuzzy topology on an -subset is introduced in [27]. Kubiak-Šostak’s -fuzzy topology and -topology are special cases of -fuzzy topology. The -fuzzy compactness of -fuzzy topology is further studied. They proved that the union of two -fuzzy compact -subsets is also -fuzzy compact, and the intersection of an -fuzzy compact -subset and -closed subset is also -fuzzy compact. Moreover, they showed that -fuzzy compactness is an invariant under -fuzzy continuous functions.

In this study, we introduce and characterize the degree of countable -fuzzy compactness and the -Lindelöf property of an -subset in -fuzzy topology based on the concepts of pseudocomplement of -subsets and the implication operator. Since -fuzzy topology in the sense of Kubiak and Šostak is a special case of -fuzzy topology, the degrees of -fuzzy compactness and the -Lindelöf property are generalization of the corresponding degrees in -fuzzy topology.

2. Preliminaries

Throughout this study, refers to a completely distributive lattice with order-reversing involution [2, 28], to a nonempty set, and to the collection of all -subsets on . The greatest and the smallest elements in and are , , and , , respectively. For , , and , we have for each , and for each . The binary operation on defined by is called the implication operator. For more properties of the implication operator, we refer the reader to [29]. An -subset is said to be valuable if . The family of all valuable -subsets on is denoted by , i.e., . For all , we define the family by . In fact, represents the powerset of -subset . If and , then the restriction of on , i.e., such that , is called the relative -fuzzy mapping (briefly, -fuzzy mapping) from to , denoted by if . The inverse image of an -subset under is given by . Clearly, we have .

If and , the pseudocomplement of relative to [26, 27], denoted by , is defined by

The following proposition lists some properties of pseudocomplement operation .

Proposition 1 (see [26, 27]). For each , , and , we have(1).(2).(3).(4). The equation holds provided that .

Definition 1. Let . A function is an -fuzzy inclusion on , defined as , which is denoted by for simplicity instead of , i.e., . Obviously, if , then is an -fuzzy inclusion function in the sense of Šostak [17].
The following lemma gives an important property for the -fuzzy mapping .

Lemma 1 (see [26]). Let , , and be -fuzzy mapping from to , and . Then, for each , we haveBased on the properties of listed in Proposition 1, the previous equation can be rewritten as follows:

An -topological space [1, 2, 28] (briefly, -ts) is a pair such that is a subcollection of which contains and and is closed for any suprema and finite infima. Moreover, is called an -topology on . Furthermore, elements of are called open -subsets, and their complements are called closed -subsets. A mapping is said to be -continuous if and only if for each .

Definition 2 (see [10, 11, 30]). An -fuzzy topology on the set is the mapping , which meets the following three conditions: (T1) . (T2) , . (T3) , .The pair is called an -fuzzy topological space (briefly, -fts). Here, can be regarded as the degree to which is an open -subset or the degree of openness of . Similarly, can be regarded as the closeness degree of an -subset . A mapping between two -fts’s and is said to be -fuzzy continuous if and only if for each .
The concept of -topology on an -subset is introduced as follows:

Definition 3 (see [26]). Let . A relative -topology on an -subset is a subcollection on that satisfies the following conditions:(1) and for all .(2) for each , .(3) for each .The pair is called a relative -topological space on (briefly, -ts). The members of are called relative open -subsets (briefly, -open subset), and an -subset is called relative -closed (briefly, -closed subset) if and only if . The family of all -closed subsets with respect to is denoted by , i.e., . Let , , and , be two -ts’s. The relative fuzzy mapping is said to be -continuous if and only if for each . It is not difficult to verify that the -topology on degenerates to -topology if .

Theorem 1 (see [26]). For any -ts , the following statements are true for :(1) and for all .(2) for each , .(3) for each .

Definition 4 (see [27]). Let . An -fuzzy topology on is a mapping such that satisfying the following statements: (R1) , , . (R2) , , . (R3) , .The pair is called an -fuzzy topological space (briefly, -fts) on . For each , the values (respectively, ) can be regarded as the degree of openness (respectively, closeness) of relative to , respectively. Moreover, if (respectively, ), then is called an -open (respectively, -closed) subset. It is easy to verify that when , the -fuzzy topology on is reduced to -fuzzy topology in the sense of Kubiak [10] and Šostak [11], that is, the -fuzzy topology on is an extension of Kubiak–Šostak’s -fuzzy topology. Moreover, is an -ts, and is a mapping defined byThen, can be seen as a special -fts. In this sense, can also be regarded as -fts.

Theorem 2 (see [27]). For any and -fts on . The mapping defined by for each satisfies the following statements: (R1)^′, , ; (R2)^′, , ; (R3)^′, .

is called an -fuzzy cotopology (briefly, -cft) on , and the pair is called an -fuzzy cotopological space (briefly, -cfts).

Definition 5 (see [27]). Let , , and , are -fts’s on and , respectively. An -fuzzy mapping is said to be -fuzzy continuous if and only if , . Furthermore, if and are corresponding -cfts’s of and , respectively; then, is called an -fuzzy continuous if and only if , .

Definition 6 (see [27]). Let and be an -fts on . An -subset is said to be an -fuzzy compact with respect to if for each , the following inequality holds:

Theorem 3 (see [27]). In the case of , the following conclusions hold:(1), .(2)-fuzzy compactness degenerates into -fuzzy compactness.(3) is -fuzzy compact if and only if is -fuzzy compact.

Theorem 4 (see [27]). For any and an -ft on , we have following conclusions:(1)If , such that and are -fuzzy compacts with respect to , then is -fuzzy compact with respect to .(2)If , such that is an -fuzzy compact with respect to and is an -closed subset, then is an -fuzzy compact.

Definition 7. Let , be an -ft on , and . Then,is called the -fuzzy compactness degree of with respect to . Obviously, an -subset is -fuzzy compact in -ts if and only if .
Based on Definition 8 in [27], we can state the following theorem:

Theorem 5. For any , , and any mapping , letIf be an -fts, then is called the -fuzzy compactness degree of with respect to . Clearly, is the -fuzzy compact in -ts iff .

3. Measure of -Countable Compactness

Let . If , be an -ts on , and . Then, is countably -compact if and only if for each , it follows that . This implies that , i.e., is countably -compact if and only if for each collection , it follows that .

Definition 8. Let , be an -ft on , and . Then,is the degree of countable -compactness of with respect to . Clearly, is the countable -compact in -ts if and only if .Based on the properties of implication operation “” ([29]), we can state the following lemma:

Lemma 2. Let , be an -fts on , and . Then, if and only iffor each .

Theorem 6. Let , be an -fts on , and . Then,The following theorem is an intuitive result from the Definitions 7 and 8.

Theorem 7. Let , be an -fts on , and . Then, .

Theorem 8. Let , be an -fts on , and . Then, .