Some Classes of Soft Functions Defined by Soft Open Sets Modulo Soft Sets of the First Category

Abstract

Soft continuity can contribute to the development of digital images and computational topological applications other than the field of soft topology. In this work, we study a new class of generalized soft continuous functions defined on the class of soft open sets modulo soft sets of the first category, which is called soft functions with the Baire property. This class includes all soft continuous functions. More precisely, it contains various classes of weak soft continuous functions. The essential properties and operations of the soft functions with the Baire property are established. It is shown that a soft continuous with values in a soft second countable space is identical to a soft function with the Baire property, apart from a topologically negligible soft set. Then we introduce two more subclasses of soft functions with the Baire property and examine their basic properties. Furthermore, we characterize these subclasses in terms of soft continuous functions. At last, we present a diagram that shows the relationships between the classes of soft functions defined in this work and those that exist in the literature.

1. Introduction

The reduction in uncertainty is one of the most crucial features that must be addressed in order to improve the robustness of the results acquired from data analysis. However, breaking down the existing uncertainty in order to remove it is frequently a difficult task. For this reason, numerous mathematical techniques created for data analysis ended up short of fulfilling these objectives. Fuzzy sets [1], rough sets [2], and probability are a few of the key mathematical techniques that try to remove uncertainty from data analysis. Divergences from classical mathematics with the goal of removing uncertainty have been made as a result of getting better results with various set types. The decision-making procedures have experienced certain issues as a result of their inability to effectively represent uncertainty. Considering that the lack of a parameterization tool is the primary cause of these challenges, the authors of [3] proposed soft sets. Soft sets represent a particularly effective mathematical model for processing decision-making procedures that focus on the selection of the best alternative since objects supplying parameters can be described in this way. This prompts the fast development of the theory of soft sets and its related field in a short measure of time and gives different applications of soft sets in real-world applications (see, [4,5,6,7,8,9]).

Then, various mathematical branches have been studied in soft set environments. Soft topology is one of the branches introduced in [10,11] as a fresh generality of classical topology. The aforementioned work was crucial to the development of soft topology. After that, in soft set contexts, many traditional topological properties have been generalized, for instance, soft separation axioms [12], soft second countable spaces [13], soft separable spaces [12], soft connected spaces [14], soft compact spaces [15], soft extremally disconnected spaces [16], soft submaximal spaces [17], and soft paracompact spaces [14].

It is understood that soft open sets are the building blocks of soft topology, but other classes of soft sets can contribute to the growth of soft topology. Namely, soft dense [18], soft codense [19], soft somewhat open [20], soft nowhere dense [18], soft meager (first category soft set) [18], soft semiopen [21], soft $\alpha$-open [22], and soft sets with the Baire property [23].

In addition to soft topology, soft continuity is useful in the development of computational topological applications and digital images [24]. Soft continuity of functions was defined by Zorlutuna et al. [25] in 2012. Afterwards, multiple generalized forms of soft continuous functions started to appear in the literature. Namely: soft $\mathcal{U}$-continuous functions [26], soft C-continuous functions [27], soft $\omega$-continuous functions [28], soft somewhat continuous functions [20], soft $\alpha$-continuous functions [22], soft semicontinuous functions [29], etc.

The concept of functions with the Baire property was studied by many mathematicians as a tool for developing several fields of mathematics, such as (descriptive) set theory, general topology, and measure theory (see [30,31,32]). In an analogous manner, studying functions with the Baire property in soft settings will have an interchangeable role in soft topology and soft measure theory. The latter statement and the rich literature on the generalized classes of soft continuous functions with their applications motivate us to investigate the so-called “soft functions with the Baire property”with two more subclasses of such soft functions.

The primary contributions of this paper are follows:

• We introduce a wide class of soft functions, named soft functions with the Baire property, via a mix of topological and algebraic structures, which includes various classes of generalized soft continuous functions.
• We find some conditions under which the class (or a subclass) of soft functions with the Baire property is identical to soft continuity.
• We characterize a subclass of soft functions with the Baire property in terms of the set of soft points of soft discontinuity.

We arrange the content of the paper as follows: Section 2 recalls some properties and operations of soft set theory and some soft topology. Section 3 collects and studies some classes of soft sets with the Baire property in soft topological spaces. In Section 4, we introduce the concept of soft functions with the Baire property and characterize them in terms of soft continuity. In Section 5, we define two subclasses of soft functions with the Baire property and study them. After that, we find their connections to some known classes of generalized soft continuous functions. In Section 6, we finish this work with a brief conclusion.

2. Preliminaries

We start with an overview of soft sets along with some operations.

Definition 1

([3]). Let F be a set-valued mapping from a subset A of a set of parameters E into the power set $2^X$ of an initial universe X. An ordered pair $(F,A)=\left\{\right(a,F\left(a\right)):a\in A\}$ is called the soft set over X.

The class of all soft subsets of X along with A is denoted by $SS(X,A)$.

Definition 2

([7]). The soft complement ${(F,A)}^c$ of a soft set $(F,A)$ is a soft set $(F^c,A)$, whereas $F^c:A\to 2^X$ is a mapping for which $F^c\left(a\right)=X-F\left(a\right)$ for each $a\in A$.

Remark 1.

One can easily extend a soft set $(F,A)$ to the soft set $(F,E)$ by assuming $F\left(a\right)=\varnothing$ for all $a\in E-A$.

Definition 3

([33]). A null soft set with respect to A, $(\Phi ,A)$, is soft set $(F,A)$ over X if $F\left(a\right)=\varnothing$ for each $a\in A$. An absolute soft set with respect to A, $(X,A)$, is a soft set $(F,A)$ such that $F\left(a\right)=X$ for each $a\in A$. The null and absolute soft sets are denoted by $(\Phi ,E)$ and $(X,E)$, respectively.

Notice that ${\left({(F,A)}^c\right)}^c=(F,A)$, ${(\Phi ,A)}^c=(X,A)$, and ${(X,A)}^c=(\Phi ,A)$.

Definition 4

([13]). A finite (resp. countable) soft set $(F,A)$ is such a soft set that $F\left(a\right)$ is finite (resp. countable) for each $a\in A$. Otherwise, it is called infinite (resp. uncountable).

Definition 5

([34]). A soft set $(F,A)$ over X is said to be a soft point, referred to $x_a$, if $F\left(a\right)=\left\{x\right\}$ and $F\left(a^{\prime }\right)=\varnothing$ for each $a^{\prime }\in A$ such that $a\ne a^{\prime }$, $a\in A$. The collection of all soft points in X associate with A is denoted by $SP(X,A)$.

Definition 6

([7,35]). Let $(F,A)$, $(G,B)$ be soft sets, where $A,B\subseteq E$. Then $(F,A)$ is a soft subset of $(G,B)$, denoted by $(F,A)\tilde{\subseteq }(G,B)$, if $A\subseteq B$ and $F\left(a\right)\subseteq G\left(a\right)$ for all $a\in A$. The two soft sets are said to be equal, denoted by $(F,A)=(G,B)$, if $(F,A)\tilde{\subseteq }(G,B)$ and $(G,B)\tilde{\subseteq }(F,A)$.

Definition 7

([33]). Let $\{(F_i,A):i\in I\}$ be a family of soft sets over X, where I is any index set.

1.

The soft union of $(F_i,A)$ is defined to be the soft set $(F,A)={\tilde{\cup }}_{i\in I}(F_i,A)$ such that $F\left(a\right)={\bigcup }_{i\in I}{F}_i\left(a\right)$ for each $a\in A$.

2.

The soft intersection of $(F_i,A)$ is defined to be the soft set $(F,A)={\tilde{\cap }}_{i\in I}(F_i,A)$ such that $F\left(a\right)={\bigcap }_{i\in I}{F}_i\left(a\right)$ for each $a\in A$.

Definition 8

([33,36]). Let $(F,A),(G,A)\in SS(X,A)$. Then

1.

The soft set difference $(F,A)$ and $(G,A)$ is defined to be the soft set $(H,A)=(F,A)-(G,A)$, where $H\left(a\right)=F\left(a\right)-G\left(a\right)$ for all $a\in A$.

2.

The soft symmetric difference of $(F,A)$ and $(G,A)$ is defined by $(F,A)\tilde{\Delta }(G,A)=[(F,A)-(G,A)]\tilde{\cup }[(G,A)-(F,A)]$.

One can easily check that $(F,A)-(G,A)=(F,A)\tilde{\cap }{(G,A)}^c$.

In what follows, by two distinct soft points $x_a,y_{a^{\prime }}$ we mean either $x\ne y$ or $a\ne a^{\prime }$ and by two disjoint soft sets $(F,A),(G,A)$ over X, we mean $(F,A)\tilde{\cap }(G,A)=(\Phi ,A)$.

Definition 9

([11]). A family $\theta \tilde{\subseteq }SS(X,A)$ is called a soft topology over X if

1.

$(\Phi ,A),(X,A)\in \theta$,

2.

$(F,A),(G,A)\in \theta$ implies $(F,A)\tilde{\cap }(G,A)\in \theta$, and

3.

$\{(F_i,A):i\in I\}\tilde{\subseteq }\theta$ implies ${\tilde{\cup }}_{i\in I}(F_n,A)\in \theta$.

The triple $(X,\theta ,A)$ is called a soft topological space. The elements of θ are called soft open sets, and their complements are called soft closed sets. The set of all soft closed sets is denoted by ${\theta }^c$.

Definition 10

([11]). Let $(Y,A)\ne (\Phi ,A)$ be a soft subset of $(X,\theta ,A)$. Then ${\theta }_{(Y,A)}=\{(G,A)\tilde{\cap }$$(Y,A):(G,A)\in \theta \}$ is called a relative soft topology over Y and $(Y,{\theta }_{(Y,A)},A)$ is a soft subspace of $(Y,\theta ,A)$.

Lemma 1

([11]). Let $(Y,{\theta }_{(Y,A)},A)$ be a soft subspace of $(Y,\theta ,A)$ and let $(F,A)\tilde{\subseteq }(Y,A)\in \theta$. Then $(F,A)\in {\theta }_{(Y,A)}$ iff $(F,A)\in \theta$.

Definition 11

([10]). A (countable) soft base for a soft topology θ is a (countable) family $\mathcal{B}\tilde{\subseteq }\theta$ such that elements of θ are soft unions of elements of $\mathcal{B}$.

Lemma 2

([11]). Let $(X,\theta ,A)$ be a soft topological space, then for each $a\in A$, the collection $\theta \left(a\right)=\left\{F\right(a):(F,A)\in \theta \}$ is a (crisp) topology on X.

Definition 12

([37]). A soft topology generated by a collection $\mathcal{C}\tilde{\subseteq }SS(X,A)$ is the intersection of all soft topologies over X including $\mathcal{C}$.

Definition 13

([38]). Let $(G,A)\tilde{\subseteq }(X,\theta ,A)$. Then $(G,A)$ is a soft neighborhood of $x_a\in SP(X,A)$ if there exists $(U,A)\in \theta \left(x_a\right)$ such that $x_a\in (U,A)\tilde{\subseteq }(G,A)$, where $\theta \left(x_a\right)$ is the family of all soft open sets containing $x_a$.

Definition 14

([11]). Let $(W,A)\tilde{\subseteq }(X,\theta ,A)$. Then

1.

$cl(W,A)=\tilde{\cap }\{(F,A):(W,A)\tilde{\subseteq }(F,A),(F,A)\in {\theta }^c\}$ is called the soft closure of $(W,A)$.

2.

$int(W,A)=\tilde{\cup }\{(F,A):(F,A)\tilde{\subseteq }(W,A),(F,A)\in \theta \}$ is called the soft interior of $(W,A)$.

Lemma 3

([39]). Let $(F,A),(G,A)\tilde{\subseteq }(X,\theta ,A)$. Then

1.

$int\left((F,A)\tilde{\cap }(G,A)\right)=int(F,A)\tilde{\cap }int(G,A)$.

2.

$cl\left((F,A)\tilde{\cap }(G,A)\right)\tilde{\subseteq }cl(F,A)\tilde{\cap }cl(G,A)$.

Lemma 4

([39]). Let $(F,A)\tilde{\subseteq }(X,\theta ,A)$. Then

$$int\left({(F,A)}^c\right)={\left(cl\left((F,A)\right)\right)}^c\mathrm{and}cl\left({(F,A)}^c\right)={\left(int\left((F,A)\right)\right)}^c.$$

Definition 15

([39,40]). Let $(F,A)\tilde{\subseteq }(X,\theta ,A)$. The soft boundary of $(F,A)$ is given by $b(F,A)=cl(F,A)-int(F,A)$.

Definition 16.

Let $(F,A),(G,A)\tilde{\subseteq }(X,\theta ,A)$. Then $(F,A)$ is called

1.

soft clopen [41] if $(F,A)$ is both soft open and soft closed.

2.

soft regular open [42] if $int\left(cl\right(F,A\left)\right)=(F,A)$.

3.

soft $G_{\delta }$[19] if $(F,A)={\tilde{\cap }}_{n=1}^{\infty }(G_n,A)$, where $(G_n,A)\in \theta$.

4.

soft $F_{\sigma }$[19] if $(F,A)={\tilde{\cup }}_{n=1}^{\infty }(F_n,A)$, where $(F_n,A)\in {\theta }^c$.

5.

soft dense in $(G,A)$[18,19] if $(G,A)\tilde{\subseteq }cl(F,A)$.

6.

soft nowhere dense [18] if $int\left(cl\right(F,A\left)\right)=(\Phi ,A)$.

7.

soft semiopen set [21] if $(F,A)\tilde{\subseteq }cl\left(int(F,A)\right)$.

8.

soft α-open set [22] if $(F,A)\tilde{\subseteq }int\left(cl\left(int(F,A)\right)\right)$.

9.

soft meager [18,43] (or a soft set of the first category) if $(F,A)={\tilde{\cup }}_{n=1}^{\infty }(F_n,A)$, where each $(F_n,A)$ is soft nowhere dense, otherwise $(F,A)$ is of the second category.

The collection of all soft sets of the first category (resp. soft sets of the second category, soft nowhere dense sets) over X is denoted by $\mathcal{M}(X,A)$ (resp. $\mathcal{S}(X,A),\mathcal{N}(X,A)$). Examples on the aforementioned classes of soft sets can be found in [43], Example 1.

Definition 17

([18,19]). A soft topological space $(X,\theta ,A)$ is called soft Baire if the soft intersection of each countable collection of soft open dense sets in $(X,\theta ,A)$ is soft dense. Equivalently, each non-null soft open set in $(X,\theta ,A)$ is of the second category.

Definition 18

([44]). A non-null class $\tilde{I}\tilde{\subseteq }SS(X,A)$ is called a soft ideal over X if $\tilde{I}$ satisfies the following conditions:

1.

If $(F,A),(G,A)\in \tilde{I}$, then $(F,A)\tilde{\cup }(G,A)\in \tilde{I}$.

2.

If $(G,A)\in \tilde{I}$ and $(F,A)\tilde{\subseteq }(G,A)$, then $(F,A)\in \tilde{I}$.

$\tilde{I}$ is called a soft σ-ideal if (1) holds for many (countable) soft sets. We denote the family of soft ideals over X by $\mathcal{I}(X,A)$.

Remark 2

([43]). For any soft topological space $(X,\theta ,A)$, $\mathcal{M}(X,A)$ forms a soft σ-ideal and $\mathcal{N}(X,A)$ forms a soft ideal.

Definition 19

([45]). A collection $\Sigma \tilde{\subseteq }SS(X,A)$ is a soft algebra over X if:

1.

$(\Phi ,A)\in \Sigma$,

2.

$(F,A)\in \Sigma$ implies ${(F,A)}^c\in \Sigma$, and

3.

$(F_n,A)\in \Sigma$, for all $n=1,2,\cdots ,k$, implies ${\tilde{\bigcup }}_{n=1}^k(F_n,A)\in \Sigma$.

If (3) holds true for many (countable) members of $\Sigma$, $\Sigma$ is said to be a soft $\sigma$-algebra on X (see [46]).

Definition 20

([47]). Let $\mathcal{F}\tilde{\subseteq }SS(X,A)$. The soft intersection of all soft σ-algebras over X containing $\mathcal{F}$ is a soft σ-algebra and it is called the soft σ-algebra generated by $\mathcal{F}$ and is referred to as $\sigma \left(\mathcal{F}\right)$.

Definition 21

([15]). A soft topological space $(X,\theta ,A)$ is called soft compact if every cover of $(X,A)$ by soft open sets has a finite subcover.

Definition 22

([11]). A soft topological space $(X,\theta ,A)$ is called soft regular if each $x_a\in SP(X,A)$ and each $(G,A)\in \theta$, there exists $(H,A)\in \theta$ such that $x_a\in (H,A)\tilde{\subseteq }cl(H,A)\tilde{\subseteq }(G,A)$.

Definition 23

([13]). A soft topological space $(X,\theta ,A)$ is called soft second countable if it has a countable soft base.

Definition 24

([25,48]). Let $SS(X,A),SS(Y,B)$ be collections of soft sets, and let $p:X\to Y,q:A\to B$ be mappings. The image of a soft set $(F,A)\tilde{\subseteq }(X,A)$ under $g:SS(X,A)\to SS(Y,B)$ is a soft subset $g(F,A)=\left(g\right(F),q(A\left)\right)$ of $(Y,B)$ which is given by

$$g\left(F\right)\left(b\right)=\left\{\begin{array}{cc}{\displaystyle \bigcup _{a\in q^{-1}\left(b\right)\cap A}}p\left(F\left(a\right)\right),\hfill & q^{-1}\left(b\right)\cap A\ne \varnothing \hfill \\ \varnothing ,\hfill & \mathrm{otherwise},\hfill \end{array}\right.$$

for each $b\in B$.

The inverse image of a soft set $(G,B)\tilde{\subseteq }(Y,B)$ under g is a soft subset $g^{-1}(G,B)=(g^{-1}\left(G\right),$$q^{-1}\left(B\right))$ such that

$$(g^{-1}\left(G\right)\left(a\right)=\left\{\begin{array}{cc}{p}^{-1}\left(G\left(q\right(a\left)\right)\right),\hfill & q\left(a\right)\in B\hfill \\ \varnothing ,\hfill & \mathrm{otherwise},\hfill \end{array}\right.$$

for each $a\in A$.

The soft mapping g is injective (resp. surjective, bijective) if both p and q are injective (resp. surjective, bijective).

Lemma 5

([25]). Let $g:SS(X,A)\to SS(Y,B)$ be a soft function and $(W,B)\in SS(Y,B)$. Then

$$g^{-1}\left({(W,B)}^c\right)={\left(g^{-1}(W,B)\right)}^c.$$

Definition 25.

A soft function $g:(X,\theta ,A)\to (Y,\vartheta ,B)$ is said to be

1.

soft continuous [25] if $g^{-1}(G,B)\in \theta$ for each $(G,B)\in \vartheta$.

2.

soft semicontinuous [29] if $g^{-1}(G,B)$ is soft semiopen for each $(G,B)\in \vartheta$.

3.

soft α-continuous [22] if $g^{-1}(G,B)$ is soft α-open for each $(G,B)\in \vartheta$.

3. Classes of Soft Sets with the Baire Property

We recall and study some properties of certain classes of soft sets that have the Baire property.

Definition 26

([23,36]). Let $(F,A)\tilde{\subseteq }(X,\theta ,A)$. It is said that $(F,A)$ is soft open modulo $\mathcal{M}(X,A)$ if there exists $(G,A)\in \theta$ such that $(F,A)\tilde{\Delta }(G,A)\in \mathcal{M}(X,A)$. Soft open sets modulo $\mathcal{M}(X,A)$ are named soft sets with the Baire property. The family of all soft sets over X with the Baire property is denoted by $\mathcal{B}(X,\theta ,A)$.

One can easily check that $(F,A)$ has the Baire property iffit is of the form $(F,A)=(G,A)\tilde{\Delta }(P,A)$, where $(G,A)\in \theta$ and $(P,A)\in \mathcal{M}(X,A)$.

Lemma 6

([23]). Let $(F,A)\tilde{\subseteq }(X,\theta ,A)$. The following properties are equivalent:

1.

$(F,A)\in \mathcal{B}(X,\theta ,A)$.

2.

if $(F,A)=(H,A)\tilde{\Delta }(P,A)$, where $(H,A)$ is soft regular open and $(P,A)\in \mathcal{M}(X,A)$.

3.

if $(F,A)=(K,A)\tilde{\Delta }(Q,A)$, where $(K,A)\in {\theta }^c$ and $(Q,A)\in \mathcal{M}(X,A)$.

4.

if $(F,A)=[(D,A)-(R,A)]\tilde{\cup }(S,A)$, where $(D,A)\in {\theta }^c$ and $(R,A),(S,A)\in \mathcal{M}(X,A)$.

5.

if $(F,A)=[(G,A)-(M,A)]\tilde{\cup }(N,A)$, where $(G,A)\in \theta$ and $(M,A),(N,A)\in \mathcal{M}(X,A)$.

6.

if $(F,A)=(U,A)\tilde{\cup }(L,A)$, where $(U,A)$ is a soft $G_{\delta }$ set and $(L,A)\in \mathcal{M}(X,A)$.

7.

if $(F,A)=(W,A)-(T,A)$, where $(W,A)$ is a soft $F_{\sigma }$ set and $(T,A)\in \mathcal{M}(X,A)$.

8.

if there exists $(V,A)\in \mathcal{M}(X,A)$ such that $(F,A)-(V,A)$ is soft clopen in ${(V,A)}^c$.

Lemma 7

([23]). Let $(F,A)\tilde{\subseteq }(X,\theta ,A)$. If $(F,A)\in \mathcal{B}(X,\theta ,A)$, then ${(F,A)}^c\in \mathcal{B}(X,\theta ,A)$.

Lemma 8

([23]). Let $(F,A),(Y,A)\tilde{\subseteq }(X,\theta ,A)$. If $(F,A)\in \mathcal{B}(X,\theta ,A)$, then $(F,A)\tilde{\cap }(Y,A)\in \mathcal{B}(X,$${\theta }_{(Y,A)},A)$.

Lemma 9

([23]). Let $(F,A),(Y,A)\tilde{\subseteq }(X,\theta ,A)$. If $(F,A)\in \mathcal{B}(X,{\theta }_{(Y,A)},A)$, $(Y,A)\in \mathcal{B}(X,\theta ,A)$, then $(F,A)\in \mathcal{B}(X,\theta ,A)$.

We are now in a position to define two subclasses of soft sets of the Baire property. We have seen that $(F,A)\in \mathcal{B}(X,\theta ,A)$ iff $(F,A)=(G,A)-(P,A)\tilde{\cup }(Q,A)$, where $(G,A)\mathrm{or}{(G,A)}^c\in \theta$ and $(Q,A)\in \mathcal{M}(X,A)$. From this representation, we introduce the following soft sets:

Definition 27.

Let $(F,A)\tilde{\subseteq }(X,\theta ,A)$. Then $(F,A)$ is said to be of the first type if $(F,A)=(G,A)-(P,A)$, where $(G,A)\mathrm{or}{(G,A)}^c\in \theta$ and $(P,A)\in \mathcal{M}(X,A)$. And it is of the second type if $(F,A)=(H,A)\tilde{\cup }(Q,A)$, where $(H,A)or{(H,A)}^c\in \theta$ and $(Q,A)\in \mathcal{M}(X,A)$. We will hereafter refer to soft sets of the first and second types as $S{T}_1$-sets and $S{T}_2$-sets, respectively.

Lemma 10.

Let $(F,A)\tilde{\subseteq }(X,\theta ,A)$. Then $(F,A)$ is an $S{T}_1$-set iff ${(F,A)}^c$ is an $S{T}_2$-set.

Proof. 

Let $(F,A)$ be an $S{T}_1$-set. Then $(F,A)=(G,A)-(P,A)$ for some $(G,A)\mathrm{or}{(G,A)}^c\in \theta$ and $(P,A)\in \mathcal{M}(X,A)$. Now, ${(F,A)}^c={\left((G,A)-(P,A)\right)}^c={(G,A)}^c\tilde{\cup }(P,A)$, where ${(G,A)}^c\mathrm{or}$$(G,A)\in \theta$ and $(P,A)\in \mathcal{M}(X,A)$. Thus, ${(F,A)}^c$ is an $S{T}_2$-set.

The converse is similar. □

Lemma 11.

Let $(F,A),(G,A)\tilde{\subseteq }(X,\theta ,A)$ such that $(G,A)\in \theta$ or $(G,A)\in {\theta }^c$. If $(F,A)$ is an $S{T}_i$-set, then $(F,A)\tilde{\cap }(G,A)$ is an $S{T}_i$-set, for $i=1,2$.

Proof. 

Straightforward. □

Lemma 12.

Let $(D,A)\tilde{\subseteq }(X,\theta ,A)$. If $(D,A)$ is soft dense in $(X,A)$, then $cl\left[(G,A)\tilde{\cap }(D,A)\right]=cl(G,A)$ for each $(G,A)\in \theta$.

Proof. 

Let $(G,A)\in \theta$. Since $(G,A)\tilde{\cap }(D,A)\tilde{\subseteq }(G,A)$, so $cl\left[(G,A)\tilde{\cap }(D,A)\right]\tilde{\subseteq }cl(G,A)$. On the other hand, we need to show $cl(G,A)\tilde{\subseteq }cl\left[(G,A)\tilde{\cap }(D,A)\right]$. Consider the following with applying Lemma 3:

$$\begin{array}{cc}\hfill (G,A)-cl\left[(G,A)\tilde{\cap }(D,A)\right]& =(G,A)\tilde{\cap }{\left(cl\left[(G,A)\tilde{\cap }(D,A)\right]\right)}^c\hfill \\ \hfill & =int(G,A)\tilde{\cap }int\left({\left[(G,A)\tilde{\cap }(D,A)\right]}^c\right)\hfill \\ \hfill & =int(G,A)\tilde{\cap }int\left[{(G,A)}^c\tilde{\cup }{(D,A)}^c\right])\hfill \\ \hfill & =int\left(\left[(G,A)\tilde{\cap }{(D,A)}^c\right]\tilde{\cup }\left[(G,A)\tilde{\cap }{(G,A)}^c\right]\right)\hfill \\ \hfill & =int\left[(G,A)\tilde{\cap }{(D,A)}^c\right]\hfill \\ \hfill & =(G,A)\tilde{\cap }int\left[{(D,A)}^c\right]\hfill \\ \hfill & =(G,A)-cl(G,A)\hfill \\ \hfill & =(G,A)-(X,A)\hfill \\ \hfill & =(\Phi ,A).\hfill \end{array}$$

This proves that $(G,A)\tilde{\subseteq }cl\left[(G,A)\tilde{\cap }(D,A)\right]$ implies $cl(G,A)\tilde{\subseteq }cl\left[(G,A)\tilde{\cap }(D,A)\right]$. Thus, $cl(G,A)=cl\left[(G,A)\tilde{\cap }(D,A)\right]$. □

Lemma 13.

Let $(F,A)\tilde{\subseteq }(X,\theta ,A)$. Then $(F,A)$ is a soft α-open set iffthen $(F,A)=(G,A)-(N,A)$, where $(G,A)\in \theta$ and $(N,A)\in \mathcal{N}(X,A)$.

Proof. 

Suppose $(F,A)$ is a soft $\alpha$-open set. Consider the identity

$$(F,A)=int\left(cl\right(int(F,A)\left)\right)-\left[int\right(cl\left(int\right(F,A\left)\right))-(F,A\left)\right].$$

Since $(F,A)\tilde{\subseteq }int\left(cl\left(int(F,A)\right)\right)$, then $int(F,A)\tilde{\subseteq }int\left(cl\left(int(F,A)\right)\right)$. Therefore, we have $int\left(cl\right(int(F,A)\left)\right)$$-(F,A)\tilde{\subseteq }int\left(cl\left(int(F,A)\right)\right)-int(F,A)$, which implies $int\left(cl\right(int(F,A)\left)\right)$$-(F,A)\in \mathcal{N}(X,A)$ since $int\left(cl\right(int(F,A)\left)\right)-int(F,A)\in \mathcal{N}(X,A)$. If we set $(G,A)=$$int\left(cl\right(int(F,A)\left)\right)$ and $(N,A)=int\left(cl\right(int(F,A)\left)\right)-(F,A)$, then we conclude that $(F,A)=(G,A)-(N,A)$ for some $(G,A)\in \theta$ and $(N,A)\in \mathcal{N}(X,A)$.

Conversely, if $(F,A)=(G,A)-(N,A)$, where $(G,A)\in \theta$ and $(N,A)\in \mathcal{N}(X,A)$. Now, $(F,A)=(G,A)\tilde{\cap }{(N,A)}^c$. Therefore, applying Lemma 3, we have $int(F,A)=int(G,A)\tilde{\cap }$$int\left({(N,A)}^c\right)=(G,A)\tilde{\cap }int\left({(N,A)}^c\right)$. Since $int\left({(N,A)}^c\right)$ is soft dense, it follows from Lemma 12, $cl\left(int(F,A)\right)=cl(G,A)\tilde{\supseteq }(G,A)$ and thus $(G,A)\tilde{\subseteq }int\left(cl\left(int(F,A)\right)\right)$. But, clearly, $(F,A)\tilde{\subseteq }(G,A)$. Hence, $(F,A)$ is soft $\alpha$-open. □

Proposition 1.

Let $(F,A)\tilde{\subseteq }(X,\theta ,A)$. If $(F,A)$ is a soft α-open set, then $(F,A)$ is an $S{T}_1$-set.

Proof. 

It follows from the fact that $\mathcal{N}(X,A)\tilde{\subseteq }\mathcal{M}(X,A)$. □

Proposition 2.

Let $(F,A)\tilde{\subseteq }(X,\theta ,A)$. If $(F,A)$ is a soft semiopen set, then $(F,A)$ is an $S{T}_2$-set.

Proof. 

Suppose $(F,A)$ is a soft semiopen set in $(X,\theta ,A)$. From Theorem 3.1 in [29], one can find $(G,A)\in \theta$ such that $(G,A)\tilde{\subseteq }(F,A)\tilde{\subseteq }cl(G,A)$. Consider, the identity $(F,A)=(G,A)\tilde{\cup }\left((F,A)-(G,A)\right)$. Since $(G,A)$ is soft open, then $cl(F,A)-(G,A)\in \mathcal{N}(X,A)$ and so $cl(F,A)-(G,A)\in \mathcal{M}(X,A)$. But $(F,A)-(G,A)\tilde{\subseteq }cl(F,A)-(G,A)$, therefore $(F,A)-(G,A)\in \mathcal{M}(X,A)$. Set $(N,A)=(F,A)-(G,A)$. Therefore, $(F,A)=(G,A)\tilde{\cup }(N,A)$. Hence, $(F,A)$ is an $S{T}_2$-set. □

This is a suitable place to illustrate the connections between the previously stated soft sets.

Generally, none of the above arrows are reversible, as is shown in the following example:

Example 1.

Let $\mathbb{R}$ be the set of real number and A be a set of parameters. Let θ be the soft topology on $\mathbb{R}$ generated by $\left\{\right(a,F\left(a\right)):F(a)=(t,s);t,s\in \mathbb{R};t<s,a\in A\}$. The soft set $(F,A)=\left\{\right(a,\left[\right(-1,0)-\mathbb{Q}]\cup \left[\right(0,1)\cap \mathbb{Q}]):a\in A\}$ has the Baire property but is neither an $S{T}_1$-set nor an $S{T}_2$-set, where $\mathbb{Q}$ is the set of rationals. The soft set $(G,A)=\left\{\right(a,\mathbb{R}-\mathbb{Q}):a\in A\}$ is an $S{T}_1$-set but not soft α-open. The soft set $(H,A)=\left\{\right(a,(-1,0)\cup (0,1)\cup \left\{2\right\}):a\in A\}$ is an $S{T}_2$-set but not soft semiopen. The soft set $(D,A)=\left\{\right(a,\mathcal{C}):a\in A\}$ is an $S{T}_2$-set but not an $S{T}_1$-set, where $\mathcal{C}$ is the ternary Cantor set. While $(G,A)$ is an $S{T}_1$-set but not an $S{T}_2$-set.

The counterexample for other cases are available in the literature.

4. Soft Functions with the Baire Property

Definition 28.

A soft function $g:(X,\theta ,A)\to (Y,\vartheta ,B)$ is said to have the Baire property if $g^{-1}(H,B)\in \mathcal{B}(X,\theta ,A)$ for each $(H,B)\in \vartheta$.

The Baire property is evidently present in all soft continuous functions. There are, on the other hand, soft functions that have the Baire property but are not soft continuous.

Example 2.

Consider the soft topological space $(X,\theta ,A)$ given in Example 1. Define a soft function $g:(X,\theta ,A)\to (X,\theta ,A)$ by

$$g\left(x_a\right)=\left\{\begin{array}{cc}{x}_a,\hfill & if{x}_a\notin \{0_a,1_a\};\hfill \\ 0_a,\hfill & if{x}_a=1_a;\hfill \\ 1_a,\hfill & if{x}_a=0_a.\hfill \end{array}\right.$$

One can easily show g has the Baire property because the inverse image of any soft open set is either a soft open set or a soft open sets union a soft set containing one of the soft points and both though are soft sets the Baire property. On the other hand g cannot be soft continuous. Take the soft open set $(G,A)=\left\{\right(a,(-\epsilon ,\epsilon )):a\in A\}$, where $\epsilon <1$. Then

$$g^{-1}(G,A)=\{(a,(-\epsilon ,0)\cup (0,\epsilon )\cup \left\{1\right\}):a\in A\}.$$

not a soft open set and hence g is not a soft continuous function.

Theorem 1.

A soft function $g:(X,\theta ,A)\to (Y,\vartheta ,B)$ has the Baire property iff $g^{-1}(R,B)\in \mathcal{B}(X,\theta ,A)$ for each $(R,B)\in {\vartheta }^c$.

Proof. 

It follows from Lemma 7. □

Proposition 3.

Let $g:(X,\theta ,A)\to (Y,\vartheta ,B)$ have the Baire property and $(F,A)\tilde{\subseteq }$$(X,\theta ,A)$. Then ${g|}_{(F,A)}$ has the Baire property.

Proof. 

Let $(H,A)\in \vartheta$. Then $g^{-1}{|}_{(F,A)}(H,A)=g^{-1}(H,A)\tilde{\cap }(F,A)$. By hypothesis, we have $g^{-1}(H,A)\in \mathcal{B}(X,\theta ,A)$ and, by Lemma 8, $g^{-1}{|}_{(F,A)}(H,A)\in \mathcal{B}(X,{\theta }_{(F,A)},A)$ □

Theorem 2.

Let $g:(X,\theta ,A)\to (Y,\vartheta ,B)$ be a soft function and let $(X,\theta ,A)$ be soft compact. If ${g|}_{(H,A)}$ has the Baire property for each $(H,A)\in \theta$, then g has the Baire property.

Proof. 

Let $\{(H_i,A):i\in I\}$ be a soft open cover of $(X,A)$. Let $(V,B)\in \vartheta$. By assumption, ${\left({g|}_{(H_i,A)}\right)}^{-1}(V,B)$ has the Baire property in $(H_i,A)$ for each i. Since each soft open set has the Baire property, by Lemma 9, ${\left({g|}_{(H_i,A)}\right)}^{-1}(V,B)=g^{-1}(V,B)\tilde{\cap }(H_i,A)\in \mathcal{B}(X,\theta ,A)$. And, by soft compactness of $(X,\theta ,A)$, one can find a finite subset $I_0\subseteq I$ such that $(X,A)={\tilde{\cup }}_{i\in I_0}(H_i,A)$. Now,

$$\begin{array}{cc}\hfill g^{-1}(V,B)& =g^{-1}(V,B)\tilde{\cap }\left({\tilde{\bigcup }}_{i\in I_0}(H_i,A)\right)\hfill \\ \hfill & ={\tilde{\bigcup }}_{i\in I_0}\left(g^{-1}(V,B)\tilde{\cap }(H_i,A)\right)\hfill \\ \hfill & ={\tilde{\bigcup }}_{i\in I_0}\left[{\left({g|}_{(H_i,A)}\right)}^{-1}(V,B)\right].\hfill \end{array}$$

Therefore, since $\mathcal{B}(X,\theta ,A)$ is closed under finite soft unions, $g^{-1}(V,B)\in \mathcal{B}(X,\theta ,A)$ and thus, g has the Baire property. □

Proposition 4.

Let $g:(X,\theta ,A)\to (Y,\vartheta ,B)$, $h:(Y,\vartheta ,B)\to (Z,\eta ,C)$ be soft functions. If g has the Baire property and h is soft continuous, then $h\circ g$ has the Baire property.

Proof. 

Let $(V,C)\in \eta$. By soft continuity of h, $h^{-1}(V,C)\in \vartheta$. Since g has the Baire property, so $g^{-1}\left(h^{-1}(V,C)\right)\in \theta$. But ${\left(h\circ g\right)}^{-1}=g^{-1}\left(h^{-1}(V,C)\right)$. Hence, $h\circ g$ has the Baire property. □

Proposition 5.

Let $(F,A)\tilde{\subseteq }(X,\theta ,A)$. Then $(F,A)\in \mathcal{B}(X,\theta ,A)$ iff the characteristic soft function ${\chi }_{(F,A)}$ of $(F,A)$ has the Baire property.

Proof. 

The characteristic soft function ${\chi }_{(F,A)}$ of $(F,A)$ is a soft function ${\chi }_{(F,A)}:(X,\theta ,A)\to (\{0,1\},{\vartheta }_{discrete},B)$, which is defined by

$${\chi }_{(F,A)}\left(x_a\right)=\left\{\begin{array}{cc}{1}_b\hfill & \mathrm{if}{x}_a\in (F,A);\hfill \\ & \\ 0_{b^{\prime }}\hfill & \mathrm{if}{x}_a\notin (F,A),\hfill \end{array}\right.$$

where ${\vartheta }_{discrete}$ is the soft discrete topology on $\{0,1\}$. Suppose $(F,A)\in \mathcal{B}(X,\theta ,A)$. Let $(V,B)\in {\vartheta }_{discrete}$. Then

$${\chi }_{(F,A)}^{-1}(V,B)=\left\{\begin{array}{cc}(X,A),\hfill & if{1}_b,0_{b^{\prime }}\in (V,B)\hfill \\ (F,A),\hfill & if{1}_b\in (V,B),0_{b^{\prime }}\notin (V,B)\hfill \\ {(F,A)}^c,\hfill & if{1}_b\notin (V,B),0_{b^{\prime }}\in (V,B)\hfill \\ (\Phi ,A),\hfill & if{1}_b,0_{b^{\prime }}\notin (V,B).\hfill \end{array}\right.$$

All those soft sets are in $\mathcal{B}(X,\theta ,A)$ since $\mathcal{B}(X,\theta ,A)$ is a soft $\sigma$-algebra. Thus, ${\chi }_{(F,A)}$ has the Baire property.

Conversely, since $\left\{1_b\right\}\in \vartheta$ and it contains $1_b$, by assumption, ${\chi }_{(F,A)}^{-1}\left(\left\{1_b\right\}\right)=(F,A)\in \mathcal{B}(X,\theta ,A)$. The proof is finished. □

Theorem 3.

Let $g:(X,\theta ,A)\to (Y,\vartheta ,B)$ be a soft function such that $(Y,\vartheta ,B)$ is soft second countable. Then g has the Baire property iffthere exists $(P,A)\in \mathcal{M}(X,A)$ such that ${g|}_{{(P,A)}^c}$ is soft continuous.

Proof. 

Assume g has the Baire property. We need to construct $(P,A)\in \mathcal{M}(X,A)$ for which ${h=g|}_{{(P,A)}^c}$ is soft continuous. Let $(H,B)\in \vartheta$ and let $\mathfrak{B}=\{(G_n,B):n=1,2,\cdots \}$ be a countable soft base of $(Y,\vartheta ,B)$. Then $(H,B)={\tilde{\cup }}_{i=1}^{\infty }(G_i,B)$ for some $(G_i,B)$ in $\mathfrak{B}$. Since g has the Baire property, so $g^{-1}(G_n,B)\in \mathcal{B}(X,\theta ,A)$ for each n. By Lemma 6,

$$g^{-1}(G_n,B)=(U_n,A)-(P_n,A)\tilde{\cup }(Q_n,A),$$

where $(U_n,A)\in \theta$, $(P_n,A),(Q_n,A)\in \mathcal{M}(X,A)$. Set $(P,A)={\tilde{\cup }}_{n=1}^{\infty }(P_n,A)\tilde{\cup }(Q_n,A)$. Then $(P,A)\in \mathcal{M}(X,A)$ since $\mathcal{M}(X,A)$ is a soft $\sigma$-ideal. It remains to show that h is soft continuous. Since

$$h^{-1}(H,B)=g^{-1}(H,B)\tilde{\cap }{(P,A)}^c,$$

then

$$\begin{array}{cc}\hfill h^{-1}(H,B)& =g^{-1}\left({\tilde{\bigcup }}_{i=1}^{\infty }(G_i,B)\right)\tilde{\cap }{(P,A)}^c\hfill \\ \hfill & ={\tilde{\bigcup }}_{i=1}^{\infty }{g}^{-1}(G_i,B)\tilde{\cap }{(P,A)}^c\hfill \\ \hfill & ={\tilde{\bigcup }}_{i=1}^{\infty }\left((U_i,A)-(P_i,A)\tilde{\cup }(Q_i,A)\right)\tilde{\cap }{(P,A)}^c.\hfill \end{array}$$

Since $(P_i,A)\tilde{\cup }(Q_i,A)\tilde{\subseteq }(P,A)$, therefore,

$$\left((U_i,A)-(P_i,A)\tilde{\cup }(Q_i,A)\right)\tilde{\cap }{(P,A)}^c=(U_i,A)\tilde{\cap }{(P,A)}^c.$$

This implies that $h^{-1}(H,B)=\left({\tilde{\cup }}_{i=1}^{\infty }(U_i,A)\right)\tilde{\cap }{(P,A)}^c$. Since ${\tilde{\cup }}_{i=1}^{\infty }(U_i,A)$ is soft open in $(X,A)$, so $h^{-1}(H,B)$ is a soft open set in ${(P,A)}^c$. Thus, h is soft continuous.

Conversely, suppose there exists $(P,A)\in \mathcal{M}(X,A)$ such that ${g|}_{{(P,A)}^c}$ is soft continuous. Let $(V,B)\in \vartheta$. By assumption, $h^{-1}(V,B)=g^{-1}(V,B)\tilde{\cap }{(P,A)}^c$. That is, $g^{-1}(V,B)\tilde{\cap }{(P,A)}^c=(U,A)\tilde{\cap }{(P,A)}^c$, where $(U,A)\in \theta$. Now,

$$\begin{array}{cc}\hfill g^{-1}(V,B)& =\left[g^{-1}(V,B)\tilde{\cap }{(P,A)}^c\right]\tilde{\bigcup }\left[g^{-1}(V,B)\tilde{\cap }(P,A)\right]\hfill \\ \hfill & =\left[(U,A)\tilde{\cap }{(P,A)}^c\right]\tilde{\bigcup }\left[g^{-1}(V,B)\tilde{\cap }(P,A)\right].\hfill \end{array}$$

Since $(P,A)\in \mathcal{M}(X,A)$, $(Q,A)=g^{-1}(V,B)\tilde{\cap }(P,A)\tilde{\subseteq }(P,A)$ implies $(Q,A)\in \mathcal{M}(X,A)$. Therefore, $g^{-1}(V,B)=(U,A)-(P,A)\tilde{\cup }(Q,A)$. By Lemma 6, $g^{-1}(V,B)\in \mathcal{B}(X,\theta ,A)$ and hence, g has the Baire property. □

5. Subclasses of Soft Functions with the Baire Property

We introduce two subclasses of soft functions with the Baire property in this section and discuss their fundamental properties.

Definition 29.

A soft function $g:(X,\theta ,A)\to (Y,\vartheta ,B)$ is said to be the Baire soft function of the first type (or shortly, $BS{T}_1$) if $g^{-1}(H,B)$ is an $S{T}_1$-set for each $(H,B)\in \vartheta$. It is the Baire soft function of the second type (or shortly, $BS{T}_2$) if $g^{-1}(H,B)$ is an $S{T}_2$-set for each $(H,B)\in \vartheta$.

By the use of Proposition 5, one can construct the following:

Example 3.

Consider the soft topological space $(\mathbb{R},\theta ,A)$ given in Example 1 and let ${\vartheta }_{discrete}$ be the soft discrete topology on $\{0,1\}$. Assume the soft function $g:(X,\theta ,A)\to (\{0,1\},{\vartheta }_{discrete},B)$ is defined by

$$g\left(x_a\right)=\left\{\begin{array}{cc}{1}_b\hfill & \mathrm{if}{x}_a\in (G,A);\hfill \\ & \\ 0_{b^{\prime }}\hfill & \mathrm{if}{x}_a\notin (G,A),\hfill \end{array}\right.$$

where $(G,A)=\left\{\right(a,\mathbb{R}-\mathbb{Q}):a\in A\}$ such that $\mathbb{Q}$ is the set of rationals. Then, g is a $BS{T}_1$-function but not $BS{T}_2$ since $g^{-1}\left(\left\{1_b\right\}\right)$$=(G,A)$ is an $S{T}_1$-set but not an $S{T}_2$-set, see Example 1. If we replace $(G,A)$ by the soft set $(D,A)=\left\{\right(a,\mathcal{C}):a\in A\}$, where $\mathcal{C}$ is the ternary Cantor set, we obtain a $BS{T}_2$-function but not $BS{T}_1$. On the other hand, if we replace $(G,A)$ by the soft set $(F,A)=\left\{\right(a,\left[\right(-1,0)-\mathbb{Q}]\cup \left[\right(0,1)\cap \mathbb{Q}]):a\in A\}$, we obtain a soft function with the Baire property but neither $BS{T}_1$ nor $BS{T}_2$.

Proposition 6.

A soft function $g:(X,\theta ,A)\to (Y,\vartheta ,B)$ is $BS{T}_i$ iff $g^{-1}(H,B)$ is an $S{T}_j$-set for each $(H,B)\in {\vartheta }^c$, $i,j=1,2$ and $i\ne j$.

Proof. 

We only prove when $i=1$ and $j=2$, the other case is the same. Suppose g is $BS{T}_1$. Let $(H,B)\in {\vartheta }^c$. Then ${(H,B)}^c\in \vartheta$. By assumption, $g^{-1}\left({(H,B)}^c\right)$ is an $S{T}_1$-set. But, by Lemma 5, $g^{-1}\left({(H,B)}^c\right)={\left(g^{-1}(H,B)\right)}^c$. By Lemma 10, $g^{-1}(H,B)$ is an $S{T}_2$-set.

The converse is clear. □

Proposition 7.

A soft function $g:(X,\theta ,A)\to (Y,\vartheta ,B)$ is $BS{T}_1$ if it is soft α-continuous.

Proof. 

Apply Proposition 1. □

Proposition 8.

A soft function $g:(X,\theta ,A)\to (Y,\vartheta ,B)$ is $BS{T}_2$ if it is soft semicontinuous.

Proof. 

Apply Proposition 2. □

The earlier two propositions and Figure 1 imply

Where an $SBP$-set means a soft set with the Baire property.

Corollary 1.

If $g:(X,\theta ,A)\to (Y,\vartheta ,B)$ is a soft continuous function, then it is $BS{T}_i$, for $i=1,2$.

Theorem 4.

Let $g:(X,\theta ,A)\to (Y,\vartheta ,B)$ be a soft function such that $(X,\theta A)$ is soft Baire and $(Y,\vartheta ,B)$ is soft regular. Then g is $BS{T}_1$ iffg is soft continuous.

Proof. 

Suppose g is $BS{T}_1$ and $x_a\in SP(X,A)$. Let $(H,B)\in \vartheta$ that contains $g\left(x_a\right)$. By soft regularity of $(Y,\vartheta ,B)$, there exists $(V,B)\in \vartheta$ such that $g\left(x_a\right)\in (V,B)\tilde{\subseteq }cl(V,B)\tilde{\subseteq }(H,B)$. Since g is $BS{T}_1$, so $g^{-1}(V,B)=(G,A)-(P,A)$ for some $(G,A)\in \theta$ and $(P,A)\in \mathcal{M}(X,A)$. Evidently, $x_a\in (G,A)$. To show the soft continuity of g, it suffices to show that $g(G,A)\tilde{\subseteq }cl(V,B)$. Suppose otherwise that $g\left(x_{a^{\prime }}^{\prime }\right)\notin cl(V,B)$ for some $x_{a^{\prime }}^{\prime }\in (G,A)$. This means that there exists $(W,B)\in \vartheta$ containing $g\left(x_{a^{\prime }}^{\prime }\right)$ such that $(V,B)\tilde{\cap }(W,B)=(\Phi ,B)$ and $g^{-1}(W,B)=(U,A)-(Q,A)$, where $(U,A)\in \theta$ and $(Q,A)\in \mathcal{M}(X,A)$. Since $x_{a^{\prime }}^{\prime }\in (G,A)\tilde{\cap }(U,A)$, so $(\Phi ,A)\ne (G,A)\tilde{\cap }(U,A)\in \theta$. Now, we have

$$\begin{array}{cc}\hfill (\Phi ,A)& =g^{-1}(V,B)\tilde{\cap }{g}^{-1}(W,B)\hfill \\ \hfill & =(G,A)-(P,A)\tilde{\cap }(U,A)-(Q,A)\hfill \\ \hfill & =\left[(G,A)\tilde{\cap }(U,A)\right]-\left[(P,A)\tilde{\cap }(Q,A)\right].\hfill \end{array}$$

This means that $(G,A)\tilde{\cap }(U,A)\tilde{\subseteq }(P,A)\tilde{\cap }(Q,A)$, which is not possible since $(X,\theta A)$ is a soft Baire space. Hence, we must have $g(G,A)\tilde{\subseteq }cl(V,B)\tilde{\subseteq }(H,B)$ which implies g is soft continuous.

Conversely, if g is soft continuous, then for each $(V,B)\in \vartheta$, $g^{-1}(V,B)\in \theta$. Clearly $g^{-1}(V,B)$ can be written as $g^{-1}(V,B)(G,A)-(\Phi ,A)$ and thus $g^{-1}(V,B)$ is an $S{T}_1$-set. Hence, g is $BS{T}_1$. □

Lemma 14.

Let $g:(X,\theta ,A)\to (Y,\vartheta ,B)$ be a soft function and let $\{(G_i,B):i\in I\}$ be a soft base of ϑ. The set $\mathcal{D}\left(g\right)$ of all soft points of soft discontinuity of g is of the form

$$\begin{array}{c}\hfill \mathcal{D}\left(g\right)={\tilde{\bigcup }}_{i\in I}\left(g^{-1}(G_i,B)-int\left(g^{-1}(G_i,B)\right)\right)\end{array}$$

(1)

Proof. 

Let $x_a\in SP(X,A)$. If $x_a$ is a soft point of soft discontinuity of g, then there exists $(H_{y_b},B)\in \vartheta$ containing $y_{{}_b}=g\left(x_a\right)$ such that $x_a$ is not a soft interior point of $g^{-1}(H_{y_b},B)$. That is, $x_a\in \left[g^{-1}(H_{y_b},B)-int\left(g^{-1}(H_{y_b},B)\right)\right]$. Since $\{(G_i,B):i\in I\}$ is a soft base of $\vartheta$, one can find some $(G_i,B)$ such that $(G_i,B)\tilde{\subseteq }(H_{y_b},B)$. Therefore, $x_a\in {\tilde{\cup }}_{i\in I}\left[g^{-1}(G_i,B)-int\left(g^{-1}(G_i,B)\right)\right]$.

Conversely, if for a soft point $x_a\in SP(X,A)$, there exists $i\in I$ such that $x_a\in \left[g^{-1}(G_i,B)-int\left(g^{-1}(G_i,B)\right)\right]$. This implies that $(G_i,B)\in \vartheta$ containing $g\left(x_a\right)$ for which $g^{-1}(G_i,B)$ is not a soft open set over X, and thus g is not soft continuous at $x_a$. □

Proposition 9.

Let $g:(X,\theta ,A)\to (Y,\vartheta ,B)$ be a soft function. If the set $\mathcal{D}\left(g\right)$ of all soft points of soft discontinuity of g is in $\mathcal{M}(X,A)$, then g is $BS{T}_2$.

Proof. 

Suppose $\mathcal{D}\left(g\right)\in \mathcal{M}(X,A)$. Let $(H,B)\in \vartheta$. Since the soft set $(F,A)=g^{-1}(H,B)-int\left(g^{-1}(H,B)\right)\tilde{\subseteq }\mathcal{D}\left(g\right)$, then $(F,A)\in \mathcal{M}(X,A)$. Therefore, $g^{-1}(H,B)=int\left(g^{-1}(H,B)\right)\tilde{\cup }$$(F,A)$ implies $g^{-1}(H,B)$ is $S{T}_2$. Hence, g is $BS{T}_2$. □

Proposition 10.

Let $g:(X,\theta ,A)\to (Y,\vartheta ,B)$ be a soft function. If g is $BS{T}_2$ and $(Y,\vartheta ,B)$ is soft second countable, then the set $\mathcal{D}\left(g\right)$ of all soft points of soft discontinuity of g is in $\mathcal{M}(X,A)$.

Proof. 

Let $\{(G_n,B):n=1,2,\cdots \}$ be a countable soft base of $\vartheta$. By (1) in Lemma 14,

$$\mathcal{D}\left(g\right)={\tilde{\bigcup }}_{n=1}^{\infty }\left(g^{-1}(G_n,B)-int\left(g^{-1}(G_n,B)\right)\right).$$

Since g is $BS{T}_2$, so $g^{-1}(G_n,B)-int\left(g^{-1}(G_n,B)\right)\in \mathcal{M}(X,A)$, and thus $\mathcal{D}\left(g\right)\in \mathcal{M}(X,A)$ as $\mathcal{M}(X,A)$ is closed under countable soft unions. □

From Propositions 9 and 10, we have the following result:

Theorem 5.

Let $g:(X,\theta ,A)\to (Y,\vartheta ,B)$ be a soft function such that $(Y,\vartheta ,B)$ is soft second countable. Then g is $BS{T}_2$ iff $\mathcal{D}\left(g\right)$ is in $\mathcal{M}(X,A)$.

We conclude this work by presenting the Figure 2 below:

Where $SBP$-function means a soft function with the Baire property.

One can derive from Proposition 5 and Examples 1 and 3 that the remaining examples demonstrating the opposites of the aforementioned arrows are untrue.

6. Conclusions and Future Work

Soft continuity is one of the most natural topics in the field of soft topology, which is a combination of topology and soft set theory. Soft continuity between soft topological spaces has a rich literature. After recalling and studying certain classes of soft sets with the Baire property, we have first started by defining the concept of soft functions having the Baire property. A soft function with the Baire property sends soft open sets back to soft sets with the Baire property. Basic operations on soft functions with the Baire property are discussed, along with some basic properties. We have seen that each soft continuous function is a soft function with the Baire property. The converse is generally false. We have shown that a soft function from a soft topological space into a soft second countable space has the Baire property if and only if there exists a soft set of the first category such that the restriction of the soft function to its complement is soft continuous. Secondly, we have introduced two subfamilies of soft functions with the Baire property called soft functions with the first and second types. We have studied these types of soft functions and established some of their characterizations. In particular, we have proved that a soft function from a soft Baire space into a soft regular space is of the first type if and only if it is soft continuous. And a soft function is of the second type if and only if the set of its discontinuous soft points is a soft set of the first category, provided that the range of the soft function is soft second countable. Moreover, we have shown that soft functions with the first type or second type are weaker than certain natural classes of generalized soft continuous functions, like soft $\alpha$-continuous and soft semicontinuous functions. Lastly, we have built the relationships between the classes of soft functions mentioned above and have offered some counterexamples that disprove the reverse of the relationships.

The conclusions in this article are preliminary, and more study will be necessary. These findings can also be seen as the foundation for researching new topics in soft topology and soft measure theory. Since the soft $\sigma$-algebra of soft set of the Baire property [43], a soft function with the Baire property can be considered a soft measurable function within the context of soft measure theory. As a result, soft functions can contribute to the growth of soft measure theory. Furthermore, by virtue of Proposition 5, one can study the determinacy of the Banach-Mazur game on a certain soft topological space when the characteristic soft function of each soft subset has the Baire property.