Research Article | Open Access

Ting Yang, Sheng-Gang Li, William Zhu, Xiao-Fei Yang, Ahmed Mostafa Khalil, "Convergence Classes of -Filters in -Fuzzy Topological Spaces", *Journal of Mathematics*, vol. 2021, Article ID 8246173, 10 pages, 2021. https://doi.org/10.1155/2021/8246173

# Convergence Classes of -Filters in -Fuzzy Topological Spaces

**Academic Editor:**Kenan Yildirim

#### Abstract

An -fuzzy topological convergence structure on a set is a mapping which defines a degree in for any -filter (of crisp degree) on to be convergent to a molecule in . By means of -fuzzy topological neighborhood operators, we show that the category of -fuzzy topological convergence spaces is isomorphic to the category of -fuzzy topological spaces. Moreover, two characterizations of -topological spaces are presented and the relationship with other convergence spaces is concretely constructed.

#### 1. Introduction

A convergence space of filters (as a generalization of a topological space based on the concept of convergence of filters as fundamental) is a pair , where is a set, is a subset of , and is the set of filters on . The pair means that is said to converge to . The convergence theory of filters is an important part in topology. It was proved that the category Conv of convergence spaces is a quasitopos which may be thought of as a nice category of spaces that includes Top (the category topological spaces) as a full subcategory (see [1â€“3] for details).

The corresponding convergence theory of filters in the fuzzy setting is also studied by many authors. E. Lowen and R. Lowen [4] gave a one-to-one correspondence between the set of limit functions on and the set of stratified -topologies on based on -filters (filters in the lattice [5]). JÃ¤ger [6] introduced another fuzzy convergence structure (called stratified -generalized convergence structure for , a frame which was extended to the case of complete residual lattices by Yao [7] later on). Here, the degree of convergence of a stratified -fuzzy filter (called also stratified -filter in [8]) is from but it converges to a crisp point.

JÃ¤ger also gave some kinds of characterizations of stratified -topological spaces (see [6, 9â€“11]). With the help of JÃ¤gerâ€™s work, Li [12] proved that there is a one-to-one correspondence between the set of all convergence functions of specific -fuzzy filters on and the set of all specific -topologies on for , a complete residual lattice. Following papers [13, 14], GÃ¼loÇ§lu and Ã‡oker [15] proved that there exists a one-to-one correspondence between the set of -fuzzy topological convergence structures on and the set of -fuzzy topologies on . As a generalization, Pang and Fang [16] proved that there is a one-to-one correspondence between the set of topological -fuzzy Q-convergence structures on and the set of -fuzzy topologies on when is a completely distributive complete lattice with an order-reversing involution â€². Here, the -fuzzy filter converges to a fuzzy point but the degree of convergence is from . Furthermore, Pang [17, 18] discussed categorical properties in some fuzzy convergence spaces, such as -fuzzifying convergence spaces and stratified -convergence tower spaces.

In [19], the convergence theory of molecular nets in -fuzzy topological spaces was discussed. As a generalization, Yao [20] presented a definition of -fuzzy nets and established a Mooreâ€“Smith convergence in -fuzzy topology. It is well known that every molecular net can induce an -filter (a filter in lattice ) and an -filter can induce a molecular net (see [21]). In addition, the degree of fuzzy topology, openness degree, and quotient degree [22] are interesting. Taking these in mind, we will use -filters different from that of [6, 8, 16] to give some axioms of filter-theoretical convergence classes in -fuzzy topological spaces. Such a convergence class on a set will be called an -fuzzy topological convergence structure, where the -filter converges to a molecular in and the degree of convergence is an element in . This kind of convergence class is proved to have nice properties.

The present paper is arranged as follows. The rest of this section contains some basic definitions and notions which will be used in this paper. In Section 2, we prove that there exists a one-to-one correspondence between -fuzzy topological convergence structures on a set and -fuzzy topologies on . In Section 3, we give two approaches to -fuzzy topological convergence structures. In Section 4, we discuss the relation between -fuzzy topological convergence structures and Liâ€™s structures in [12].

#### 2. Preliminaries

Now, we review some basic notions and results which will be used in this paper. Unless other explanations are given, always stands for a complete lattice with a smallest element 0 and a largest element 1 in this paper. Obviously, for every set , (the set of all -subsets of ) is also a complete lattice with the pointwise order. We will denote the -set taking constant value on by , and we write for each (, , and can be defined analogously for ). For the sake of convenience, we denote the orders of and and their restrictions by the same symbol . is called a join irreducible element if, for any finite subset satisfying (the supremum of ), there exists a such that . is called a coprime element if, for any finite subset satisfying , there exists a such that . is called a prime element of iff it is a coprime element of , where denotes the opposite lattice of . The set of all join irreducible elements (resp., all coprime elements and all prime elements) of will be denoted by (resp., Copr () and Pr ()). Clearly, Copr, where is the -subset taking value at and 0 elsewhere, and we call the members of -points. It is well known that if is a distributive lattice. is called a completely distributive complete lattice if it satisfies the following completely distributive laws:(i)(CD1) (ii)(CD2)

It is also well known that when is a completely distributive complete lattice, where (, , and can be defined analogously for ), and is called the wedge-below relation on which is defined as follows: iff for any satisfying , there exists an such that ( can be defined analogously).

An order-reversing involution â€² on is a self-map on such that, for any , the following hold: (1) implies ; (2) . The following hold for any subset :(1)(2)

For each mapping , we have a mapping (called -forward powerset operator) which is defined by . The right adjoint to (called -backward powerset operator) is denoted by (therefore, the pair is a Galois connection on and ). It can be easily verified that preserves arbitrary supremal, preserves arbitrary supremal and arbitrary infimal (and also complements if has an order-reversing involution â€²), and .

For other undefined notions, refer to [8, 23].

#### 3. Relationship between -Fuzzy Topological Spaces and -Fuzzy Topological Convergence Spaces

In this section, we assume that (resp., ) is a completely distributive complete lattice with an order-reversing involution â€² (resp., ); in this case, , , and . Our main task is to give a one-to-one correspondence between -fuzzy topological convergence structures (resp., -fuzzy principle convergence structures) on a set and -fuzzy topologies (resp., -fuzzy preinterior operators) on , which gives rise to an isomorphism between the category of -fuzzy topological convergence spaces (resp., the category of -fuzzy principle convergence spaces) and the category of -fuzzy topological spaces (resp., the category of -fuzzy preinterior spaces).

*Definition 1 (See [13, 14]). *(1)(See [13, 14]). An -fuzzy filter on is a mapping , which satisfies the following conditions:(i)(FF1) , (ii)(FF2) If and , then (iii)(FF3) (iv)An -fuzzy filter is said to be stratified [8] if it satisfies the following condition: (Fs) .â€‰The set of all -fuzzy filters (resp., stratified -fuzzy filters) on is denoted as (resp., ). Apparently, the mapping , defined by , is a stratified -fuzzy filter ().(2)(See [5]). An -filter on is a family which satisfies the following conditions:(i)(F1) , (ii)(F2) If and , then (iii)(F3) If , then â€‰The set of all -filters on is denoted as (where ). Obviously, every -filter may be looked upon as an -fuzzy filter.

*Remark 1. *Apparently, . Moreover, for every mapping and every -filter on , defineand then and , where .

*Definition 2 (see [24]). *(1)An -fuzzy topology on *X* is a mapping , which satisfies the following conditions:(i)(LMFT1) (ii)(LMFT2) (iii)(LMFT3) â€‰In this case, can be interpreted as the degree for to be an open set , and the pair is called an -fuzzy topological space. An -fuzzy topology is also called -topology.(2)A continuous mapping from one -fuzzy topological space to another -fuzzy topological space is a mapping , which satisfies . The category of -fuzzy topological spaces and continuous mappings between them is denoted by -FTop.

*Definition 3 (see [25]). *(1)An -fuzzy neighborhood operator on a set is a mapping , which satisfies the following conditions:(i)(LMFN1) , (ii)(LMFN2) (iii)(LMFN3) (iv)(LMFN4) â€‰In this case, is called an -fuzzy neighborhood space.(2)A mapping (where and are both -fuzzy neighborhood spaces) is said to be continuous if holds for any and any . The category of -fuzzy neighborhood spaces and continuous mappings between them will be denoted by -**FNS.** In [25], Shi proved that -**FNS** is isomorphic to -**FTop.** Furthermore, it is easily proved that -**FNS** is isomorphic to -**FTop.**

*Definition 4. *(1)An -fuzzy preinterior operator on is a mapping , which satisfies the following conditions:(i)(LMPI1) (ii)(LMPI2) (iii)(LMPI3) â€‰In this case, is called an -fuzzy preinterior space.(2)A mapping (where and are both -fuzzy preinterior spaces) is said to be continuous if holds for each and each . The category of -fuzzy preinterior spaces and continuous mappings between them will be denoted by -**FPIS.**

*Definition 5. *(1)An -fuzzy convergence structure on is a mapping , which satisfies the following conditions:(i)(LM1) (ii)(LM2) If , then â€‰In this case, is called an -fuzzy convergence space.(2)A mapping (where and are both -fuzzy convergence spaces) is said to be continuous if for any and any . The category of -fuzzy convergence spaces and continuous mappings between them will be denoted by -**FCS.**(3)For an -fuzzy convergence space , we define a mapping as follows:

*Definition 6. *(1)An -fuzzy convergence structure on is said to be an -fuzzy principle convergence structure if it satisfies the following condition:â€‰(LM3)â€‰In this case, is called an -fuzzy pretopological convergence space.(2)The category of -fuzzy pretopological convergence spaces and continuous mappings (see Definition 5) between them will be denoted by -**FPCS.**

*Definition 7. *(1)An -fuzzy topological convergence structure on is an -fuzzy principle convergence structure , which satisfies the following condition:â€‰(LM4)â€‰In this case, is called an -fuzzy topological convergence space.(2)The category of -fuzzy topological convergence spaces and continuous mappings (see Definition 5) between them will be denoted by -**FTCS.**In the rest of this section, we will show that -**FNS** is isomorphic to -**FTCS** (thus, -**FTop** is isomorphic to -**FTCS** by [25]) and that -**FPIS** is isomorphic to -**FPCS**.

Proposition 1. *For an -fuzzy topological convergence structure , the mapping is an -fuzzy neighborhood operator on .*

*Proof. *Since , we have . is an -filter on and , by (LM1). Thus, (LMFN1) is true. Again for each and each , we have since . (LMFN2) is also true.

Let and . Then, holds. Thus, by definition of ,which means that (LMFN3) is true. (LMFN4) follows from (LM4).

Proposition 2. *For each -fuzzy neighborhood operator on , the mapping , defined byis an -fuzzy topological convergence structure on .*

*Proof. *For any and any , we have by (LMFN2). Hence, , which means that (LM1) is true.

Let and . It can be easily checked that . Thus, for any , which means that (LM2) is true.

Since satisfies (LM1) and (LM2), the mapping is well-defined (see Definition 5). In order to prove (LM3), it suffices to prove the equality by definition of . By Definition 5 (3),For each satisfying , we have . As is arbitrary, the inequality holds. To prove the other inequality , let and . As is a completely distributive complete lattice,First, we show that , where . As is obvious, we only need to show that is an -filter.(i)(F1) As and , , and .(ii)(F2) Assume that and ; then . Further, , so we have , and thus, .(iii)(F3) Let . Then we declare that (i.e., ). Otherwise, or since , which is a contradiction.Next, it follows from that , and thus, (particularly, ). As is arbitrary and is a completely distributive complete lattice, . As is arbitrary, .

Since and is an -fuzzy neighborhood operator, (LM4) holds.

Proposition 3. *If is an - FNS morphism, then is an -FTCS morphism.*

*Proof. *For any and any , as is an -FNS morphism, . Further, is an order-reversing mapping; . It follows thatTherefore, is an -**FTCS** morphism.

Proposition 4. *If is an - FTCS morphism, then is an -FNS morphism.*

*Proof. *For each and , as is an -**FTCS** morphism, . Further, is an order-reversing mapping; . For all , it follows thatThe first inequality holds since . Therefore, is an -**FNS** morphism.

Proposition 5. *For any -fuzzy topological convergence structure on , one has .*

*Proof. *Let and . We need to prove . By Proposition 2 and Definition 5,For each with , we haveand then the inequality holds since is arbitrary. To show the other inequality , put and . As is a completely distributive complete lattice,It suffices to proveFor each , by (LM3), we have . As , from (LM2), we have , which meansBy Propositions 1â€“5, we have the following.

Theorem 1. (1)*-FNS is isomorphic to -FTCS.*â€‰*Similar to [25], it is easy to check that -FNS is isomorphic to -FTop*(2)*Thus -FTop is isomorphic to -FTCS.*

*Remark 2. *(1)If is an -fuzzy topological convergence structure on , then it satisfies the following: if , then ().(2)From Proposition 5, the following is true due to (LM3):(3)Now we turn our attention to the Mooreâ€“Smith convergence theory; for basic notions, refer to [19, 21, 23].For a molecule net (, and is a directed set), we define an -filter associated with the net as follows: . Conversely, for an -filter , define a molecule net , where is a directed set, on which the order is equipped with the relation : if and only if , and the mapping is .

Now define convergence of asIn an -fuzzy topological space , the following hold. The proof is simple and is missing:(i)(1) for each molecule and .(ii)(2) for each and .(iii)(4) In [16], Pang and Fang proposed the concept of topological -fuzzy Q-convergence spaces (the corresponding category is denoted by -QFTCS). In these convergence spaces, the value of an -fuzzy filter converging to a fuzzy point is from . Thus, Pang and Fangâ€™s spaces are different from ours. But -QFTCS is isomorphic to -FTCS when is a completely distributive lattice with an order-reversing involution.(iv)(5) For a given set , let be a set of all -fuzzy topological convergence structures on , and the relation on is defined by if and only if .The set of all -fuzzy neighborhood operators on is written as , and the relation on is defined by if and only if .

From Propositions 2 and 5, we have and for each , for each . This implies that there exists a bijection between and . Similar to paper [26], it is easily checked that and are complete lattices and they are isomorphic.

By Theorem 1 (1), there exists an isomorphic functor between -**FNS** and -**FTCS**. Furthermore, the restriction of this functor to is an order isomorphism.

Now we prove that -**FPIS** is isomorphic to -**FPCS**. Since its proof is similar to Theorem 1, we only give some propositions as follows.

Proposition 6. *For each -fuzzy preinterior operator on , the mapping , defined byis an -fuzzy principal convergence structure on .*

Proposition 7. *For each -fuzzy principal convergence structure on , the mapping , defined byis an -fuzzy preinterior operator on .*

Proposition 8. *For any -fuzzy principal convergence structure on , one has .*

Proposition 9. *For any -fuzzy preinterior operator on , one has .*

Proposition 10. *If is an -FPIS morphism, then *