Soft Open Bases and a Novel Construction of Soft Topologies from Bases for Topologies — TEI-tagged view

José Carlos R. AlcantudpersName (2020date). “Soft Open Bases and a Novel Construction of Soft Topologies from Bases for Topologies.” Mathematicstitle·j.

Object of study — soft topologies - a topology-like structure on the collection of all soft setsterm over a fixed set of alternatives - together with the general procedures that generate them from crisp topologies or from bases for topologies

Research question — how to construct soft topologies from standard (crisp) topologies or their bases, what a soft counterpart of topological separabilityterm should be, and when the generated soft topologyterm is soft second-countableterm·def

Contribution — extends the known topology-to-soft-topology construction to a family of topologies and to a family of bases (one per attribute), proposes two soft-separability notions, characterises soft second-countability, and exhibits the first non-trivial soft second-countableterm·def soft topologies

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Soft Open Basesterm and a Novel Construction of Soft Topologies from Bases for Topologies Abstract Abstract Soft topologyterm studies a structure on the collection of all soft setsterm on a given set of alternatives (the relevant attributes being fixed). It is directly inspired by the axioms of a topological space. This paper contributes to the theoretical bases of soft topologyterm in various ways. We extend a general construction of soft topologies from topologies on the set of alternatives in two different directions. An extensive discussion with criteria about what a soft counterpart of “topological separabilityterm” should satisfy is also given. The interactions of the properties that arise with separabilityterm, and of second-countability and its soft counterpart, are studied under the general mechanisms that generate soft topological spaces. The first non-trivial examples of soft second-countableterm·def soft topological spaces are produced as a consequence. Keywords: soft open baseterm; soft topologyterm; topology; separabilityterm; second countability axiom 1. Introduction Soft topologyterm stands at the junction of soft setterm theory [1] and topology [2,3]. It is concerned with a structure on the set of all soft setsterm, and is inspired by the standard axioms of a topological space.

Soft setsterm were introduced to describe a universe of alternatives by a set of relevant attributes. Soft setterm theory has been a thriving field of research and interaction with other disciplines ever since its establishment in 1999. Specifically, the field of soft topologyterm was independently set up by the works of Çağman, Karataş and Enginoglu [4] and Shabir and Naz [5]. Although many sudies followed their directions, the growth of the literature has been irregular and there is a scant collection of keystone articles and results. We acknowledge the foremost role of [6,7,8,9,10,11] among others.

This article has several objectives. Its driving force is the construction of soft topologies on the set of alternatives from elements pertaining to standard topology theory (either a collection of topologies, or bases of topologies). We also propose some axioms of ‘soft separabilityterm’ and study their behavior and the theoretical properties of soft second-countability.

Let us give a more precise list of the research goals of this paper.

First, we improve a known process that produces a soft topologyterm from one topology by the recourse to a family of topologies (one for each attribute). Despite its generality, we will show that this procedure interacts nicely with interesting axioms.

Secondly, we perform a specific analysis of what ‘soft separabilityterm’ should mean, as an extension of the usual concept of topological separabilityterm. Three criteria for the suitability of the soft counterpart(s) of topological separabilityterm are proposed. Then, two related concepts are put forward. They rely on completely different approaches to the idea of ‘soft separabilityterm’, however, they are not independent. Their relationship with topological separabilityterm is analyzed through the (improved) process that passes topologies on to soft topologies. Soft second-countability, which is quite indisputable as an axiom in soft topologyterm, is strictly stronger than these concepts, and we prove by example that it is indeed different from them.

From a theoretical standpoint, we prove a characterization of soft second-countableterm·def topologies that are generated by several topologies from the aforementioned procedure. When the set of attributes is at most countable, soft second-countability is equivalent to the fact that all the topologies associated with the attributes are second-countableterm.

Last but not least, we introduce a practical new procedure that produces soft topologies and depends upon a family of bases for topologies (one for each attribute). It allows us to visualize soft topologies in a simpler manner because bases are simpler than topologies and the same is true in the field of soft topologies. Another advantage is that it is a very natural way for the analysis of soft second-countability. Actually, with the help of this construction we put forward several non-trivial classes of (countable and uncountable) soft topological spaces with the soft second-countability property.

The remainder of this paper is organized as follows. Section 2 briefly recalls some basic notions concerning soft setsterm and soft topologies. We also show how a family of topologies (one for each attribute) can be used to generate a soft topologyterm. Section 3 discusses the elusive concept of ‘soft separabilityterm’. Here we make some new proposals and discuss why they are nice candidates for being accepted as suitable extensions of the concept of separabilityterm in topology. Section 4 addresses our characterization of soft second-countability of soft topologies generated by families of topologies. Section 5 introduces a novel construction of soft topologies from bases for a crisp topologyterm. Finally, the last section summarizes this study and suggests possible future works.

2. Preliminaries Henceforth, 𝑃 ( 𝑋 ) denotes the set of parts of a set X, i.e., 𝑃 ( 𝑋 ) is the set formed by all the subsets of X. In this section, we briefly recall the rudiments of soft setterm theory and soft topologyterm. To emphasize the distinction between topologies and soft topologies, we sometimes use the term “crisp topologyterm” to refer to a standard topology on a set. For the basic notions in (crisp) topology that we use, the reader can refer to Munkres [2] and Willard [3].

In this paper, X is a fixed nonempty set, often called a universe of discourse, and E is a set of attributes. Unless otherwise stated, henceforth X and E are unrestricted sets, i.e., they are possibly very large. We underline this feature because in almost all examples in the literature about soft topologies, both X and E are finite.

2.1. Soft Setsterm and Their Operations According to [1], a soft setterm on X is a pair ( 𝐹 , 𝐸 ) where F is a mapping 𝐹 : 𝐸 ⟶ 𝑃 ( 𝑋 ) . It is assumed that set E consists of all relevant attributes. Thus, a soft setterm over X is regarded as a parameterized family of subsets of X. The collection of all soft setsterm on X with attributes E is denoted as 𝑆 𝑆 𝐸 ( 𝑋 ) , or simply 𝑆 𝑆 ( 𝑋 ) when E is common knowledge.

When 𝑒 ∈ 𝐸 , the subset 𝐹 ( 𝑒 ) ⊆ 𝑋 is called the set of e-approximate elements of X, or alternatively, the subset of X approximated by e.

Examples include the null 𝛷 and the absolute 𝑋 ˜ soft setsterm on X. They satisfy 𝛷 ( 𝑒 ) = ⌀ for each 𝑒 ∈ 𝐸 , and 𝑋 ˜ ( 𝑒 ) = 𝑋 for each 𝑒 ∈ 𝐸 , respectively.

Soft setsterm on X with 𝐹 ( 𝑒 ) finite, resp., countable, for each 𝑒 ∈ 𝐸 are called finite, resp., countable, soft setsterm on X [8,9].

Soft pointsterm are also special soft setsterm. Two concepts have appeared in the literature under the same term [8,11], but we only need one of them. Thus, in this paper, soft pointsterm are soft setsterm ( 𝐹 , 𝐸 ) for which there exists 𝑥 ∈ 𝑋 such that 𝐹 ( 𝑒 ) = { 𝑥 } for all 𝑒 ∈ 𝐸 , and we denote by ( { 𝑥 } , 𝐸 ) this soft pointterm [11]. Note that in [8] and elsewhere, the term soft pointsterm has also been used for soft setsterm ( 𝐹 , 𝐸 ) for which there exists 𝑥 ∈ 𝑋 and 𝑒 ∈ 𝐸 such that 𝐹 ( 𝑒 ) = { 𝑥 } and 𝐹 ( 𝑒 ′ ) = ⌀ for all 𝑒 ′ ∈ 𝐸 ∖ { 𝑒 } .

Two notations make it easier to operate with soft setsterm.

Unions, intersections, inclusions and complements are standard set-theoretic operations on soft setsterm [12]. Unions and intersections of two soft setsterm on a common set X are defined as follows (and their extensions to arbitrary collections of soft setsterm are immediate): when ( 𝐹 1 , 𝐸 ) , ( 𝐹 2 , 𝐸 ) , ( 𝐹 1 , 𝐸 ) ⊔ ( 𝐹 2 , 𝐸 ) is ( 𝐹 3 , 𝐸 ) ∈ 𝑆 𝑆 𝐸 ( 𝑋 ) such that 𝐹 3 ( 𝑒 ) = 𝐹 1 ( 𝑒 ) ∪ 𝐹 2 ( 𝑒 ) for each 𝑒 ∈ 𝐸 , and ( 𝐹 1 , 𝐸 ) ⊓ ( 𝐹 2 , 𝐸 ) is ( 𝐹 4 , 𝐸 ) ∈ 𝑆 𝑆 𝐸 ( 𝑋 ) such that 𝐹 4 ( 𝑒 ) = 𝐹 1 ( 𝑒 ) ∩ 𝐹 2 ( 𝑒 ) for each 𝑒 ∈ 𝐸 .

We write ( 𝐹 1 , 𝐸 ) ⊑ ( 𝐹 2 , 𝐸 ) when 𝐹 1 ( 𝑒 ) ⊆ 𝐹 2 ( 𝑒 ) for each 𝑒 ∈ 𝐸 .

Finally, the complement of ( 𝐹 , 𝐸 ) ∈ 𝑆 𝑆 𝐸 ( 𝑋 ) is ( 𝐹 , 𝐸 ) 𝑐 = ( 𝐹 𝑐 , 𝐸 ) ∈ 𝑆 𝑆 𝐸 ( 𝑋 ) with the property that 𝐹 𝑐 ( 𝑒 ) = 𝑋 ∖ 𝐹 ( 𝑒 ) for each 𝑒 ∈ 𝐸 . In the tabular representation, the complement swaps 0’s and 1’s elsewhere.

Aygünoğlu and Aygün [7], Theorem 2.1 summarize results on the behavior of these operators from [12,13]. They endow 𝑆 𝑆 𝐸 ( 𝑋 ) with the structure of a Boolean lattice.

2.2. Soft Topologies, Soft Basesterm and Soft Open Basesterm The main concept in this section was independently proposed by Çağman, Karataş and Enginoglu [4] and Shabir and Naz [5].

([4,5]). A soft topologyterm τ on X is defined as a collection of soft setsterm on X, 𝜏 ∈ 𝑃 ( 𝑆 𝑆 𝐸 ( 𝑋 ) ) , with the following properties: (1) 𝛷 , 𝑋 ˜ ∈ 𝜏 ; (2) the union of soft setsterm in τ belongs to τ; and (3) the intersection of a finite number of soft setsterm in τ belongs to τ.

In this definition, union means ⊔ and intersection means ⊓.

The literature has produced few examples of soft topologies. Beyond the trivial cases ( 𝜏 𝑖 𝑑 = { 𝛷 , 𝑋 ˜ } , the indiscrete soft topologyterm, and 𝜏 𝑑 = 𝑆 𝑆 ( 𝑋 ) , the discrete soft topologyterm [5], Definitions 21 and 22) and some specific examples, the only general construction we are aware of appeared in Terepeta [11].

([11]). Suppose that Σ is a crisp topologyterm on X.

The soft topologyterm on X generated by Σ, 𝜏 ( 𝛴 ) , is the collection of all soft setsterm ( 𝐹 , 𝐸 ) on X for which every 𝐹 ( 𝑒 ) lies in Σ. Formally: 𝜏 ( 𝛴 ) = { ( 𝑒 , 𝐹 ( 𝑒 ) ) : 𝑒 ∈ 𝐸 } ∈ 𝑆 𝑆 𝐸 ( 𝑋 ) 𝑠 𝑢 𝑐 ℎ 𝑡 ℎ 𝑎 𝑡 𝐹 ( 𝑒 ) ∈ 𝛴 𝑓 𝑜 𝑟 𝑒 𝑎 𝑐 ℎ 𝑒 ∈ 𝐸 . (1) The single set soft topologyterm on X generated by Σ, 𝜏 ^ ( 𝛴 ) , is the collection of all soft setsterm ( 𝐹 , 𝐸 ) on X for which there is 𝑇 ∈ 𝛴 such that every 𝐹 ( 𝑒 ) coincides with T. Formally: 𝜏 ^ ( 𝛴 ) = { ( 𝑒 , 𝐹 ( 𝑒 ) ) : 𝑒 ∈ 𝐸 } ∈ 𝑆 𝑆 𝐸 ( 𝑋 ) 𝑠 𝑢 𝑐 ℎ 𝑡 ℎ 𝑎 𝑡 𝐹 ( 𝑒 ) = 𝐹 ( 𝑒 ′ ) ∈ 𝛴 𝑓 𝑜 𝑟 𝑒 𝑎 𝑐 ℎ 𝑒 , 𝑒 ′ ∈ 𝐸 . (2) The next example demonstrates the applicability of the procedure in Definition 2.

Suppose that 𝛴 𝑐 is the cofinite topology on X, also called the topology of finite complements ([3], Example 3.8 a). This is the topology that declares open those subsets of X whose complements are finite, together with ⌀. When X is finite, it is the discrete topology. When X is uncountably infinite, the topology is separable but not first-countable, therefore not second-countableterm [14], II.19.

The soft topologyterm on X generated by 𝛴 𝑐 , namely, 𝜏 ( 𝛴 𝑐 ) , is called the cofinite soft topologyterm on X.

It can be characterized as follows: 𝜏 ( 𝛴 𝑐 ) = { ( 𝑒 , 𝐹 ( 𝑒 ) ) : 𝑒 ∈ 𝐸 } ∈ 𝑆 𝑆 𝐸 ( 𝑋 ) 𝑠 𝑢 𝑐 ℎ 𝑡 ℎ 𝑎 𝑡 𝑋 ∖ 𝐹 ( 𝑒 ) 𝑖 𝑠 𝑓 𝑖 𝑛 𝑖 𝑡 𝑒 , 𝑓 𝑜 𝑟 𝑒 𝑎 𝑐 ℎ 𝑒 ∈ 𝐸 = ( 𝐹 , 𝐸 ) ∈ 𝑆 𝑆 𝐸 ( 𝑋 ) 𝑠 𝑢 𝑐 ℎ 𝑡 ℎ 𝑎 𝑡 ( 𝐹 , 𝐸 ) 𝑐 𝑖 𝑠 𝑎 𝑓 𝑖 𝑛 𝑖 𝑡 𝑒 𝑠 𝑜 𝑓 𝑡 𝑠 𝑒 𝑡 (3) It is in this guise that the cofinite soft topologyterm was introduced by [9], Definition 3.2. Here we just present it as an application of Definition 2.

Observe that the general construction of soft topologies given by Definition 2 can be improved by the recourse to several topologies, one for each attribute: Suppose that 𝛴 = { 𝛴 𝑒 } 𝑒 ∈ 𝐸 is a family of crisp topologies on X, indexed by a set E. Then 𝜏 ( 𝛴 ) = { ( 𝑒 , 𝐹 ( 𝑒 ) ) : 𝑒 ∈ 𝐸 } ∈ 𝑆 𝑆 𝐸 ( 𝑋 ) 𝑠 𝑢 𝑐 ℎ 𝑡 ℎ 𝑎 𝑡 𝐹 ( 𝑒 ) ∈ 𝛴 𝑒 𝑓 𝑜 𝑟 𝑒 𝑎 𝑐 ℎ 𝑒 ∈ 𝐸 (4) is a soft topologyterm on X. When 𝛴 𝑒 = 𝛴 𝑒 ′ = 𝛴 for each 𝑒 , 𝑒 ′ ∈ 𝐸 , 𝜏 ( 𝛴 ) = 𝜏 ( 𝛴 ) .

We say that 𝜏 ( 𝛴 ) is the soft topologyterm on X generated by the family of topologies 𝛴 . It is the collection of all soft setsterm ( 𝐹 , 𝐸 ) on X for which every 𝐹 ( 𝑒 ) lies in 𝛴 𝑒 . When all the 𝛴 𝑒 coincide (to a common topology 𝛴 ) then the soft topologies on X generated by the family 𝛴 and the topology 𝛴 coincide as well.

Just like bases are collections of subsets that generate topologies, soft basesterm are collections of soft setsterm that produce a soft topologyterm by taking the set formed by all unions of their elements. The abstract concept of a soft baseterm for a given soft topologyterm has been known since [4].

([4]). Suppose that τ is a soft topologyterm on X. Then 𝐵 ⊆ 𝜏 is a soft baseterm for τ when every ( 𝐹 , 𝐸 ) ∈ 𝜏 can be expressed as a union of soft setsterm from 𝐵 .

However Definition 3 cannot be used to enlarge the catalogue of actual soft topologies. Roy and Samanta [10] produced a related concept that can be eventually used to this purpose (but we believe that it has never been used to create distinguished soft topologies).

([10]). Suppose that 𝐵 ∈ 𝑃 ( 𝑆 𝑆 𝐸 ( 𝑋 ) ) is a collection of soft setsterm on X. It is a soft open baseterm for a soft topologyterm on X, when 𝐵 satisfies the following conditions: (1) 𝛷 ∈ 𝐵 ; (2) the union of all soft setsterm in 𝐵 is 𝑋 ˜ , i.e., for each 𝑒 ∈ 𝐸 and 𝑥 ∈ 𝑋 , there exists ( 𝐹 𝑥 , 𝐸 ) ∈ 𝐵 such that 𝑥 ∈ 𝐹 𝑥 ( 𝑒 ) ; and (3) when ( 𝐹 1 , 𝐸 ) , ( 𝐹 2 , 𝐸 ) ∈ 𝐵 and 𝑥 ∈ ( ( 𝐹 1 , 𝐸 ) ⊓ ( 𝐹 2 , 𝐸 ) ) ( 𝑒 ) = 𝐹 1 ( 𝑒 ) ∩ 𝐹 2 ( 𝑒 ) , there exists ( 𝐺 , 𝐸 ) ∈ 𝐵 such that ( 𝐺 , 𝐸 ) ⊑ ( 𝐹 1 , 𝐸 ) ⊓ ( 𝐹 2 , 𝐸 ) and 𝑥 ∈ 𝐺 ( 𝑒 ) .

Roy and Samanta [10], Theorem 13 establish the soft topologyterm generated by a soft open baseterm 𝐵 , denoted 𝜏 𝐵 . Their Theorem 16 proves that 𝐵 ⊆ 𝜏 𝐵 is a soft baseterm for 𝜏 𝐵 in the sense of Definition 3. Thus, the members of 𝜏 𝐵 are exactly the soft setsterm that can be written as a union of soft setsterm from 𝐵 . In other words, a soft setterm ( 𝐹 , 𝐸 ) belongs to 𝜏 𝐵 if and only if there is a family { ( 𝐹 𝑖 , 𝐸 ) } 𝑖 ∈ 𝐼 of members of 𝐵 such that ( 𝐹 , 𝐸 ) = ⊔ 𝑖 ∈ 𝐼 ( 𝐹 𝑖 , 𝐸 ) . And conversely, their Theorem 17 proves that any soft baseterm for a soft topologyterm 𝜏 on X (in the sense of Definition 3) must be a soft open baseterm for a soft topologyterm (in the sense of Definition 4).

Quite naturally, we say that a soft topologyterm is soft second-countableterm·def (S2Cterm) when there is a soft baseterm for it with countably many elements [8], Definition 4.32. This definition is a faithful replication of the pattern of a second-countableterm topological space. In contrast, the notion of soft separabilityterm is controversial. The next section discusses this issue in depth.

3. Soft Separabilityterm: New Proposal In the case of crisp topologies, separabilityterm amounts to the existence of a countable dense subset. Remember that when 𝛴 is a crisp topologyterm on X, 𝑈 ⊆ 𝑋 is dense in X if its topological closure is X, i.e., if 𝐶 𝑙 ( 𝑈 ) = 𝑋 .

Attempts have been made to define soft separabilityterm of a soft topologyterm on X. The only reliable source for a definition was given by Das and Samanta [8] in the context of soft metric spaces. It mimics the standard definition in crisp topologies. However this comes at the cost of being rather difficult to handle. It may also be possible to put forward alternative definitions with a different principle. Whereas [8] builds on countable soft setsterm whose soft closure is the absolute soft setterm 𝑋 ¯ , one might instead rely on the existence of countable collections of soft pointsterm with a similar behavior.

At any rate, a nice concept of soft separabilityterm is requested to satisfy three desirable characteristics: (1) It should, to some extent, replicate the spirit of separabilityterm in crisp topologies. (2) It must be implied by S2Cterm, which is universally accepted. (3) It should be easy to understand and amenable to technical operations.

In our view Das and Samanta’s soft separabilityterm meets the first two characteristics. It is a matter of opinion whether the third one is met, since it is highly subjective as a foundation for a concept of soft separabilityterm. Nevertheless, separabilityterm of crisp topologies is indeed simple to explain and we would expect a similarly undemanding notion in the soft context.

All things considered, we proceed to study soft separabilityterm with two principles in mind. First, the distinction between soft separabilities based on countable soft setsterm versus countable collections of ‘soft pointsterm’. We dedicate separate subsections to each position. Secondly, a simpler approach to the concept that understands density (thus separabilityterm) as follows: Suppose that Σ is a crisp topologyterm on X, and 𝑈 ⊆ 𝑋 . Then U is dense in X (i.e., 𝐶 𝑙 ( 𝑈 ) = 𝑋 ) if and only if each non-empty 𝑉 ∈ 𝛴 intersects U (i.e., 𝑈 ∩ 𝑉 ≠ ⌀ for each 𝑉 ∈ 𝛴 ∖ { ⌀ } ).

Therefore Σ is a separable crisp topologyterm on X if and only if there is a countable 𝑈 ⊆ 𝑋 such that 𝑈 ∩ 𝑉 ≠ ⌀ for each 𝑉 ∈ 𝛴 , 𝑉 ≠ ⌀ .

It is this alternative (and equivalent) definition of separabilityterm that we intend to extend to the realm of soft topologies. To avoid further confusion with existing literature, our two adapted versions of topological separabilityterm will avoid this term. In this we follow Munkres [2], Chapter 4, §30, who claims that the name separabilityterm is “an unfortunate choice of terminology” and “a good example of how a word can be overused”. Indeed, separation of a space and separation axioms in topology are based on very different ideas. And the same goes for soft topologyterm. In addition, we will give persuasive arguments to recommend these ideas as convenient extensions of the idea of separabilityterm to soft topological spaces.

For further use we define a special type of soft setsterm. They are vaguely related to the idea of soft pointsterm in [8].

For any 𝑈 ⊆ 𝑋 and 𝑒 ∈ 𝐸 , we define ( 𝐹 𝑈 𝑒 , 𝐸 ) ∈ 𝑆 𝑆 𝐸 ( 𝑋 ) such that 𝐹 𝑈 𝑒 ( 𝑒 ) = 𝑈 and 𝐹 𝑈 𝑒 ( 𝑒 ′ ) = ⌀ when 𝑒 ′ ≠ 𝑒 .

3.1. Soft Separabilityterm as Countable Soft-Set Density In view of Lemma 1, it is natural to extend the spirit of topological separabilityterm as follows: Suppose that τ is a soft topologyterm on X. We say that ( 𝐹 , 𝐸 ) ∈ 𝑆 𝑆 𝐸 ( 𝑋 ) is τ-soft dense in X if ( 𝐹 , 𝐸 ) ⊓ ( 𝐺 , 𝐸 ) ≠ 𝛷 , for every ( 𝐺 , 𝐸 ) ∈ 𝜏 , ( 𝐺 , 𝐸 ) ≠ 𝛷 ; and τ is countably soft-set-denseterm·def (henceforth CS 2 Dterm) if there is a countable soft setterm that is τ-soft dense in X.

Thus, CS 2 Dterm is similar to Das and Samanta’s soft separabilityterm in that it postulates the existence of a countable soft setterm with suitable characteristics. The difference is that our concept bypasses the appeal to additional definitions, like soft closures or soft limit points, in order to replicate the behavior of a countable dense set in crisp topologies.

To avail CS 2 Dterm as a sensible separability-type concept, we do two things. We confirm that it is implied by S2Cterm in the next section (cf., Proposition 1). In addition, Section 3.3 establishes a strong link of CS 2 Dterm with separabilityterm in the context of standard topologies.

3.2. Soft Separabilityterm as Soft-Points Countable Density An alternative way of extending topological separabilityterm—defined by Lemma 1—to soft topologies uses countable collections of ‘soft pointsterm’ instead of one countable soft setterm.

We propose to capture the spirit of topological separabilityterm in the following additional form: Suppose that τ is a soft topologyterm on X. We say that 𝐹 ∈ 𝑃 ( 𝑆 𝑆 𝐸 ( 𝑋 ) ) is soft τ-dense in X if for every ( 𝐺 , 𝐸 ) ∈ 𝜏 , ( 𝐺 , 𝐸 ) ≠ 𝛷 , it must be the case that ( 𝐹 , 𝐸 ) ⊓ ( 𝐺 , 𝐸 ) ≠ 𝛷 for some ( 𝐹 , 𝐸 ) ∈ 𝐹 ; and τ is soft-points countably-denseterm·def (henceforth SPCDterm) if there is a countable collection 𝐹 of soft pointsterm that is soft τ-dense in X.

Observe that SPCDterm implies CS 2 Dterm: Suppose that τ is a SPCDterm soft topologyterm on X. Then τ is CS 2 Dterm.

There exists a countable family of soft pointsterm, 𝐹 = { ( { 𝑥 𝑖 } , 𝐸 ) | 𝑖 ∈ 𝐼 } , with I countable and the property that for every ( 𝐺 , 𝐸 ) ∈ 𝜏 , ( 𝐺 , 𝐸 ) ≠ 𝛷 , there is 𝑖 ∈ 𝐼 such that ( { 𝑥 𝑖 } , 𝐸 ) ⊓ ( 𝐺 , 𝐸 ) ≠ 𝛷 .

Consider ( 𝐹 , 𝐸 ) such that 𝐹 ( 𝑒 ) = { 𝑥 𝑖 | 𝑖 ∈ 𝐼 } , a countable subset of X, for each 𝑒 ∈ 𝐸 . We have thus defined ( 𝐹 , 𝐸 ) ∈ 𝑆 𝑆 𝐸 ( 𝑋 ) , a countable soft setterm. It is obvious that ( 𝐹 , 𝐸 ) is 𝜏 -soft dense. □ More importantly, SPCDterm (thus CS 2 Dterm) is implied by S2Cterm: Let τ be a S2Cterm soft topologyterm τ on X. Then τ is SPCDterm, thus CS 2 Dterm.

There exists a countable soft baseterm 𝐵 = { ( 𝐹 𝑖 , 𝐸 ) | 𝑖 ∈ 𝐼 } for 𝜏 . Therefore I is countable and ( 𝐹 𝑖 , 𝐸 ) ∈ 𝑆 𝑆 𝐸 ( 𝑋 ) for each 𝑖 ∈ 𝐼 .

For each 𝑖 ∈ 𝐼 , if ( 𝐹 𝑖 , 𝐸 ) ≠ 𝛷 = ( 𝐹 𝑖 0 , 𝐸 ) there is 𝑒 ∈ 𝐸 such that 𝐹 𝑖 ( 𝑒 ) ≠ ⌀ , thus we can select 𝑥 𝑖 ∈ 𝐹 𝑖 ( 𝑒 ) . We let 𝐼 0 = 𝐼 ∖ { 𝑖 0 } .

Consider the countable family of soft pointsterm 𝐹 = { ( { 𝑥 𝑖 } , 𝐸 ) | 𝑖 ∈ 𝐼 0 } thus defined. We claim that it is soft 𝜏 -dense in X, which proves that 𝜏 is SPCDterm.

Fix ( 𝐺 , 𝐸 ) ∈ 𝜏 , ( 𝐺 , 𝐸 ) ≠ 𝛷 . Because 𝐵 is a soft baseterm for 𝜏 , ( 𝐺 , 𝐸 ) = ⊔ 𝑖 ∈ 𝐽 ( 𝐹 𝑖 , 𝐸 ) for some 𝐽 ⊆ 𝐼 . Now we select 𝑗 ∈ 𝐽 for which there exists 𝑒 ∈ 𝐸 with 𝐹 𝑗 ( 𝑒 ) ≠ ⌀ . By construction, ( { 𝑥 𝑗 } , 𝐸 ) ∈ 𝐹 . As 𝑥 𝑗 ∈ 𝐹 𝑗 ( 𝑒 ) ⊆ 𝐺 ( 𝑒 ) , we can guarantee ( { 𝑥 𝑗 } , 𝐸 ) ⊓ ( 𝐺 , 𝐸 ) ≠ 𝛷 . □ At the end of Section 3.3 we prove that S2Cterm is not implied by SPCDterm.

3.3. Relationships with Crisp Separabilityterm The next two results show both that Corollary 1 is useful, and that our new notions in this section, are conveniently linked to the standard notion of separabilityterm in topology: Suppose that 𝛴 = { 𝛴 𝑒 } 𝑒 ∈ 𝐸 is a family of topologies on X, such that 𝛴 𝑒 ′ is separable for some 𝑒 ′ ∈ 𝐸 . Then the soft topologyterm 𝜏 ( 𝛴 ) is SPCDterm, thus CS 2 Dterm.

In particular, if Σ is separable topology on X then the soft topologyterm 𝜏 ( 𝛴 ) on X is SPCDterm (and CS 2 Dterm).

Let us fix { 𝑥 𝑗 | 𝑗 ∈ 𝐼 } , a countable dense subset of 𝛴 𝑒 ′ . We prove that { ( { 𝑥 𝑗 } , 𝐸 ) | 𝑗 ∈ 𝐼 } is soft 𝜏 ( 𝛴 ) -dense in X. To see that this countable collection of soft pointsterm satisfies the requirements of Definition 7, let us fix an arbitrary ( 𝐹 , 𝐸 ) ∈ 𝜏 ( 𝛴 ) . Since 𝐹 ( 𝑒 ′ ) ∈ 𝛴 𝑒 ′ , there must exist 𝑖 ∈ 𝐼 such that 𝑥 𝑖 ∈ 𝐹 ( 𝑒 ′ ) , which implies ( { 𝑥 𝑖 } , 𝐸 ) ⊓ ( 𝐹 , 𝐸 ) ≠ 𝛷 . Therefore 𝛴 is SPCDterm.

The consequence on 𝜏 ( 𝛴 ) when 𝛴 is separable is now obvious. □ It is also possible to prove a converse of Proposition 2: Suppose that 𝛴 = { 𝛴 𝑒 } 𝑒 ∈ 𝐸 is a family of topologies on X, such that 𝜏 ( 𝛴 ) is CS 2 Dterm. Then 𝛴 𝑒 is separable for each 𝑒 ∈ 𝐸 . In particular, if Σ is a topology on X such that the soft topologyterm 𝜏 ( 𝛴 ) on X is CS 2 Dterm (or SPCDterm) then Σ is separable.

Let us fix ( 𝐹 , 𝐸 ) , a countable 𝜏 ( 𝛴 ) -soft dense set in X. We claim that for each 𝑒 ∈ 𝐸 , 𝐹 ( 𝑒 ) is dense in X with the 𝛴 𝑒 topology. Note that by definition, it is a countable subset of X.

We appeal to Lemma 1. Let us fix 𝑈 ∈ 𝛴 𝑒 . We define the soft setterm ( 𝐹 𝑈 𝑒 , 𝐸 ) ∈ 𝜏 ( 𝛴 ) (cf., Definition 5). Because ( 𝐹 , 𝐸 ) is 𝜏 ( 𝛴 ) -soft dense, it must be the case that ( 𝐹 , 𝐸 ) ⊓ ( 𝐹 𝑈 𝑒 , 𝐸 ) ≠ 𝛷 . This necessarily entails 𝐹 ( 𝑒 ) ∩ 𝑈 ≠ ⌀ .

The consequence on 𝜏 ( 𝛴 ) is now obvious. □ Finally, Example 1 help us to check that S2Cterm is indeed different from SPCDterm. Suppose that 𝐸 = { 𝑒 } and that X is uncountable. Then 𝜏 ( 𝛴 𝑐 ) , the cofinite soft topologyterm on X, is SPCDterm because 𝛴 𝑐 , the crisp cofinite topology on X, is separable and we can invoke Proposition 2. The fact that 𝛴 𝑐 is not second-countableterm easily proves that 𝜏 ( 𝛴 𝑐 ) cannot be S2Cterm. Nevertheless this fact can also be established by an appeal to Corollary 2 below.

4. A Characterization of Soft Second-Countability for Soft Topologies Second-countableterm crisp topologies are those that have countable bases. It is straightforward to extend the concept to soft topologies: remember that a soft topologyterm with a countable soft baseterm is a soft second-countableterm·def or S2Cterm soft topologyterm. This concept is superfluous when both X and E are finite because in this case, all soft topologies are finite (i.e., formed by a finite number of soft setsterm) thus countable; and countable soft topologies are obviously S2Cterm. In this section we avail ourselves of the ideas in Section 5.1 in order to produce the first non-trivial soft second-countableterm·def soft topologies in the literature.

Let Σ be a second-countableterm topology on X. Then when E is finite, 𝜏 ( 𝛴 ) is S2Cterm.

Let 𝛽 be a countable base for 𝛴 . Theorem 3 ensures that 𝜏 ( 𝛴 ) = 𝜏 𝐵 ( 𝛽 ) thus 𝐵 ( 𝛽 ) is a soft open baseterm for 𝜏 ( 𝛴 ) by [10], Theorem 16. We just need to observe that 𝐵 ( 𝛽 ) is countable when E is finite. □ In fact, a direct arguments proves an improvement of Theorem 1: Let 𝛴 = { 𝛴 𝑒 } 𝑒 ∈ 𝐸 be a family of topologies on X, such that 𝛴 𝑒 is second-countableterm for each 𝑒 ∈ 𝐸 . Then when E is countable, 𝜏 ( 𝛴 ) is S2Cterm.

Let { 𝛽 𝑒 } be a countable base for 𝛴 𝑒 , for each 𝑒 ∈ 𝐸 . We claim that 𝐵 = { ( 𝐹 𝑈 𝑒 , 𝐸 ) 𝑠 𝑢 𝑐 ℎ 𝑡 ℎ 𝑎 𝑡 𝑈 ∈ 𝛽 𝑒 𝑓 𝑜 𝑟 𝑒 𝑎 𝑐 ℎ 𝑒 ∈ 𝐸 } = ∪ 𝑒 ∈ 𝐸 { ( 𝐹 𝑈 𝑒 , 𝐸 ) 𝑠 𝑢 𝑐 ℎ 𝑡 ℎ 𝑎 𝑡 𝑈 ∈ 𝛽 𝑒 } (5) is a countable soft baseterm for 𝜏 ( 𝛴 ) . It is a countable family because E is countable, each { 𝛽 𝑒 } is countable, and the countable union of countable sets is countable. Its members belong to 𝜏 ( 𝛴 ) by definition of this soft topologyterm. To prove that 𝐵 is a soft baseterm of 𝜏 ( 𝛴 ) , let us fix ( 𝐹 , 𝐸 ) ∈ 𝜏 ( 𝛴 ) , i.e., 𝐹 ( 𝑒 ) ∈ 𝛴 𝑒 for each 𝑒 ∈ 𝐸 .

Then there exist { 𝑈 𝑖 } 𝑖 ∈ 𝐼 ( 𝑒 ) , respective subfamilies of open sets in 𝛽 𝑒 with the property that 𝐹 ( 𝑒 ) = ∪ 𝑖 ∈ 𝐼 ( 𝑒 ) 𝑈 𝑖 for each 𝑒 ∈ 𝐸 . Note that for each 𝑈 ∈ { 𝑈 𝑖 } 𝑖 ∈ 𝐼 ( 𝑒 ) and 𝑒 ∈ 𝐸 , the soft setterm ( 𝐹 𝑈 𝑒 , 𝐸 ) belongs to 𝐵 by definition.

The reader can easily check that 𝐹 = { ( 𝐹 𝑈 𝑒 , 𝐸 ) 𝑠 𝑢 𝑐 ℎ 𝑡 ℎ 𝑎 𝑡 𝑈 ∈ { 𝑈 𝑖 } 𝑖 ∈ 𝐼 ( 𝑒 ) 𝑓 𝑜 𝑟 𝑒 𝑎 𝑐 ℎ 𝑒 ∈ 𝐸 } ⊆ 𝐵 satisfies ( 𝐹 , 𝐸 ) = ⊔ { ( 𝐹 𝑈 𝑒 , 𝐸 ) | ( 𝐹 𝑈 𝑒 , 𝐸 ) ∈ 𝐹 } . □ We can also prove a converse to this result. Its proof follows that of Proposition 3 closely.

Suppose that 𝛴 = { 𝛴 𝑒 } 𝑒 ∈ 𝐸 is a family of topologies on X, such that 𝜏 ( 𝛴 ) is S2Cterm. Then 𝛴 𝑒 is second-countableterm for each 𝑒 ∈ 𝐸 . In particular, if Σ is a topology on X such that 𝜏 ( 𝛴 ) is S2Cterm then Σ is second-countableterm.

Let us select { ( 𝐹 𝑖 , 𝐸 ) } 𝑖 ∈ 𝐼 , a countable soft baseterm of 𝜏 ( 𝛴 ) in X. We claim that for each 𝑒 ∈ 𝐸 , the countable collection { 𝐹 𝑖 ( 𝑒 ) } 𝑖 ∈ 𝐼 is a base of the 𝛴 𝑒 topology. Note that 𝐹 𝑖 ( 𝑒 ) ∈ 𝛴 𝑒 for each 𝑖 ∈ 𝐼 and 𝑒 ∈ 𝐸 , by definition of 𝜏 ( 𝛴 ) .

Let us fix 𝑈 ∈ 𝛴 𝑒 . We define the soft setterm ( 𝐹 𝑈 𝑒 , 𝐸 ) as in Definition 5, then ( 𝐹 𝑈 𝑒 , 𝐸 ) ∈ 𝜏 ( 𝛴 ) . Because { ( 𝐹 𝑖 , 𝐸 ) } 𝑖 ∈ 𝐼 is a soft baseterm of 𝜏 ( 𝛴 ) , ( 𝐹 𝑈 𝑒 , 𝐸 ) = ⊔ 𝑗 ∈ 𝐽 ( 𝐹 𝑗 , 𝐸 ) for some subfamily 𝐽 ∈ 𝐼 . This necessarily entails 𝐹 𝑈 𝑒 ( 𝑒 ) = 𝑈 = ∪ 𝑗 ∈ 𝐽 𝐹 𝑗 ( 𝑒 ) . We conclude that { 𝐹 𝑖 ( 𝑒 ) } 𝑖 ∈ 𝐼 is a base of 𝛴 𝑒 .

The particular consequence for 𝜏 ( 𝛴 ) is now obvious. □ If we combine Theorem 2 and Proposition 4 we readily deduce a very appealing property of the soft topologyterm defined in Corollary 1: Suppose that 𝛴 = { 𝛴 𝑒 } 𝑒 ∈ 𝐸 is a family of topologies on X, and E is a countable set of attributes. Then 𝜏 ( 𝛴 ) is a S2Cterm soft topologyterm on X if and only if 𝛴 𝑒 is second-countableterm for each 𝑒 ∈ 𝐸 .

5. A New Constructive Method of Soft Topologies This section introduces a novel procedure for the construction of soft topologies. It builds on the concept of a base for a crisp topologyterm. We investigate some fundamental properties of this mechanism.

5.1. Soft Topologies Generated by Bases for Crisp Topologies Our first result in this section shows how any base for a crisp topologyterm on X generates a soft topologyterm on X, for any fixed set of attributes E.

Let β be a base for a topologyterm Σ on X. Then the collection of soft setsterm on X 𝐵 ( 𝛽 ) = { ( 𝑒 , 𝐹 ( 𝑒 ) ) : 𝑒 ∈ 𝐸 } ∈ 𝑆 𝑆 𝐸 ( 𝑋 ) 𝑠 𝑢 𝑐 ℎ 𝑡 ℎ 𝑎 𝑡 𝐹 ( 𝑒 ) ∈ 𝛽 ∪ { ⌀ } 𝑓 𝑜 𝑟 𝑒 𝑎 𝑐 ℎ 𝑒 ∈ 𝐸 (6) is a soft open baseterm for a soft topologyterm on X.

We prove that 𝐵 ( 𝛽 ) satisfies the three requirements of Definition 4. Actually, 𝐵 ( 𝛽 ) agrees with (1) and (3) in that Definition, and also with a property implying (2).

Note that 𝛷 = { ( 𝑒 , 𝛷 ( 𝑒 ) ) : 𝑒 ∈ 𝐸 } satisfies 𝛷 ( 𝑒 ) = ⌀ , therefore it is clearly a member of 𝐵 ( 𝛽 ) by construction. This ensures (1).

We claim that for each 𝑥 ∈ 𝑋 , there exists ( 𝐹 𝑥 , 𝐸 ) ∈ 𝐵 ( 𝛽 ) such that ( { 𝑥 } , 𝐸 ) ⊑ ( 𝐹 𝑥 , 𝐸 ) . Observe that this property of 𝐵 ( 𝛽 ) is stronger than (2). Let us fix 𝑥 ∈ 𝑋 . There exists 𝑥 ∈ 𝐵 𝑥 ∈ 𝛽 therefore if we define ( 𝐹 𝑥 , 𝐸 ) by 𝐹 𝑥 ( 𝑒 ) = 𝐵 𝑥 for each 𝑒 ∈ 𝐸 , then ( { 𝑥 } , 𝐸 ) ⊑ ( 𝐹 𝑥 , 𝐸 ) ∈ 𝐵 ( 𝛽 ) by definition.

In order to prove (3), suppose that ( 𝐹 1 , 𝐸 ) , ( 𝐹 2 , 𝐸 ) ∈ 𝐵 ( 𝛽 ) and 𝑥 ∈ ( ( 𝐹 1 , 𝐸 ) ⊓ ( 𝐹 2 , 𝐸 ) ) ( 𝑒 ) = 𝐹 1 ( 𝑒 ) ∩ 𝐹 2 ( 𝑒 ) . For each 𝑒 ′ ∈ , 𝑒 ′ ≠ 𝑒 , we define 𝐺 ( 𝑒 ′ ) = ⌀ . Note that 𝐹 1 ( 𝑒 ) , 𝐹 2 ( 𝑒 ) ∈ 𝛽 ∪ { ⌀ } by construction, 𝛽 is a base for a topologyterm on X by assumption, and 𝑥 ∈ 𝐹 1 ( 𝑒 ) ∩ 𝐹 2 ( 𝑒 ) ≠ ⌀ . Therefore there exists 𝐺 ( 𝑒 ) ∈ 𝛽 such that 𝑥 ∈ 𝐺 ( 𝑒 ) ⊆ 𝐹 1 ( 𝑒 ) ∩ 𝐹 2 ( 𝑒 ) . We have thus defined ( 𝐺 , 𝐸 ) ∈ 𝐵 ( 𝛽 ) with the properties ( 𝐺 , 𝐸 ) ⊑ ( 𝐹 1 , 𝐸 ) ⊓ ( 𝐹 2 , 𝐸 ) and 𝑥 ∈ 𝐺 ( 𝑒 ) . □ Actually, a straightforward modification of the proof of Proposition 5 assures that we can produce a more general construction than the procedure that it proposes.

Suppose that 𝛽 ¯ = { 𝛽 𝑒 } 𝑒 ∈ 𝐸 is a family of bases for respective topologies on X, indexed by a set E. Then 𝐵 ( 𝛽 ¯ ) = { ( 𝑒 , 𝐹 ( 𝑒 ) ) : 𝑒 ∈ 𝐸 } ∈ 𝑆 𝑆 𝐸 ( 𝑋 ) 𝑠 𝑢 𝑐 ℎ 𝑡 ℎ 𝑎 𝑡 𝐹 ( 𝑒 ) ∈ 𝛽 𝑒 ∪ { ⌀ } 𝑓 𝑜 𝑟 𝑒 𝑎 𝑐 ℎ 𝑒 ∈ 𝐸 is a soft open baseterm for a soft topologyterm on X.

To demonstrate the implementability of the above Corollary, we consider the following toy example.

Let 𝑋 = { 𝑥 , 𝑦 , 𝑧 } . Two bases for respective topologies on X are 𝛽 𝑒 1 = { ⌀ , 𝑋 , { 𝑦 } } and 𝛽 𝑒 2 = { ⌀ , 𝑋 , { 𝑦 , 𝑧 } } . Define 𝐸 = { 𝑒 1 , 𝑒 2 } . Then Corollary 3 assures that a soft open baseterm for a soft topologyterm on X is defined as the collection of the nine soft setsterm represented by Table 1. The first one is 𝛷 𝐸 whereas the last one is 𝑋 ˜ .

We are ready to formulate a problem whose solution constitutes the main result of this section. Suppose that 𝛽 is a base for a (crisp) topology 𝛴 on X. From these items we can produce two apparently different soft topologies on X, namely, 𝜏 ( 𝛴 ) and 𝜏 𝐵 ( 𝛽 ) . Definition 2 gives 𝜏 ( 𝛴 ) . Roy and Samanta [10], Theorems 13,16 give 𝜏 𝐵 ( 𝛽 ) by virtue of Proposition 5. Our next theorem proves that both soft topologies on X coincide.

Let β be a base for a (crisp) topology Σ on X. Then 𝜏 ( 𝛴 ) = 𝜏 𝐵 ( 𝛽 ) .

We prove by double inclusion the set identity 𝜏 ( 𝛴 ) = 𝜏 𝐵 ( 𝛽 ) .

Recall that Roy and Samanta [10], Theorem 16 establish that 𝐵 ( 𝛽 ) is a soft baseterm for 𝜏 𝐵 ( 𝛽 ) in the sense of Definition 3. Thus, the members of 𝜏 𝐵 ( 𝛽 ) are exactly the soft setsterm that can be written as a union of soft setsterm from 𝐵 ( 𝛽 ) . In other words, a soft setterm ( 𝐹 , 𝐸 ) belongs to 𝜏 𝐵 ( 𝛽 ) if and only if there is a family { ( 𝐹 𝑖 , 𝐸 ) } 𝑖 ∈ 𝐼 of members of 𝐵 ( 𝛽 ) such that ( 𝐹 , 𝐸 ) = ⊔ 𝑖 ∈ 𝐼 ( 𝐹 𝑖 , 𝐸 ) .

In order to prove 𝜏 ( 𝛴 ) ⊆ 𝜏 𝐵 ( 𝛽 ) , suppose ( 𝐹 , 𝐸 ) ∈ 𝜏 ( 𝛴 ) . If we write ( 𝐹 , 𝐸 ) = { ( 𝑒 , 𝐹 ( 𝑒 ) ) : 𝑒 ∈ 𝐸 } then the definition of 𝜏 ( 𝛴 ) implies 𝐹 ( 𝑒 ) ∈ 𝛴 for each 𝑒 ∈ 𝐸 . Because 𝛽 is a base for 𝛴 , with each 𝑒 ∈ 𝐸 we can associate a family { 𝐺 𝑖 ( 𝑒 ) } 𝑖 ∈ 𝐼 ( 𝑒 ) of members of 𝛽 such that 𝐹 ( 𝑒 ) = ∪ 𝑖 ∈ 𝐼 ( 𝑒 ) 𝐺 𝑖 ( 𝑒 ) . Let us define a family of soft setsterm on X that belong to 𝐵 ( 𝛽 ) , as follows: { ( 𝐹 𝑖 , 𝐸 ) } 𝑖 ∈ 𝐼 = { ( 𝐹 ′ , 𝐸 ) = { ( 𝑒 , 𝐹 ′ ( 𝑒 ) ) : 𝑒 ∈ 𝐸 } ∈ 𝑆 𝑆 𝐸 ( 𝑋 ) 𝑠 𝑢 𝑐 ℎ 𝑡 ℎ 𝑎 𝑡 𝐹 ′ ( 𝑒 ) = 𝐺 𝑖 ( 𝑒 ) , 𝑠 𝑜 𝑚 𝑒 𝑖 ∈ 𝐼 ( 𝑒 ) } . (7) Because each ( 𝐹 ′ , 𝐸 ) in this family belongs to 𝐵 ( 𝛽 ) by construction, and it is easy to check for the equality ( 𝐹 , 𝐸 ) = ⊔ 𝑖 ∈ 𝐼 ( 𝐹 𝑖 , 𝐸 ) , we can conclude ( 𝐹 , 𝐸 ) ∈ 𝜏 𝐵 ( 𝛽 ) .

Conversely, let us fix ( 𝐹 , 𝐸 ) ∈ 𝜏 𝐵 ( 𝛽 ) . There is { ( 𝐹 𝑖 , 𝐸 ) } 𝑖 ∈ 𝐼 ⊆ 𝐵 ( 𝛽 ) such that ( 𝐹 , 𝐸 ) = ⊔ 𝑖 ∈ 𝐼 ( 𝐹 𝑖 , 𝐸 ) . By definition of 𝐵 ( 𝛽 ) , 𝐹 𝑖 ( 𝑒 ) ∈ 𝛽 ∪ { ⌀ } for each 𝑒 ∈ 𝐸 and 𝑖 ∈ 𝐼 . By definition of ⊔, ( 𝐹 , 𝐸 ) must satisfy 𝐹 ( 𝑒 ) = ∪ 𝑖 ∈ 𝐼 𝐹 𝑖 ( 𝑒 ) ∈ 𝛴 for each 𝑒 ∈ 𝐸 . This in turn assures ( 𝐹 , 𝐸 ) ∈ 𝜏 ( 𝛴 ) by definition of 𝜏 ( 𝛴 ) . □ The diagram in Figure 2 summarizes the elements that produce Theorem 3.

5.2. New Non-Trivial Soft Second-Countableterm·def Soft Topologies Let us now show that the combination of Theorems 1 and 3 permits to define non-trivial examples of S2Cterm soft topologies. The next example focuses on countable sets X.

Let 𝑋 = 𝑁 . Bases for respective crisp topologies on X are 𝛽 1 = { 𝑛 , 𝑛 + 1 , … } | 𝑛 ∈ 𝑁 [3], Problem 5.B, Chapter 2, which is clearly countable (thus the topology 𝛴 1 that it generates is second-countableterm); and 𝛽 𝐹 = 𝐴 ⊂ 𝑁 | 𝑒 𝑖 𝑡 ℎ 𝑒 𝑟 1 ∉ 𝐴 𝑜 𝑟 𝑁 ∖ 𝐴 𝑖 𝑠 𝑓 𝑖 𝑛 𝑖 𝑡 𝑒 . This is a base for a countable Fort topology on X, 𝛴 𝐹 , which is second-countableterm [14], II.24.

Let 𝑌 = { 2 , 3 , 4 … } ⊆ 𝑁 . Consider for each 𝑛 ⩾ 2 , the set of the form 𝐵 𝑛 = { 𝑥 ∈ 𝑁 | 𝑥 𝑑 𝑖 𝑣 𝑖 𝑑 𝑒 𝑠 𝑛 } . The family 𝛽 𝐷 = { 𝑈 𝑛 } 𝑛 ⩾ 2 is a base for the divisor topology on Y, 𝛴 𝐷 , which is second-countableterm [14], II.57 because all elements in the base are finite thus the base itself is countable.

Thus, for any countable set of attributes E, S2Cterm soft topologies on 𝑁 are 𝜏 ( 𝛴 1 ) = 𝜏 𝐵 ( 𝛽 1 ) and 𝜏 ( 𝛴 𝐹 ) = 𝜏 𝐵 ( 𝛽 𝐹 ) . And a S2Cterm soft topologyterm on Y is 𝜏 ( 𝛴 𝐷 ) = 𝜏 𝐵 ( 𝛽 𝐷 ) . The argument in the proof of Theorem 2 provides formal expressions of respective countable soft basesterm of these soft topologies.

Actually, we can assure that when E is finite, 𝐵 ( 𝛽 1 ) is a countable soft open baseterm of 𝜏 ( 𝛴 1 ) , and that 𝐵 ( 𝛽 𝐷 ) is a countable soft open baseterm of 𝜏 ( 𝛴 𝐷 ) . The argument in the proof of Theorem 1 proves this claim.

Table 2 displays a summary of the tabular representations of the elements in 𝐵 ( 𝛽 1 ) , the countable soft open baseterm of the soft topologyterm 𝜏 ( 𝛴 1 ) on X, when 𝐸 = { 𝑒 1 , 𝑒 2 } .

One can also use the procedure above to produce S2Cterm soft topologies on uncountable sets of alternatives. For example, the Euclidean topology 𝛴 𝑒 on 𝑋 = 𝑅 is second-countableterm so with any countable set of attributes E we can assure that 𝜏 ( 𝛴 𝑒 ) is a S2Cterm soft topologyterm on 𝑅 . In fact with each countable base for 𝛴 𝑒 , e.g., the collection 𝛽 𝑒 of open intervals with rational extremes, we can associate a countable soft open baseterm for 𝜏 ( 𝛴 𝑒 ) (see the argument in the proof of Theorem 2). When E is finite, 𝐵 ( 𝛽 𝑒 ) for 𝜏 ( 𝛴 𝑒 ) is another countable soft open baseterm for 𝜏 ( 𝛴 𝑒 ) .

6. Conclusions This paper shows that the research about soft topological spaces is an active field, where fundamental contributions can still be made. Consequently, it presents strictly theoretical considerations on soft topologies.

The development of topology has been nurtured by the continuous supply of examples and classes of topological spaces. It is, therefore, important to enlarge the list of soft topological spaces, and of their properties and relations. We have contributed to this constituent of soft topologyterm with some new classes of soft topological spaces. We have produced non-trivial examples of soft second-countableterm·def soft topologies. In fact, countable soft open basesterm of these soft topologies have been identified.

Relatedly, two new concepts of ‘soft separabilityterm’ have been proposed. They are implied by (but different from) soft second-countability, and they are simple to understand and operate with. Further technical arguments show that they can play a part in the field of soft topologyterm.

We have reexamined a general procedure for generating soft topologies from crisp topologies, that has been known since [11]. It has been extended in two directions. First, by allowing that different attributes are associated with different crisp topologies. Secondly, by the recourse to the simpler notion of a base, and again we use one base for each attribute. The first procedure preserves separabilityterm under the sole assumption that one of the attributes is associated with a separable crisp topologyterm. Its behavior with respect to second-countability is still better, and an ‘if-and-only-if’ result has been achieved. The second procedure gives rise to a commutativity diagram involving the original procedure: if we generate a soft topologyterm from a base for a topologyterm, and we generate a soft topologyterm from the topology that the base produces, both soft topologies are the same. Thus, it may be argued that the second procedure allows to operationalize the well-established procedure for the generation of soft topologies in [11].

Practical applications are beyond the scope of this article. The new strategies for the production of soft topologies may be applied in the future in the analysis of other properties, like the soft separation axioms (soft normality, soft regularity, 𝑇 0 , 𝑇 1 , 𝑇 2 , 𝑇 3 , 𝑇 4 , ...) in [5,11].