On some generated soft topological spaces and soft homogeneity.

We introduce soft homogeneity as an extension of homogeneity in ordinary topological spaces. Based on the generated soft topology of a given indexed family of classical topologies inspite of a one topology given by Terepeta in [16], we investigate soft minimal open set and homogeneity relation between the generated soft topology and the given indexed family of topologies. We introduce several results, examples and counterexamples.

Introduction

The classical mathematical theories have difficulties for solving complicated problems which include uncertain data in many areas such as engineering, environment, economics, medical science, social science, etc. Theories of probability, fuzzy sets [1], rough sets [2], intuitionistic fuzzy sets [3] and vague sets [4] are considered as mathematical tools for dealing with uncertainties. Molodtsov [5] justified that each of these theories has its deep-seated difficulty. These difficulties are mainly come from the inadequacy of the parameterization tool of the theories. For dealing with uncertainties away from these difficulties, Molodtsov [5] defined soft sets as follows: Let X be an initial universe and A be a set of parameters and denote the power set of X by , a soft set over X relative to A is a function . The theory of soft sets has been introduced and studied by several researchers (see [6], [7]). Authors [5], [8] applied soft sets in many areas such as Riemann integration, Perron integration, smoothness of function, operation research, game theory, probability and theory of measurements. Authors [9] applied soft sets in decision-making problems. The notion of soft topological spaces was introduced in [10]. Then, Mathematicians modified several concepts of classical topological spaces to include soft topological spaces (see [11], [12], [13], [14], [15], [16], [17], [18], [19], [20], [21]). A topological space is homogeneous if for all , there is a homeomorphism such that . Since homogeneity concepts are of importance in general topology and still a hot area of research, as appears in [22], [23], [24], [25], [26], [27], [28], [29], [30], [31], [32], [33], we see that it is suitable to extend the homogeneity concept to include soft topological spaces. One of our main goals of the present work is to show how the definition of homogeneity in ordinary topological spaces can be modified in order to define its extension in soft topological spaces. This paper is organized as follows.

In section two, we introduce some basic definitions and results which we use them in our research.

In section three, for a given collection of topologies on a set X, we introduce a generated soft topology τ over X with a set of parameters A such that for all ; moreover, for a given topological space and a given set of parameters A, we show that generates a soft topology τ on X with the set of parameters A.

In section four, we continue the study of minimal soft open sets; in particular, we give a mapping regarding minimal soft open sets, also we give a link between the minimal soft open sets of a soft topological space and the minimal open sets of its generated topological spaces and vice versa.

In section five, we introduce and investigate soft homogeneity in soft topological spaces; in particular, we study the relation between soft homogeneity of a soft topological space and some ordinary topological spaces generated by this soft topological space, we study the relation between homogeneity of an ordinary topological space and some soft topological spaces generated by this ordinary topological space, and we introduce some properties of homogeneous soft topological spaces that contains a minimal soft open set.

Preliminaries

In this section, we introduce some basic definitions and results which we use them in our research.

[5] Let X be an initial universe and A be a set of parameters. A soft set over X relative to A is a function . The family of all soft sets over X relative to A will be denoted by .

[6] Let .

(1) F is a soft subset of G, denoted by , if for each .

(2) F and G are said to be soft equal, denoted by if and .

(3) Union of F and G is denoted by and defined to be the soft set where for each .

(4) Intersection of F and G is denoted by and defined to be the soft set where for each .

[11] Let Δ be an arbitrary index set and .

(a) The union of these soft sets is the soft set denoted by and defined by for each .

(b) The intersection of these soft sets is the soft set denoted by and defined by for each .

[6] Let .

(a) F is called a null soft set over X relative to A, denoted by , if for each .

(b) F is called an absolute soft set over X relative to A, denoted by , if for each .

[12] Let . F is called a soft point over X relative to A if there exist and such that

We denote F by . The family of all soft points over X relative to A is denoted by .

Let . Then iff and .

[12] Let and . Then is said to belong to F (notation: ) if or equivalently: iff .

iff there is such that .

Straightforward. □

Let . Then the following are equivalent:

(a) .

(b) For all , implies .

[12] Let . The set will be denoted by .

It is clear that .

Let X be an initial universe and A be a set of parameters. Then for any , iff .

[7] Let and be families of soft sets. Let and be functions. Then a soft mapping is defined as:

(a) Let . The image of F under , written as is defined by

(b) Let . The inverse of G under , written as is defined by

[7] Let be a soft mapping. Then is called:

(a) Injective if p and u are injective.

(b) Surjective if p and u are surjective.

(c) Bijective if p and u are bijective.

Let be a soft mapping. Then for all , .

Straightforward. □

Let be a soft mapping. Then for all , iff and .

Straightforward. □

A soft mapping is injective iff is injective.

Straightforward. □

A soft mapping is surjective iff is surjective.

Straightforward. □

Let be a soft mapping. Let and . Then iff there is such that .

Necessity. Assume that . Then . So, there is such that . Choose such that . Thus, we have and .

Sufficiency. Assume that there is such that . Since , then and so . □

Let be a soft mapping. Let and . Then iff .

Necessity. Assume that . Then . So, . Thus, .

Sufficiency. Assume that . Then and . Thus, . Hence, . □

[10] Let τ ⊆ . Then τ is called a soft topology on X relative to A if

(1) ,

(2) the union of any number of soft sets in τ belongs to τ,

(3) the intersection of any two soft sets in τ belongs to τ.

The triplet is called a soft topological space (STS) over X relative to A. The members of τ are called soft open sets in and their complements are called soft closed sets in .

A soft mapping is called:

(1) Soft continuous [15] if for every .

(2) Soft open [15] if for every .

(3) A soft homeomorphism [14] if is soft continuous, soft open and bijective.

[15] Let and be functions, then the composition of and is soft mapping from onto denoted by and defined by .

If and are soft continuous, then is soft continuous.

A soft mapping is a soft homeomorphism iff is bijective, and and are soft continuous.

Let be a STS. Then the set of all soft homeomorphisms from onto forms a group under the operation .

Straightforward. □

Let be a STS. Then the collection defines a topology on X for every . This topology will be denoted by .

[28] Let be a topological space. A non-empty open subset is called a minimal open set if the only open subsets of A are A and ∅. The set of all minimal open sets of will be denoted by .

[19] Let be a soft topological space. A subcollection of τ is called a soft base of τ if every member of τ can be expressed as a union of members of .

Let be a STS and let . Then is a soft base for τ if for every and every F, there exists such that F.

A collection is a soft base for some soft topology on X relative to A iff the following conditions hold:

i. .

ii. For every and for every , there is such that .

Soft topologies generated by ordinary topologies

In this section, for a given collection of topologies on a set X, we introduce a generated soft topology τ over X with a set of parameters A such that for all ; moreover, for a given topological space and a given set of parameters A we show that generates a soft topology τ on X with the set of parameters A.

Let X be an initial universe and let A be a set of parameters. Let be an indexed family of topologies on X and let

Then is a STS.

For all , and so . Let , then for all , and . So for all , and hence . Let , then for all and and . So for all , and hence . □

Let X be an initial universe and let A be a set of parameters. Let be an indexed family of topologies on X. Then the soft topology will be denoted by .

For any topological space and any set of parameters A, denote the family by .

For any topological space and any set of parameters A, is a STS.

For each , set . Then and by Theorem 3.1 we get the result. □

Let X be an initial universe and let A be a set of parameters. For any and , the soft set defined by will be denoted by .

Let X be an initial universe and let A be a set of parameters. Let be an indexed family of topologies on X. Then the family is a soft base of .

If , then . □

Let X be an initial universe and let A be a set of parameters. Let be an indexed family of topologies on X. If is a base for for all , then is a soft base of .

Let X be an initial universe and let A be a set of parameters. Let be an indexed family of topologies on X. Then for all .

Let , then there is such that . By the definition of , and thus . Conversely, if , then and so . □

If is a topological space and A is any set of parameters, then for all .

For each , set . Then and by Theorem 3.7 we get the result. □

If is a STS, then .

Let , then for all and so . □

The following example shows that the equality in Theorem 3.9 does not hold in general:

Let , , , and . Then , and . So .

If is a STS, then for all .

Theorem 3.7. □

Let X be an initial universe and A be a set of parameters. A soft set such that for all will be called a constant soft set and will be denoted by . The family of all constant soft sets in will be denoted by .

Let X be an initial universe and A be a set of parameters. Then and .

Let . Then and .

Straightforward. □

If is a topological space, then for any set of parameters A, the collection is a soft topology on X relative to A.

Suppose that is a topological space. Since ∅, , then , and so , . Let , where . Then and so . By Lemma 3.14, and thus . Let where , then and so . By Lemma 3.14, and thus . This ends the proof that is a STS. □

If is a topological space and A is a set of parameters, then we will denote the soft topology by .

If is a topological space and A is a set of parameters, then for all .

Clear. □

Let be a STS with , then is a topology on X.

Suppose that is a STS with . Let . Since , , then ∅, . Let . Then , and so . By Lemma 3.14, . Thus, . Let . Then and so . By Lemma 3.14, . Thus, . This ends the proof that is a topological space. □

If is a STS with , then the topology on X will be denoted by .

If is a STS with , then for all .

Let be a topological space and let A be a set of parameters. Then .

Let be a STS with . Then .

Minimal soft open sets

In this section, we continue the study of minimal soft open sets; in particular, we give a mapping regarding minimal soft open sets, also we give a link between the soft minimal open sets of a soft topological space and the minimal open sets of its generated topological spaces and vice versa.

[21] Let be a STS. A soft set is said to be a minimal soft open set of if and for all with either or .

The set of all minimal soft open sets of a STS will be denoted by .

Let be a STS. If and , then or .

Let be a STS. If , then or .

Let be a soft continuous function. If with , then .

Since , then there is . So, and hence . Suppose with . Choose . Then . So, there is such that . Since is soft continuous, then . Since we have , then . So by Proposition 4.2, . Hence, . It follows that . Hence, . □

Let be bijective and soft open. If with , then .

Let be a STS and let be a soft base for τ. Then .

Let . Choose . Since is a soft base for τ, then by Proposition 2.29, there exists such that F. Since and , then . Hence, . □

If , then for every with , .

Suppose and let such that . Suppose with . Take such that . Choose . Then . Since , then by Proposition 4.2 we have . Therefore, . Hence, . This shows that . □

Let be a STS and let be a soft base for τ. Then for every , the family forms a base for the topology on X.

The following example shows that the converse of Theorem 4.7 is not true in general:

Let and . Let

Let and let τ be the soft topology on X relative to A having as a soft base. For every , set . By Lemma 4.8, , is a base for and is a base for . Now and , however, because with but .

A collection is called a soft partition of if the following three conditions hold:

(a) .

(b) If , then or .

(c) .

For any STS , the following are equivalent:

(a) .

(b) is a soft partition of .

(c) is a soft base of .

(a) ⟹ (b): By the definition of minimal soft open sets, . Also, by Proposition 4.3 we have or for all . The assumption that ends the proof that is a soft partition of .

(b) ⟹ (c): We apply Proposition 2.29. By the definition of minimal soft open sets we have . Let . Choose . Then . Since, by (b), is a soft partition of , then , and so there is such that . Since , then by Proposition 4.2 we have . This shows that is a soft base of .

(c) ⟹ (a): Obvious. □

If is a soft partition of , then for every the set is a soft partition of X.

Straightforward. □

Let X be an initial universe and let A be a set of parameters. Let be an indexed family of topologies on X. Then .

Let . By Theorems 3.5 and 4.6, there is and such that . Since , then it is clear that . The other inclusion is straightforward. □

If is a topological space and A is a set of parameters, then .

Let where . Let with . Then we have with and so . Thus, and hence . Conversely, let . Then . Let where and . So, we have with and hence . It follows that and hence . □

If is a STS with , then .

Similar to the proof of Theorem 4.14. □

Soft homogeneity

In this section, we introduce and investigate soft homogeneity in soft topological spaces; in particular, we study the relation between soft homogeneity of a soft topological space and some ordinary topological spaces generated by this soft topological space, we study the relation between homogeneity of an ordinary topological space and some soft topological spaces generated by this ordinary topological space, and we introduce some properties of homogeneous soft topological spaces that contains a minimal soft open set.

A STS is called soft homogeneous if for any , there exists a soft homeomorphism such that .

Let be a STS. Then the following are equivalent:

(a) is soft homogeneous.

(b) For any , there is a soft homeomorphism such that and .

(c) For any two pairs , there is a soft homeomorphism such that and .

Follows directly from the definition. □

Let , , , and . Then is soft homogeneous.

Let .

Case 1. and . Take and . Then is a bijection with .

is soft open: and .

is soft continuous: .

Case 2. and . Take and . Then is a bijection with .

is soft open: and .

is soft continuous: .

Case 3. and . Take and . Then is a bijection with .

is soft open: and .

is soft continuous: .

Case 4. and . Take and . Then is a bijection with .

is soft open: and .

is soft continuous: .

Case 5. and . Take and . Then is a bijection with .

is soft open: and .

is soft continuous: .

Case 6. and . Take and . Then is a bijection with .

is soft open: and .

is soft continuous: . □

Each of the rest cases is similar to one of the above six cases.

Let and . Let and . The soft topological space which has as a soft base is soft homogeneous.

Let .

Case 1. and . Take and . Then is a bijection with .

is soft open: , , and .

is soft continuous: .

Case 2. and . Take and . Then is a bijection with .

is soft open: , , and .

is soft continuous: .

Case 3. and . Take and . Then is a bijection with .

is soft open: , , and .

is soft continuous: .

Case 4. and . Take and . Then is a bijection with .

is soft open: , , and .

is soft continuous: .

Case 5. and . Take and . Then is a bijection with .

is soft open: , , and .

is soft continuous: . □

Each of the rest cases is similar to one of the above five cases.

Let and . Let and . The soft topological space which has as a soft base is soft homogeneous.

Let .

Case 1. and . Take and . Then is a bijection with .

is soft open: , , .

is soft continuous: .

Case 2. and . Take and . Then is a bijection with .

is soft open: , , .

is soft continuous: , , .

Case 3. and . Take and . Then is a bijection with .

is soft open: , , .

is soft continuous: .

Case 4. and . Take and . Then is a bijection with .

is soft open: , , .

is soft continuous: .

Case 5. and . Take and . Then is a bijection with .

is soft open: , , .

is soft continuous: .

Case 6. and . Take and . Then is a bijection with .

is soft open: , , .

is soft continuous: . □

Each of the rest cases is similar to one of the above six cases.

For any non-empty set X and for any set of parameters A, the STS is soft homogeneous.

Let . Define and as follows: and

Then is a soft homeomorphism with . □

Let be a homogeneous STS. If there is , then .

It is sufficient to see that . Let . Since is soft homogeneous, then there is a soft homeomorphism such that . Since , then . It follows that . □

Let be a STS such that there is . Then is soft homogeneous iff .

Follows directly from Theorems 5.6 and 5.7. □

Let X be a non-empty set and let α be a cardinal number. A partition of X is called an α-partition of X if for all .

Let be a topological space, be a base of and let α be a cardinal number. Then is called an α-partition base of if is an α-partition of X.

Let be a topological space which contains . Then is homogeneous iff has an -partition base.

Let be soft continuous. Then for every , is continuous.

Let be a soft homeomorphism. Then for every , is a homeomorphism.

If is soft homogeneous, then for all , is homogeneous.

Let . Let . Since is soft homogeneous, then there is a soft homeomorphism such that . Then and . By Corollary 5.13, is a homeomorphism with . It follows that is homogeneous. □

If is soft homogeneous, then for all , and are homeomorphic.

Let . Choose . Since is soft homogeneous, then there is a soft homeomorphism such that . Then . By Corollary 5.13, is a homeomorphism. It follows that and are homeomorphic. □

Each of the following two examples show that the converse of Theorem 5.14 is not true in general:

Let and . Let

, and let be the STS that has as a soft base.

1. is the discrete topology and , and by Theorem 5.11 both of and are homogeneous.

2. Since it is clear that and are not homeomorphic, then by Theorem 5.15, is not soft homogeneous.

Let and . Let and be the usual and the discrete topologies on , respectively. Let .

1. and . It is well known that both of and are homogeneous.

2. Since and are not homeomorphic, then by Theorem 5.15, is not soft homogeneous.

Let be a soft mapping. Then for every , .

Let and let . Then

Therefore, . □

Let be a soft mapping with and . Then is soft continuous iff is continuous.

Necessity. Suppose that is soft continuous. Let . Then and so . By Lemma 5.18, . Thus, . Hence, is continuous.

Sufficiency. Suppose that is continuous. Let . Then and so . Thus, we have . By Lemma 5.18, . Thus, . Therefore, is soft continuous. □

Let be a soft mapping with and . Let be a bijection. Then is a soft homeomorphism iff is a homeomorphism.

Let be a STS with . Then is soft homogeneous iff is homogeneous.

Necessity. Suppose that is soft homogeneous. Let . Choose . Then there is a soft homeomorphism such that . So, . Also by Corollary 5.20, is a homeomorphism with . Therefore, is homogeneous.

Sufficiency. Suppose that is homogeneous. Let . Choose a homeomorphism such that . Choose a bijection such that . By Corollary 5.20, is a soft homeomorphism. Therefore, by Theorem 5.2, it follows that is soft homogeneous. □

Let be a function. Let be a function between two sets of parameters A and B. Then is soft continuous iff p is continuous.

Necessity. Suppose that is soft continuous. Choose . Then by Theorem 5.12, is continuous. By Theorem 3.17, and . It follows that is continuous.

Sufficiency. Suppose that is continuous. Let . Then and so . Thus, we have . By Lemma 5.18, . Thus, . Therefore, is soft continuous. □

Let be a function. Let be a bijection between two sets of parameters A and B. Then is a soft homeomorphism iff p is a homeomorphism.

Let be a topological space and let A be a set of parameters. Then is soft homogeneous iff is homogeneous.

Necessity. Suppose that is soft homogeneous. Let . Choose . Then there is a soft homeomorphism such that . So, . Also by Corollary 5.23, is a homeomorphism with . Therefore, is homogeneous.

Sufficiency. Suppose that is homogeneous. Let . Choose a homeomorphism such that . Choose a bijection such that . By Corollary 5.23, is a soft homeomorphism. Therefore, by Theorem 5.2, it follows that is soft homogeneous. □

Consider the topological space where ℑ is the usual topology on and let . It is well known that is homogeneous. Then by Theorem 5.24, is soft homogeneous.

Consider the topological space where ℑ is the usual topology on and let . It is well known that is not homogeneous. Then by Theorem 5.24, is not soft homogeneous.

Consider the topological space where and ℑ is the usual topology on , and let . It is well known that is homogeneous. Then by Theorem 5.24, is soft homogeneous.

Consider the topological space where and ℑ is the usual topology on X, and let . It is well known that is not homogeneous. Then by Theorem 5.24, is not soft homogeneous.

Let X be an initial universe and let A be a set of parameters. Let be an indexed family of topologies on X. If is soft homogeneous, then

(a) is homogeneous for all .

(b) is homeomorphic to for all .

(a) Let . “Then by Theorem 5.14, is homogeneous. By Theorem 3.7 we have and hence is homogeneous.

(b) Let . Then by Theorem 5.15, and are homeomorphic. By Theorem 3.7 we have and . It follows that is homeomorphic to . □

Let be an indexed family of topological spaces such that

(a) is homogeneous for all .

(b) is homeomorphic to for all .

Is it true that is soft homogeneous?

Let be a function between two topological spaces and let be a function. Then is soft continuous iff is continuous.

Necessity. Suppose that is soft continuous. Choose . Then by Theorem 5.12, is continuous. By Corollary 3.8, and . Thus, is continuous.

Sufficiency. Suppose that is continuous. Let and let . Then . Since , then . Since is continuous, then and so . Therefore, and hence is soft continuous. □

Let be a function between two topological spaces and let be a bijection. Then is a soft homeomorphism iff is a homeomorphism.

Necessity. Suppose that is a soft homeomorphism. Then is a bijection. By Theorem 5.31, and are continuous. It follows that is a homeomorphism.

Sufficiency. Suppose that is a homeomorphism. Then is a bijection. Since we have p and u are bijections, then is a bijection. Since is continuous, then by Theorem 5.31, is soft continuous. Since is continuous, then again by Theorem 5.31, is soft continuous. If follows that is a soft homeomorphism. □

The following result answers Question 5.30 partially:

Let be a topological space and let A be a set of parameters. Then is soft homogeneous iff is homogeneous.

Necessity. Suppose that is soft homogeneous. For all , put . Then and so is soft homogeneous. By Theorem 5.29 (a), it follows that is homogeneous.

Sufficiency. Suppose that is homogeneous. Let . Choose a homeomorphism such that . Choose a bijection such that . By Theorem 5.32, is a soft homeomorphism. Therefore, by Theorem 5.2, it follows that is soft homogeneous. □

Consider the topological space where ℑ is the usual topology on and let . It is well known that is homogeneous. Then by Theorem 5.33, is soft homogeneous.

Consider the topological space where X is the Cantor set and ℑ is the usual topology on X, and let . It is well known that is homogeneous. Then by Theorem 5.33, is soft homogeneous.

For every , the set is called the support of F.

Let be a soft mapping and let . Then .

To see that , let . Then . Note that . So, there is such that . Thus, we have and . Therefore, . This ends the proof that . To see that , let . Then there is such that . Thus, we have and . It follows that . This ends the proof that . □

Let be a soft mapping with is an injection, and let . Then .

By Theorem 5.36, and so . Since is an injection, then . It follows that . □

Let be a homogeneous STS which contains . Then

(a) is a soft base for .

(b) For all and for all with and , .

(c) For all , .

(a) We apply Proposition 2.29. Clearly that . Let and let . Choose . Since is soft homogeneous, then there is a soft homeomorphism such that . So, . Since , then by Theorem 4.4, . Thus, by Proposition 4.2, . Therefore, is a soft base for .

(b) Let and let such that and . Choose and . Then and . Since is soft homogeneous, then there is a soft homeomorphism such that . Since , by Theorem 4.4, . Since , then . Since and , then by Proposition 4.3, . Note that . Since p is bijective, then . Therefore, .

(c) Choose and . Since is soft homogeneous, then there is a soft homeomorphism such that . By Theorem 4.4, . Since , then by Proposition 4.3, . Thus, by Theorem 5.37, it follows that . □

Let be a homogeneous STS with the property that for all . If , then we have the following:

(a) is a soft base for .

(b) For all and for all , .

Conclusion

In this paper, homogeneity as an ordinary topological property is extended to include soft topological spaces. Also, the study of soft minimal open sets is continued. The results deals mainly with the relation between the generated soft topology and the given indexed family of topologies defined in [17]. Also, some properties of soft homogeneous soft topological spaces that contains a minimal soft open set are given. In our future study, the following topics could be considered: 1) To define soft homogeneity components; 2) To extend countable dense homogeneity to include soft topological spaces.

Declarations

Author contribution statement

Samer Al Ghour: Conceived and designed the experiments; Analyzed and interpreted the data; contributed reagents, materials, analysis tools or data; Wrote the paper.

Awatef Bin-Saadon: Performed the experiments; Analyzed and interpreted the data; contributed reagents, materials, analysis tools or data; Wrote the paper.

Funding statement

This research did not receive any specific grant from funding agencies in the public, commercial, or not-for-profit sectors.

Competing interest statement

The authors declare no conflict of interest.

Additional information

No additional information is available for this paper.