A non-continuous soft mapping that preserves some structural soft sets

Abstract

As everyday problems contain a lot of data and ambiguity, it has become necessary to develop new mathematical approaches to address them and soft set theory is the best tool to deal with such problems. Hence, in this article, we introduce a non-continuous mapping in soft settings called soft $U$ -continuous. We mainly focus on studying soft $U$ -continuity and its connection to soft continuity. We further show that soft $U$ -continuity preserves soft compact sets and soft connected sets. The later sets have various applications in computing science and decision making theory. In the end, we show that if each soft $U$ -continuous mapping f from a soft space X into a soft T₀-space Y is soft continuous, then Y is soft T₁.

Keywords

soft continuous soft compact soft connected soft separable

1 Introduction

The data used in most real-world problems in social sciences, engineering, medical sciences, economics, and other fields is imprecise. The usage of mathematical ideas based on uncertainty and imprecision is used to solve such difficulties. As a result, classical set theory, which is predicated on the crisp and accurate case, may not be totally suitable for dealing with such uncertainty concerns. A variety of theories have been presented for effectively dealing with uncertainties. Namely: the theory of fuzzy sets [39], theory of intuitionistic fuzzy sets [10], theory of vague sets [20], theory of rough sets [33], and others. Each of these theories has its own set of benefits and drawbacks when it comes to dealing with uncertainty. One key issue that all of these ideas have in common is their incompatibility with parameterization tools. Molodtsov [31] presented a whole new idea for modeling uncertainty, in 1999, called soft set theory, which associates a set with a set of parameters and thereby avoids the challenges imposed by the identified problem. Soft set theory has been shown to have a wide range of applications in areas such as function smoothness, Riemann integration, decision making, measurement theory, and game theory. The link between soft sets and information systems was studied by Pei and Miao [34]. They demonstrated that soft sets are a type of special information system. Then Roy et al. [35] gave the notion of fuzzy soft set theory. Shabir and Naz [36] and Çağman et al. [15] separately established the concepts of soft topological spaces and soft open sets in 2011. The concepts of compactness [11], connectedness [26], and separation axioms [14, 36] have all been introduced in soft contexts in a similar way. Then, in the literature, several generalized forms of soft continuity [11] appeared. For example: soft semicontinuity [27], soft β-continuity [38], soft somewhat continuity [9], soft somewhere dense continuity [5], and so on.

In this paper, we study the so called soft $U$ -continuous mappings. Generally speaking, a soft $U$ -continuous mapping need not be soft continuous. We find some soft topological properties under which soft $U$ -continuity and soft continuity are identical. Then, we prove that soft $U$ -continuity preserves soft compactness and soft connectedness but not soft separability. It is worth saying that soft compact sets and soft connected sets have their applications in decision making theory [23, 25]. Finally, we characterize soft T₁-spaces with the help of $U$ -continuous mappings.

2 Preliminaries

This section introduces some key terms and definitions that will be utilized in the sequel. We will refer to X as an initial universe, E as a set of parameters, and P (X) as the power set of X from now on.

Definition 2.1. [31] A collection F (E) = {(e, F (e)) : e ∈ E} is said to be a soft set over X, where F : E → ¶ (X) is a (crisp) map. The class of all soft sets on X is represented by $P (X, E)$ . If L ⊆ E, then it will be represented by $P (X, L)$ .

Definition 2.2. [8, 32] A soft set F (E) over X is called

a soft element if F (e) = {x} for all e ∈ E, where x ∈ X. It is denoted by ({x} , E).

a soft point if e ∈ E and x ∈ X such that F (e) = {x} and F (e′) =∅ for each e′ ≠ e, e′ ∈ E. It is denoted by x (e). A statement $x (e) \tilde{\in} F (E)$ means that x ∈ F (e).

By {x (e)} = {(e, {x}) , (e′, ∅) , ⋯ : ∀ e′ ∈ E, e′ ≠ e} we mean a singleton soft set. If E = {e₁, e₂, e₃} and X = {a, b}, the singleton soft set {a (e₁)} = {(e₁, {a}) , (e₂, ∅) , (e₂, ∅)}.

Definition 2.3. [7] The complement of F (E) is a soft set X (E) ∖ F (E) (or simply F (E) ^c), where F^c : E → ¶ (X) is given by F^c (e) = X ∖ F (e) for all e ∈ E.

Definition 2.4. [31] A soft subset F (E) over X is called

null if F (e) =∅ for any e ∈ E.

absolute if F (e) = X for any e ∈ E.

The null and absolute soft sets are respectively represented by

\tilde{Φ}

and

\tilde{X}

. Clearly,

{\tilde{X}}^{c} = \tilde{Φ}

and

{\tilde{Φ}}^{c} = \tilde{X}

Definition 2.5. [28] Let E₁, E₂ ⊆ E. It is said that G (E₁) is a soft subset of H (E₂) (written by $G (E_{1}) \tilde{\subseteq} H (E_{2}))$ if E₁ ⊆ E₂ and F (e) ⊆ G (e) for any e ∈ E₁. We say G (E₁) = H (E₂) if $G (E_{1}) \tilde{\subseteq} H (E_{2})$ and $H (E_{2}) \tilde{\subseteq} G (E_{1})$ .

Maji et al. [28] gave definitions of the soft union and soft intersection of two soft sets with respect to arbitrary subsets of E. However, as Ali et al. [7] and Terepeta [37] report, these definitions are inaccurate and confusing. As a result, we stick to Terepeta’s [37] definitions.

Definition 2.6. Let {F_α (E) : α ∈ Λ} be a collection of soft sets over X, where Λ is any index set.

The intersection of F_α (E), for α ∈ Λ, is a soft set G (E) such that G (e) = ⋂ _α∈ΛF_α (e) for each e ∈ E and denoted by $G (E) = {⋂^{˜}}_{α \in Λ} F_{α} (E)$ .

The union of F_α (E), for α ∈ Λ, is a soft set G (E) such that G (e) = ⋃ _α∈ΛF_α (e) for each e ∈ E and denoted by $G (E) = {⋃^{˜}}_{α \in Λ} F_{α} (E)$ .

Definition 2.7. [36] A subfamily $T$ of $P (X, E)$ is called a soft topology on X if

$\tilde{Φ}$ and $\tilde{X}$ belong to $T$ ,

finite intersection of sets from $T$ belongs to $T$ , and

any union of sets from $T$ belongs to $T$ .

Terminologically, we call

(X, T, E)

a soft topological space on X. The elements of

T

are called soft open sets, and their complements are called soft closed sets.

In what follow, by $(X, T, E), (Y, §, E^{'})$ we mean soft topological spaces.

Definition 2.8. [15] A subfamily $B \subseteq T$ is called a soft base for the soft topology $T$ if each element of $T$ is a union of elements of $B$ .

Definition 2.9. [1, Definition 3.4] Let $G$ be a collection of soft subset of (X, E). The soft topology on X generated by $G$ is the intersection of all soft topologies on X containing $G$ .

Definition 2.10. [36] Let Y (E) be a non-null soft subset of $(X, T, E)$ . Then $T_{Y} :$ $= {G (E) ⋂^{˜} Y (E) : G (E) \tilde{\in} T}$ is called a soft relative topology on Y and $(Y, T_{Y},$ E) is a soft subspace of $(X, T, E)$ .

Definition 2.11. [36] Let F (E) be a soft subset of $(X, T, E)$ . The soft interior of F (E) is the largest soft open set contained in F (E) and denoted by int_X (F (E)) (or shortly int (F (E))). The soft closure of F (E) is the smallest soft closed set which contains F (E) and denoted by cl_X (F (E)) (or simply cl (F (E))).

Lemma 2.12. [24] For a soft subset G (E) of a space $(X, T, E)$ , we have int (G (E) ^c) = (cl (G (E))) ^c, cl (G (E) ^c) = (int (G (E))) ^c.

Definition 2.13. [14, 19] A soft topological space $(X, T, E)$ is called

soft T₀ if for each x (e) , y (e′) over X with x (e) ≠ y (e′), there exist soft open sets G (E) , H (E) such that $x (e) \tilde{\in} G (E)$ , $y (e^{'}) \tilde{\notin} G (E)$ or $y (e^{'}) \tilde{\in} H (E)$ , $x (e) \tilde{\notin} H (E)$ .

soft T₁ if for each x (e) , y (e′) over X with x (e) ≠ y (e′), there exist soft open sets G (E) , H (E) such that $x (e) \tilde{\in} G (E)$ , $y (e^{'}) \tilde{\notin} G (E)$ and $y (e^{'}) \tilde{\in} H (E)$ , $x (e) \tilde{\notin} H (E)$ ,

soft T₂ (soft Hausdorff) if for each x (e) , y (e′) over X with x (e) ≠ y (e′), there exist soft open sets G (E) , H (E) containing x (e) , y (e′) respectively such that $G (E) ⋂^{˜} H (E) = \tilde{Φ}$ .

soft regular if for each soft closed set F (E) and each soft point x (e) with $x (e) \tilde{\notin} F (E)$ , there exist soft open sets G (E) , H (E) such that $x (e) \tilde{\in} G (E)$ , $F (E) \tilde{\subseteq} H (E)$ and $G (E) ⋂^{˜} H (E) = \tilde{Φ}$ .

The above soft separation axioms have been defined by Sabir and Naz [36] for the first time with respect to soft elements.

Definition 2.14. A space $(X, T, E)$ is called

soft compact [11] if each soft open cover of $\tilde{X}$ has a finite subcover, (c.f. compact structural softsets in [25]).

soft locally compact [13] if each soft point has a soft compact neighborhood.

soft connected [26] if it cannot be written as a union of two disjoint soft open sets.

3 Soft $U$ -continuity and its properties

Definition 3.1. A mapping $f : (X, T, E) \to (Y, §, E^{'})$ is said to be soft continuous with respect to a soft cover $U$ (shortly, $U$ -continuous) if for each soft open cover $V$ of $\tilde{Y}$ , there exists a soft open cover $U$ of $\tilde{X}$ such that for each $U (E) \tilde{\in} U$ , there exists $V (E^{'}) \tilde{\in} V$ and $f (U (E)) \tilde{\subseteq} V (E^{'})$ .

From the definition, one can conclude the following remark:

Remark 3.2. (1) A mapping $f : (X, T, E) \to (Y, §, E^{'})$ is soft $U$ -continuous if and only if for each $x (e) \tilde{\in} \tilde{X}$ and each soft open cover $V$ of $\tilde{Y}$ , there is a soft open set U (E) over X containing x (e) and a soft open set $V (E^{'}) \tilde{\in} V$ such that $f (U (E)) \tilde{\subseteq} V (E^{'})$ .

(2) A crisp mapping $f (e) : (X, T) \to (Y, §)$ is $U$ -continuous if $f : (X, T, E)$ → (Y, § , E′) is soft $U$ -continuous for each e ∈ E.

Proposition 3.3. Each soft continuous mapping is soft $U$ -continuous.

Proof. The proof is simple, given a soft continuous mapping $f : (X, T, E) \to (Y, §, E^{'})$ , since for each soft open cover $V$ of $\tilde{Y}$ , $f^{- 1} (V) = {f^{- 1} (V (E^{'})) : V (E^{'}) \tilde{\in} V}$ is a soft open cover of $\tilde{X}$ by soft continuity and $f (f^{- 1} (V (E^{'}))) \tilde{\subseteq} V (E^{'})$ for each f^-1 (V (E′)) in $f^{- 1} (V)$ . □

The next example shows that a soft $U$ -continuous mapping need not be soft continuous.

Example 3.4. Let $X = ℝ$ be the set of real numbers and E = {e₁, e₂} be a set of parameters. Let $T$ be a soft topology on X generated by ${(e_{1}, B (e_{1})), (e_{2}, B (e_{2}))},$ where B (e₁) = (a, b) , B (e₂) = (c, d) for $a, b, c, d \in ℝ; a < b, c < d$ and let § be another soft topology on X generated by ${(e_{1}, B (e_{1})), (e_{2}, B (e_{2}))},$ where B (e₁) = (a, ∞) , B (e₂) = [b, ∞) for $a, b \in ℝ$ The identity mapping $f : (X, T, E) \to (X, §, E)$ is soft $U$ -continuous but not soft continuous.

Proposition 3.5. If $f : (X, T, E) \to (Y, §, E^{'})$ is a soft $U$ -continuous mapping and A (E) is any soft subset of $\tilde{X}$ . Then $f |_{A (E)} : (A, T_{A}, E) \to (Y, §, E^{'})$ is soft $U$ -continuous.

Proof. Apply Theorem 1 in [36]. □

Theorem 3.6. A mapping $f : (X, T, E) \to (Y, §, E^{'})$ is soft $U$ -continuous if and only if for each soft open cover $V$ of $\tilde{Y}$ the family ${int (f^{- 1} (V)) : V \tilde{\in} V}$ forms a cover for $\tilde{X}$ .

Proof. Suppose that f is soft $U$ -continuous and let $V$ be any soft open cover of $\tilde{Y}$ . Choose any $x (e) \tilde{\in} \tilde{X}$ , then there is a soft open set U (E) containing x (e) and a soft open set $V (E^{'}) \tilde{\in} V$ such that $f (U (E)) \tilde{\subseteq} V (E^{'})$ . Therefore $x (e) \tilde{\in} U (E) \tilde{\subseteq} int (f^{- 1} (V (E^{'})) \tilde{\subseteq} ⋃^{˜} int (f^{- 1} (V (E^{'})) .$ Hence ${int (f^{- 1} (V (E^{'}))) : V (E^{'}) \tilde{\in} V}$ covers $\tilde{X}$ .

Conversely, given $x (e) \tilde{\in} \tilde{X}$ and let $V$ be a soft open cover of $\tilde{Y}$ . By assumption ${int (f^{- 1} (V (E^{'}))) : V (E^{'}) \tilde{\in} V}$ covers $\tilde{X}$ , so there is $V (E^{'}) \tilde{\in} V$ such that $x (e) \tilde{\in} int (f^{- 1} (V (E^{'}))$ and so $f (int (f^{- 1} (V (E^{'})))) \tilde{\subseteq} V (E^{'}) .$ Hence f is soft $U$ -continuous. □

Theorem 3.7. For any soft mapping $f : (X, T, E) \to (Y, §, E^{'})$ , there is a family ${U_{α} (E) \in T : α \in Δ}$ of soft open subsets over X such that $C_{U} (f) = ⋂^{˜} {U_{α} (E) : α \in Δ},$ where $C_{U} (f)$ is the set of all soft $U$ -continuity points of f.

Proof. For any soft open cover $V$ of $\tilde{Y}$ , we set $W_{V} (E) = {x (e) \tilde{\in} \tilde{X} : x (e) \tilde{\in} U (E), V (E^{'}) \tilde{\in} V, f (U (E)) \tilde{\subseteq} V (E^{'})} .$ We can easily verify that $W_{V} (E)$ is a soft open set over X. We now prove that $C_{U} (f) = ⋂^{˜} {W_{V} (E) : V$ is a soft open cover of $\tilde{Y}} .$

If $x (e) \tilde{\in} C_{U} (f)$ , then for each soft open cover $V$ of $\tilde{Y}$ , there is a soft open set U (E) over X containing x (e) and a soft open set $V (E^{'}) \tilde{\in} V$ such that $f (U (E)) \tilde{\subseteq} V (E^{'})$ . This implies that $x (e) \tilde{\in} W_{V} (E)$ for each soft open cover $V$ of $\tilde{Y}$ .

On the other hand, if $x (e) \tilde{\in} W_{V} (E)$ for each soft open cover $V$ of $\tilde{Y}$ , then obviously f is soft $U$ -continuous at x (e). □

From this result, we conclude that

Corollary 3.8. If $f : (X, T, E) \to (Y, §, E^{'})$ is a soft mapping and (Y, § , E′) has a countable soft base, then the set of all soft $U$ -continuity points of f forms a soft G_δ set (=countable intersection of soft open sets).

Theorem 3.9. Let $f : (X, T, E) \to (Y, §, E^{'})$ be a soft mapping and let $(X, T, E)$ be a soft locally compact space. The following are equivalent:

f is soft $U$ -continuous;

for each soft open cover $V$ of $\tilde{Y}$ , there is a cover $K$ of $\tilde{X}$ containing soft compact neighborhoods such that for each $K (E) \tilde{\in} K$ , there is $V (E^{'}) \tilde{\in} V$ such that $f (K (E)) \tilde{\subseteq} V (E^{'})$ .

Proof. (i)→(ii) Suppose that f is soft $U$ -continuous and let $V$ be any soft open cover of $\tilde{Y}$ . Then for each $x (e) \tilde{\in} \tilde{X}$ , there exist soft open U (E) containing x (e) and soft open $V (E^{'}) \in V$ such that $f (U (E)) \tilde{\subseteq} V (E^{'})$ . Therefore $x (e) \tilde{\in} U (E) \tilde{\subseteq} int (f^{- 1} (V (E^{'}))$ and so $x (e) \tilde{\in} int (f^{- 1} (V (E^{'}))) .$ Since X is soft locally compact, there exists a soft compact neighborhoods K_x(e) (E) such that $x (e) \tilde{\in} K_{x (e)} (E) \tilde{\subseteq} int (f^{- 1} (V (E^{'}))) .$ Hence $K = {K_{x (e)} (E) : K_{x (e)} (E) \tilde{\subseteq} \tilde{X}}$ is a soft compact neighborhoods cover of $\tilde{X}$ and $\begin{matrix} f (K_{x (e)} (E)) & \tilde{\subseteq} & f (int (f^{- 1} (V (E^{'}))) \\ \tilde{\subseteq} & f ((f^{- 1} (V (E^{'}))) \\ \tilde{\subseteq} & V (E^{'}) . \end{matrix}$

(i)→(ii) Let $V$ be any soft open cover of $\tilde{Y}$ . By (ii), there exists a cover $K$ of $\tilde{X}$ containing soft compact neighborhoods such that for each $K (E) \tilde{\in} K$ , there is $V (E^{'}) \tilde{\in} V$ such that $f (K (E)) \tilde{\subseteq} V (E^{'})$ . Since $K$ covers a soft locally compact space $\tilde{X}$ , then for each $x (e) \tilde{\in} \tilde{X}$ and each $K (E) \tilde{\in} K$ , there exists a soft open set G (E) such that $x (e) \tilde{\in} G (E) \tilde{\subseteq} K (E)$ . Therefore the family $G$ of all G (E)s’ covers $\tilde{X}$ and $f (G (E)) \tilde{\subseteq} f (K (E)) \tilde{\subseteq} V (E^{'})$ and so f is soft $U$ -continuous. □

Theorem 3.10. Let $f : (X, T, E) \to (Y, §, E^{'})$ be a soft mapping and let each singleton soft set in (Y, § , E′) be soft closed. If f is soft $U$ -continuous, then it is soft continuous.

Proof. Pick an arbitrary $x (e) \tilde{\in} \tilde{X}$ and let V (E′) be a soft open set containing f (x (e)). By assumption, $\tilde{Y} ∖ {f (x (e))}$ is soft open over Y. Therefore $V = {V (E^{'}), \tilde{Y} ∖ {f (x (e))}}$ is a soft open cover of $\tilde{Y}$ . Since f is soft $U$ -continuous, then there exists a soft open cover $U$ of $\tilde{X}$ such that for each $U (E) \tilde{\in} U$ , there is $H (E^{'}) \tilde{\in} V$ such that $f (U (E)) \tilde{\subseteq} H (E^{'})$ . Since $U$ covers $\tilde{X}$ , then there is $U (E) \tilde{\in} U$ containing x (e) and so f (U (E)) containing f (x (e)). Therefore $f (U (E)) \tilde{\subseteq} V (E^{'})$ . This proves that f is soft continuous. □

The following example shows the soft closedness of each singleton soft set in (Y, § , E′) is important in Theorem 3.10.

Example 3.11. The mapping $f : (X, T, E) \to (X, §, E)$ given in Example 3.4 is soft $U$ -continuous but not soft continuous because not each singleton soft set in (X, § , E) is soft closed.

Theorem 3.12. Let $f : (X, T, E) \to (Y, §, E^{'})$ be a soft mapping and let (Y, § , E′) be a soft regular T₀-space. If f is soft $U$ -continuous, then it is soft continuous.

Proof. Let $x (e) \tilde{\in} \tilde{X}$ and let V (E′) be a soft open set containing f (x (e)). Since Y is soft T₀, then for each y (e′) with f (x (e)) ≠ y (e′), there is a soft open set H (E′) containing f (x (e)) but not y (e′). Therefore $y (e^{'}) \tilde{\in} \tilde{Y} ∖ H (E^{'})$ and so $cl ({y (e^{'})}) \tilde{\subseteq} \tilde{Y} ∖ H (E^{'})$ . Then $f (x (e)) \tilde{\notin} cl ({y (e^{'})})$ . Since (Y, § , E′) is soft regular, then for each $y (e^{'}) \tilde{\in} \tilde{Y}$ with y (e′) ≠ f (x (e)), there are soft open sets R (E′) , S (E′) such that $f (x (e)) \tilde{\in} R (E^{'})$ , $y (e^{'}) \tilde{\in} S (E^{'})$ and $R (E^{'}) ⋂^{˜} S (E^{'}) = \tilde{Φ}$ . Set $V = {V (E^{'}) ⋂^{˜} R (E^{'})} ⋃^{˜} {S (E^{'})},$ where $y (e^{'}) \tilde{\in} \tilde{Y}$ , $y (e^{'}) \tilde{\in} S (E^{'})$ , y (e′) ≠ f (x (e)). Then $V$ is a soft open cover of $\tilde{Y}$ . Since f is soft $U$ -continuous, then there exists a soft open cover $U$ of $\tilde{X}$ such that for each $U (E) \tilde{\in} U$ , there is $H (E^{'}) \tilde{\in} V$ such that $f (U (E)) \tilde{\subseteq} H (E^{'})$ . Since $U$ covers $\tilde{X}$ , then there is $U (E) \tilde{\in} U$ containing x (e) and so f (U (E)) containing f (x (e)). But $V (E^{'}) \tilde{\cap} R (E^{'})$ is the only soft open set that includes f (x (e)), hence $f (U (E)) \tilde{\subseteq} V (E^{'}) ⋂^{˜} R (E^{'}) \tilde{\subseteq} V (E^{'})$ . Hence f is soft continuous.□

The following examples show the soft regularity and soft T₀ of (Y, § , E′) cannot be removed in Theorem 3.12.

Example 3.13. The mapping $f : (X, T, E) \to (X, §, E)$ given in Example 3.4 is soft $U$ -continuous but not soft continuous as (X, § , E) is not soft regular.

Example 3.14. Let $X = ℝ$ be the set of real numbers and E = {e₁, e₂} be a set of parameters. Let $T$ be a soft topology on X generated by {(e₁, B (e₁)) , (e₂, B (e₂)) : B (e₁) = [a, b) , B (e₂) = (c, d) ;} , where $a, b, c, d \in ℝ; a < b, c < d$ and let § be another soft topology on X generated by {(e₁, B (e₁)) , (e₂, B (e₂)) : B (e₁) = [n, n + 1) , B (e₂) = [a, b)} , where $a, b \in ℝ, a < b, n \in ℤ$ . Clearly (X, § , E) is soft regular but not soft T₀ and the identity mapping $f : (X, T, E) \to (X, §, E)$ is soft $U$ -continuous but not soft continuous.

Theorem 3.15. Let $f : (X, T, E) \to (Y, §, E^{'})$ be a soft $U$ -continuous surjective mapping. If $(X, T, E)$ is a soft compact space, then (Y, § , E′) is soft compact.

Proof. Let $H$ be any soft open cover of $\tilde{Y}$ . Since f is soft $U$ -continuous, then there is a soft open cover $G$ of $\tilde{X}$ such that for each $G (E) \tilde{\in} G$ , there is $H (E^{'}) \tilde{\in} H$ such that $f (G (E)) \tilde{\subseteq} H (E^{'})$ . Since X is soft compact, then $G$ has a finite subcover. Namely: there exist G₁ (E) , G₂ (E) , ⋯ , G_n (E) such that $\tilde{X} = {⋃^{˜}}_{i = 1}^{n} G_{i} (E) .$ By assumption, for each G_i (E) with i = 1, 2, . . . , n, there is H_i (E′) such that $f (G_{i} (E)) \tilde{\subseteq} H_{i} (E^{'})$ . Therefore $\begin{matrix} \tilde{Y} = f (\tilde{X}) & = & f ({⋃^{˜}}_{i = 1}^{n} G_{i} (E)) \\ = & {⋃^{˜}}_{i = 1}^{n} f (G_{i} (E)) \\ \tilde{\subseteq} & {⋃^{˜}}_{i = 1}^{n} H_{i} (E^{'}) . \end{matrix}$ Thus Y is a soft compact space.□

Theorem 3.16. Let $f : (X, T, E) \to (Y, §, E^{'})$ be a soft $U$ -continuous surjective mapping. If $(X, T, E)$ is a soft connected space, then (Y, § , E′) is soft connected.

Proof. Suppose otherwise that Y is not soft connected. Then there are two non-null soft open sets H (E′) , W (E′) such that $\tilde{Y} = H (E^{'}) ⋃^{˜} W (E^{'}) & H (E^{'}) ⋂^{˜} W (E^{'}) = \tilde{Φ} .$ This implies $V = {H (E^{'}), W (E^{'})}$ forms a soft open cover for $\tilde{Y}$ . Since f is soft $U$ -continuous, there is a soft open cover $U$ of $\tilde{X}$ such that for each $U (E) \tilde{\in} U$ , there is $H (E^{'}) \tilde{\in} V$ such that $f (U (E)) \tilde{\subseteq} H (E^{'})$ . Set $R (E) = ⋃^{˜} {U (E) : U (E) \tilde{\in} U, f (U (E)) \tilde{\subseteq} H (E^{'})}$ and $S (E) = ⋃^{˜} {U (E) : U (E) \tilde{\in} U, f (U (E)) \tilde{\subseteq} W (E^{'})} .$ Clearly R (E) , S (E) are non-null. Since $H (E^{'}) ⋂^{˜} W (E^{'}) = \tilde{Φ},$ then R (E) $⋂^{˜} S (E)$ $= \tilde{Φ}$ . Furthermore, $R (E) ⋂^{˜} S (E)$ are soft open sets and $\tilde{X} = R (E) ⋃^{˜} S (E)$ (because $U$ covers $\tilde{X}$ ). But this is a contradiction to the assumption that X is soft connected. Hence Y must be soft connected.□

The following example shows that the image of a soft separable space under soft $U$ -continuity need not be soft separable.

Example 3.17. Let $X = ℝ$ be the set of real numbers and E = {e} be a set of parameters. Let $T$ be a soft topology on X generated by ${(e, B (e)) : B (e) = (a, b); a, b \in ℝ; a < b}$ and let § be another soft topology on X defined by ${A (E) \tilde{\subseteq} \tilde{X} : 0 (e) \tilde{\notin} A (E)$ or $A (E) = \tilde{X}} .$ The identity mapping $f : (X, T, E) \to (X, §, E)$ is soft $U$ -continuous and $(X, T, E)$ is soft separable but (X, § , E) not soft separable.

Theorem 3.18. If each soft $U$ -continuous mapping from a soft space $(X, T, E)$ into a soft T₀-space (Y, § , E′) is soft continuous, then (Y, § , E′) is a soft T₁-space.

Proof. Suppose for contradiction that Y is not soft T₁. Then there are y₁ (e^*), $y_{2} (e^{* *}) \in \tilde{Y}$ with y₁ (e^*) ≠ y₂ (e^**) such that each soft open set including y₁ (e^*) includes y₂ (e^**). Set X = {y₁, y₂} and E = {e^*, e^**}. Let $T = {\tilde{Φ}, \tilde{X}}$ . Define $f : (X, T, E) \to (Y, §, E^{'})$ to be the identity soft mapping. If $V$ is any soft open cover of $\tilde{Y}$ , the cover $U = {\tilde{X}}$ always exists and $f (\tilde{X}) \tilde{\subseteq} V (E^{'})$ for some $V (E^{'}) \tilde{\in} V$ as both y₁ (e^*) , y₂ (e^**) belong to some soft open set $\tilde{X}$ . This proves that the mapping f is soft $U$ -continuous. Since Y is soft T₀, so there is a soft open set, say, U₀ (E) such that $y_{1} (e^{*}) \tilde{\in} U_{0} (E)$ but $y_{2} (e^{* *}) \tilde{\notin} U_{0} (E)$ . This implies that f^-1 (U₀ (E)) = {y₁ (e^*)} which is not soft open over X and so f is not soft continuous. Hence (Y, § , E′) is soft T₁.□

The next example illustrates that the assumption soft T₀-space of (Y, § , E′) is essential.

Example 3.19. Let X = {1, 2} and E = {e₁, e₂}. Define the soft topologies $I$ and $D$ on X to be indiscrete and discrete, respectively. One can easily conclude that each mapping $f : (X, D, E) \to (X, I, E)$ is soft continuous, but $(X, I, E)$ is not a soft T₁-space.

Remark 3.20. At this place, it is time to notice that throughout the paper, we have used the soft point approach given in [8, 14] but we assert that all the obtained results are also true for the soft element approach defined in [36].

Furthermore, the assumption "each singleton soft set in (Y, § , E′) is soft closed" in Theorem 3.10 is equivalent to a soft T₁-space of (Y, § , E′) for soft points (see, Theorem 4.1 in [14]). Therefore, Theorem 3.12 can be followed from Theorem 3.10, but this is not the case for soft elements that is why we have kept both Theorems (c.f., Theorem 3.19 in [30] and Note 4.1 in [14]). The topic of soft separation axioms is indeed controversial, see [2, 19].

Lastly, if the cardinality of sets of parameters is one, so all the above results are true for classical topology by Theorem 1 in [37].

Conclusion and future work

In this paper, we defined a class of soft non-continuous mapping concerning some soft open cover of its domain under the name of soft $U$ -continuous. We studied soft $U$ -continuous mappings and determined their relationship with soft continuous mappings. We further showed that soft $U$ -continuity can preserve soft compact sets and soft connected sets, but not soft separable sets. Then we proved that if each soft $U$ -continuous mapping $f : (Y, T, E) \to (Y, §, E^{'})$ is soft continuous and (Y, § , E′) is soft T₀, then Y is soft T₁. We also discussed the obtained consequences through different theories of soft points.

The following tasks are expected to be completed as part of future work:

Different soft point theories can be applied to all the results presented in this paper.

Soft open covering is the key component in the definition of $U$ -continuity. One can take soft covers of weak types of soft open sets and study the consequences.

Since soft $U$ -continuity is nicely related to soft separation axioms, examining the soft $U$ -continuity with different types of soft separation axioms (see [18]) can be a worthwhile research topic to study.

On soft topological spaces, $U$ -continuity is defined. Different topological structures can be chosen to define $U$ -continuity in the same way. Namely, fuzzy topology [16], supra topology [29], infra soft topology [4], generalized topology [17], and so on.

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A non-continuous soft mapping that preserves some structural soft sets

Abstract

Keywords

1 Introduction

2 Preliminaries

3 Soft U -continuity and its properties

Conclusion and future work

References

3 Soft $U$ -continuity and its properties