Soft Path, Soft-Path-Connectedness, and the Category of Soft-Topological Groups

Abstract

In this study, the soft usual topology compatible with the usual topology of $\mathbb{R}$ is defined, and using its subspace topology on the interval $[0,1]$, the concept of a soft path is introduced. Within this context, the notions of soft-connectedness and soft-path-connectedness are developed, their relationship is analyzed, and it is shown that these properties are preserved under soft-continuous mappings. Moreover, the behavior of these concepts within soft-topological groups is investigated in detail. Finally, the category of soft-topological groups is constructed, its morphisms are identified, and it is shown that this category forms a symmetric monoidal category.

1. Introduction

With the development of fuzzy logic and uncertainty theories, the variety of mathematical structures used to model uncertain data has significantly increased [1,2]. In 1999, Molodtsov introduced the theory of soft sets as a parameter-based approach, extending the applicability of the classical set theory to uncertain environments [3]. His aim was to overcome some of the difficulties encountered in traditional approaches and to provide a parameter-oriented solution method for problems in fields such as physics, economics, and engineering. Consequently, this theory has found applications in operations research, game theory, and probability theory [4].

Following the emergence of the soft set concept, both topological and algebraic notions on this structure developed rapidly. Maji, Biswas, and Roy, in [5], studied the fundamental properties of soft sets; Aktaş and Çağman, in [6], introduced the concept of soft groups, while Shabir and Naz [7] as well as Çağman, Karataş, and Enginoğlu [8] independently defined soft-topological spaces. Kharal and Ahmad, in [9] proposed the notion of soft mappings. Soft continuity has been defined using different methodologies by Aygünoğlu and Aygün [10], Zorlutuna et al. [11], Yang et al. [12], and many other authors. However, Hida [13] defined soft continuity with the same mathematical rigor as in classical topological spaces, defining it for a soft element taken from the domain. Hida proved that under this definition, a soft mapping being soft-continuous does not necessarily imply that the preimage of every soft-open set in the codomain is soft-open in the domain. This result is the most fundamental difference that distinguishes Hida’s definition from the other definitions. Throughout this study, Hida’s notion of soft continuity is employed.

Category theory, first introduced in the 1940s by Samuel Eilenberg [14] and Saunders Mac Lane [15], plays a role in mathematics comparable to that of set theory as a unifying language across nearly all disciplines. More specifically, it identifies similarities across different fields of mathematics and provides a common framework for unification. A category consists of objects and morphisms between them, where composition and identity laws hold. Category theory has been particularly influential in topology, especially in the development of homotopy theory and homology theory. Monoidal categories were independently introduced by Bénabou [16] as “categories with multiplication” and by Mac Lane [17] under the name “bicategories” (not to be confused with two-categories). These structures include additional data explaining how objects and morphisms can be combined in parallel.

Monoidal categories extend beyond basic category theory and find significant applications in modeling the multiplicative fragment of intuitionistic linear logic, in establishing the mathematical foundation of topological order in condensed matter physics, and in quantum information, quantum field theory, and string theory. Every Cartesian monoidal category is also a symmetric monoidal category [18]. These structures are further employed in modeling open games in game theory [4]. Recently, Alemdar and Arslan [19] constructed the category SGp, consisting of soft groups as objects and soft group homomorphisms as morphisms, and proved that it forms a symmetric monoidal category.

Topological group theory is an important field where algebraic and topological structures are studied together. In classical topological groups, the fundamental group ${\pi }_1(X,x_0)$ is among the most powerful tools of algebraic topology, often used to investigate topological properties of spaces [20]. If X is a connected topological space with a universal covering and $x_0\in X$, then for a subgroup G of ${\pi }_1(X,x_0)$, according to ([20] Theorem 10.42), there exists a covering map $p:(\widetilde{X_G},\widetilde{x_0})\to (X,x_0)$ with characteristic group G. Thus, a new structure is obtained that preserves both the group structure and the covering property. This method has been generalized to topological R-modules [21], topological groups with operations [22], and irresolute topological groups [23], and was also studied in the context of groupoids in [24]. Through this approach, invariants such as connectedness and the number of holes of a topological space are analyzed using the fundamental group. Moreover, the collection of fundamental groups in a space can be interpreted as the objects of a groupoid, where every morphism is an isomorphism.

Also, Hida, in [13], established the notion of soft-topological groups. The structure of soft-topological groups has also been examined in a different way in [25]. This structure reinterprets classical topological groups within the framework of soft topology, allowing the study of algebraic and topological properties under parameterized uncertainty. As pointed out by Hida, every classical topological group is a soft-topological group with a single-element parameter set $\xi =\left\{e\right\}$; in this case, the conditions of soft continuity reduce to the classical continuity conditions. This compatibility makes Hida’s definition particularly well-suited for transferring classical notions such as paths and connectedness to the soft setting. As a consequence of this approach, the notion of soft-connectedness introduced in this paper differs from the existing definitions in the literature.

The definition of soft-disconnectedness, which uses soft disjoint open sets that do not include elements of the initial universe as soft elements, is available in many studies [12,26,27]. Soft-connectedness via soft somewhere dense sets and related operators was investigated in [28], while infra soft-connectedness and infra soft locally connected spaces were introduced and studied in [29]. In this study, the concept of soft-connectedness, defined in terms of set decomposition, was introduced, and its fundamental properties were investigated in soft-topological spaces. This definition has guaranteed the existence of soft disjoint soft-open sets, which constitute soft-disconnectedness and include the elements of the initial universe as soft elements. Subsequently, the soft usual topology compatible with the usual topology of $\mathbb{R}$ was defined. Based on this definition, by considering the soft subspace topology on the interval $[0,1]$, the notion of a soft path was established, and the concept of soft-path-connectedness was introduced. Two fundamental consequences of connectedness in general topology are its preservation under continuous mappings and the fact that every path-connected space is connected. Since the definition of soft continuity adopted in our study necessitates soft element membership, we have introduced a definition of soft-connectedness that explicitly accounts for soft element membership to maintain consistency. This construction provides the theoretical foundation necessary for extending the concepts of homotopy and the fundamental group from classical topology to the context of soft topology. Furthermore, the behavior of soft-connectedness and soft-path-connectedness within soft-topological groups was analyzed, and it was proven that these properties are preserved under soft-continuous mappings.

In this study, the notions of soft-disconnectedness and soft path are presented in coherence with soft continuity, that is, with attention to the soft element, which shows that these notions are different from those in the literature. From this, it can be seen that the concepts and results obtained are not redundant when considering the redundancy theory of soft-topological spaces proposed by Shi and Pang [30].

Finally, the category of soft-topological groups was constructed, its morphisms were identified, and the existence of terminal and product objects within this category was demonstrated. The results show that the category of soft-topological groups forms a symmetric monoidal category.

2. Preliminaries

Definition 1.

Let X be the initial universe, $\mathcal{P}\left(X\right)$ be the power set of X, and $\xi$ be the set of parameters. Let $W:\xi \to \mathcal{P}\left(X\right)$ be a set-valued function. Then, the set

$$W_{\xi }=\{(e,W\left(e\right)):e\in \xi ,W\left(e\right)\in \mathcal{P}\left(X\right)\}$$

is called a soft set over X [3].

Definition 2.

Let $W_{\xi }$ and $W_{{\xi }^{\prime }}^{\prime }$ be two soft sets over X.

1. 

If $W\left(e\right)=\varnothing$, for every $e\in \xi$, then the soft set $W_{\xi }$ is called a soft empty set, and it is denoted by ${\varnothing }_{\xi }$.

2. 

If $W\left(e\right)=X$, for every $e\in \xi$, then the soft set $W_{\xi }$ is called an absolute soft set, and it is denoted by $X_{\xi }$.

3. 

If $\xi \subseteq {\xi }^{\prime }$ and $W\left(e\right)\subseteq W^{\prime }\left(e\right)$, for every $e\in \xi$, then $W_{\xi }$ is called a soft subset of $W_{{\xi }^{\prime }}^{\prime }$, and it is denoted by $W_{\xi }\widetilde{⊑}{W}_{{\xi }^{\prime }}^{\prime }$.

4. 

If $W_{\xi }\widetilde{⊑}{W}_{{\xi }^{\prime }}^{\prime }$ and $W_{{\xi }^{\prime }}^{\prime }\widetilde{⊑}{W}_{\xi }$, then $W_{\xi }$ is said to be soft equal to $W_{{\xi }^{\prime }}^{\prime }$ and is denoted by $W_{\xi }=W_{{\xi }^{\prime }}^{\prime }$.

5. 

Let ${\xi }^{″}=\xi \cap {\xi }^{\prime }$. If, for each $e\in {\xi }^{″}$, $W^{″}\left(e\right)=W\left(e\right)\cap W^{\prime }\left(e\right),$ then the soft set $W_{{\xi }^{″}}^{\prime \prime }$ is called the soft intersection of $W_{\xi }$ and $W_{{\xi }^{\prime }}^{\prime }$, and is denoted by $W_{\xi }\tilde{\sqcap }{W}_{{\xi }^{\prime }}^{\prime }$.

6. 

Let ${\xi }^{‴}=\xi \cup {\xi }^{\prime }$. If, for each $e\in {\xi }^{‴}$,

$$W^{‴}\left(e\right)=\left\{\begin{array}{cc}W\left(e\right),\hfill & \mathit{if}e\in \xi \setminus {\xi }^{\prime }\hfill \\ W^{\prime }\left(e\right),\hfill & \mathit{if}e\in {\xi }^{\prime }\setminus \xi \hfill \\ W\left(e\right)\cup W\left(e\right),\hfill & \mathit{if}e\in \xi \cap {\xi }^{\prime }\hfill \end{array}\right.$$

then $W_{{\xi }^{‴}}^{‴}$ is called the soft union of the soft sets $W_{\xi }$ and $W_{{\xi }^{\prime }}^{\prime }$ and is denoted by $W_{\xi }\tilde{\bigsqcup }{W}_{{\xi }^{\prime }}^{\prime }$.

7. 

For each $e\in \xi$, define $W^c\left(e\right)=X\setminus W\left(e\right)$. Then $W_{\xi }^c$ is called the soft complement of the soft set $W_{\xi }$ [5].

Definition 3.

Let $W_{\xi }$ be a soft set over X and let $x\in X$. If $x\in W\left(e\right)$ for every $e\in \xi$, then x is called a soft element of the soft set $W_{\xi }$, and is denoted by $x\tilde{\in }{W}_{\xi }$. Otherwise, if there exists at least one $e\in \xi$ such that $x\notin W\left(e\right)$, then x is not a soft element of $W_{\xi }$, and is denoted by $x\tilde{\notin }{W}_{\xi }$ [7].

By convention, throughout this paper, we denote by $P^S\left(X\right)$ the collection of all soft sets over the initial universe X.

Definition 4.

Let $W_{\xi }\in P^S\left(X\right)$ and $U_{{\xi }^{\prime }}\in P^S\left(Y\right)$, and let $\varrho :X\to Y$ and $\phi :\xi \to {\xi }^{\prime }$ be two functions.

(i) 

$(\phi ,\varrho ):W_{\xi }\to U_{{\xi }^{\prime }}$ is said to be a soft mapping from $W_{\xi }$ to $U_{{\xi }^{\prime }}$ if and only if

$$\varrho \left(W_{\xi }\left(e\right)\right)=U_{{\xi }^{\prime }}\left(\phi \left(e\right)\right),$$

for all $e\in \xi .$

(ii) 

The soft image of the soft set $W_{\xi }\in P^S\left(X\right)$ under the mapping ϱ with respect to the mapping φ is the soft set $\varrho {\left(W_{\xi }\right)}_{\phi \left(\xi \right)}\in P^S\left(Y\right)$, defined by

$$\varrho {\left(W_{\xi }\right)}_{\phi \left(\xi \right)}\left(d\right)=\bigcup _{\phi \left(e\right)=d}\varrho \left(W_{\xi }\left(e\right)\right),$$

for all $d\in \phi \left(\xi \right).$

(iii) 

The inverse soft image of the soft set $U_{{\xi }^{\prime }}\in P^S\left(Y\right)$ under the mapping ϱ with respect to the mapping φ is the soft set ${\varrho }^{-1}{\left(U_{{\xi }^{\prime }}\right)}_{\xi }\in P^S\left(X\right)$, defined by

$${\varrho }^{-1}{\left(U_{{\xi }^{\prime }}\right)}_{\xi }\left(e\right)={\varrho }^{-1}\left(U_{{\xi }^{\prime }}\left(\phi \left(e\right)\right)\right),$$

for all $e\in \xi$ [9].

This definition was arranged by Tahat et al. [25], as in the remark below:

Remark 1.

Let $W_{\xi }\in P^S\left(X\right)$ and $U_{{\xi }^{\prime }}\in P^S\left(Y\right)$, and let $\varrho :X\to Y$ and $\phi :\xi \to {\xi }^{\prime }$ be two functions.

(i) 

Let $(\phi \times \varrho ):\xi \times X\to {\xi }^{\prime }\times Y$ be a soft mapping defined by $(\phi \times \varrho )(e,x)=\left(\phi \right(e),\varrho (x\left)\right)$ for all $e\in \xi ,x\in X.$ Then,

$${\left(\varrho \left(W_{\xi }\right)\right)}_{\phi \left(\xi \right)}=(\phi \times \varrho )\left(W_{\xi }\right).$$

(ii) 

Let $(\phi \times \varrho ):\xi \times X\to {\xi }^{\prime }\times Y$ be a soft mapping defined by $(\phi \times \varrho )(e,x)=\left(\phi \right(e),\varrho (x\left)\right)$ for all $e\in \xi ,x\in X.$ Then, the inverse image of $U_{{\xi }^{\prime }}$ under ϱ with respect to φ satisfies

$${\left({\varrho }^{-1}\left(U_{{\xi }^{\prime }}\right)\right)}_{\xi }={(\phi \times \varrho )}^{-1}\left(U_{{\xi }^{\prime }}\right).$$

(iii) 

Let $\xi ={\xi }^{\prime }$ and $\phi ={\mathbb{I}}_{\xi }$ be the identity mapping on $\xi$. Then, for all $e\in \xi$, the image of the soft set $W_{\xi }$ under ϱ with respect to φ satisfies

$${\left(\varrho \left(W_{\xi }\right)\right)}_{\phi \left(\xi \right)}\left(e\right)=\varrho \left(W_{\xi }\left(e\right)\right).$$

Definition 5.

Let X be the initial universe and $\xi$ be a set of parameters. A collection ζ of soft sets over X is called a soft topology on X if

• ${\varnothing }_{\xi }$ and $X_{\xi }$ belong to ζ;
• The soft union of any family of soft sets in ζ belongs to ζ;
• The soft intersection of any two soft sets in ζ belongs to ζ;

The pair ${(X,\zeta )}_{\xi }$ is called a soft-topological space. The members of ζ are called soft-open sets. A soft set W over X is called soft-closed if the soft complement $W^c$ is soft-open [7].

Example 1.

Let X be an initial universal set and $\xi$ be a parameter set. The collection $\{X_{\xi },{\varnothing }_{\xi }\}$, consisting of the soft empty set and soft absolute set, is a soft topology on X, called the soft trivial topology or soft indiscrete topology on X [7].

Proposition 1.

Let ${(X,\zeta )}_{\xi }$ be a soft-topological space. Then, for each $e\in \xi$,

$${\zeta }_e=\{W\left(e\right)\mid W\in \zeta \}$$

is a topology on X [7].

Definition 6.

Let ${(X,\zeta )}_{\xi }$ be a soft-topological space, and let $A\subseteq X$ be a nonempty subset. Define a mapping $W_A:\xi \to \mathcal{P}\left(X\right)$ by $W_A\left(e\right)=W\left(e\right)\cap A$ for all $W\in \zeta$. Then the collection

$${\zeta }_A=\{W_A\mid W\in \zeta \}$$

is called the soft subtopology on A, and the pair ${(A,{\zeta }_A)}_{\xi }$ is called a soft subspace of ${(X,\zeta )}_{\xi }$ [7].

Definition 7.

Let ${(X,\zeta )}_{\xi }$ be a soft-topological space, $x\in X$, and U be a soft set on X such that $x\tilde{\in }U$. If there exists a soft-open set W such that $x\tilde{\in }W\widetilde{⊑}U$, then the soft set U is called a soft neighborhood of x [7].

Definition 8.

Let ${(X,\zeta )}_{\xi }$ be a soft-topological space. A subcollection $\mathcal{B}$ of ζ is said to be a base for ζ if every member of ζ can be expressed as a union of members of $\mathcal{B}$. Here,

$$\zeta =\{{\varnothing }_{\xi },X_{\xi }\}\tilde{\bigsqcup }\left\{\widetilde{⨆}B|B\subseteq \mathcal{B}\right\}$$

is the soft topology induced by the basis $\mathcal{B}$ [10].

Definition 9.

Let ${(X,\zeta )}_{\xi }$ be a soft-topological space. A subcollection $\mathcal{S}$ of ζ is said to be a subbase for ζ if the family of all finite intersections of members of $\mathcal{S}$ forms a base for ζ [10].

Definition 10.

Let ${(X,\zeta )}_{\xi }$ and ${(X^{\prime },{\zeta }^{\prime })}_{{\xi }^{\prime }}$ be two soft-topological spaces. A soft mapping

$$(\phi ,\varrho ):{(X,\zeta )}_{\xi }\to {(X^{\prime },{\zeta }^{\prime })}_{{\xi }^{\prime }}$$

is soft-continuous at the point $x\in X$ if for every soft-open neighborhood $W^{\prime }$ of $\varrho \left(x\right)$ there exists a soft-open neighborhood W of x such that $\varrho \left(x\right)\tilde{\in }(\phi ,\varrho )\left(W\right)\widetilde{⊑}{W}^{\prime }$. If $(\phi ,\varrho )$ is soft-continuous at every point of X, then $(\phi ,\varrho )$ is called a soft-continuous mapping [13].

If $\xi ={\xi }^{\prime }$ and $\phi ={\mathbb{I}}_{\xi }$ (the identity mapping on $\xi$), then the definition of soft continuity is as follows:

Definition 11.

Let $\varrho :X\to X^{\prime }$ be a function. Then $({\mathbb{I}}_{\xi },\varrho )$ is called a soft-continuous mapping from ${(X,\zeta )}_{\xi }$ to ${(X^{\prime },{\zeta }^{\prime })}_{\xi }$ if the following condition holds [13]:

• For every $x\in X$, and for every soft-open neighborhood $W^{\prime }$ of $\varrho \left(x\right)$, there exists a soft-open neighborhood W of x such that $\varrho \left(W\right)\widetilde{⊑}{W}^{\prime }.$

Proposition 2.

Let ${(X,\zeta )}_{\xi }$ and ${(X^{\prime },{\zeta }^{\prime })}_{\xi }$ be two soft-topological spaces and let $\varrho :X\to X^{\prime }$ be a function. Then the soft mapping $({\mathbb{I}}_{\xi },\varrho ):{(X,\zeta )}_{\xi }\to {(X^{\prime },{\zeta }^{\prime })}_{\xi }$ is soft-continuous if for every soft-open set $W^{\prime }\in {\zeta }^{\prime }$, the inverse image ${({\mathbb{I}}_{\xi },\varrho )}^{-1}\left(W^{\prime }\right)\in \zeta$ [13].

The converse of this proposition is not true in general. Hida [13] gives the following example which shows that this is not always true.

Example 2.

Let $X=\left\{v\right\}$ and $\xi =\{e_1,e_2\}$, and consider the soft topologies, $\zeta =\{X_{\xi },{\varnothing }_{\xi }\}$ and ${\zeta }^{\prime }=\{X_{\xi },{\varnothing }_{\xi },\left\{(e_2,\left\{v\right\})\right\}\}$. Then, the soft mapping $({\mathbb{I}}_{\xi },{\mathbb{I}}_X):{(X,\zeta )}_{\xi }\to {(X,{\zeta }^{\prime })}_{\xi }$ is soft-continuous. Although $\left\{(e_2,\left\{v\right\})\right\}$ is soft-open in ${\zeta }^{\prime }$, ${({\mathbb{I}}_{\xi },{\mathbb{I}}_X)}^{-1}\left(\left\{(e_2,\left\{v\right\})\right\}\right)=\left\{(e_2,\left\{v\right\})\right\}$ is not soft-open in ζ. Therefore, in a soft-continuous mapping, the inverse image of a soft-open set is not always soft-open.

Definition 12.

Let ${(X,\zeta )}_{\xi }$ and ${(X^{\prime },{\zeta }^{\prime })}_{{\xi }^{\prime }}$ be two soft-topological spaces. Then the collection

$$\{W\tilde{\times }{W}^{\prime }\mid W\in \zeta ,W^{\prime }\in {\zeta }^{\prime }\}$$

induces a soft topology ${\zeta }_{\times }$ on the Cartesian product set $X\times X^{\prime }$. The soft-topological space ${(X\times X^{\prime },{\zeta }_{\times })}_{\xi \times {\xi }^{\prime }}$ is called the soft product space of ${(X,\zeta )}_{\xi }$ and ${(X^{\prime },{\zeta }^{\prime })}_{{\xi }^{\prime }}$ [13].

Definition 13.

Let $(G,\ast )$ be a group. If there exists a soft topology ζ on G with the parameter set $\xi$ such that

• For every $(a,b)\in G\times G$, and for each soft-open neighborhood W of $a\ast b$, there exist soft-open neighborhoods U of a and V of b such that: $U\ast V\widetilde{⊑}W.$
• The inverse map $\iota :G\to G$, defined by $\iota \left(a\right)=a^{-1}$, is soft-continuous,

then the pair ${(G,\zeta )}_{\xi }$ is called a soft-topological group [13].

Proposition 3.

${(G,\zeta )}_{\xi }$ is a soft-topological group if and only if for every $a,b\in G$ and for every soft-open set W containing $a\ast b^{-1}$, there exist soft-open neighborhoods U of a and V of b such that $U\ast {\left(V\right)}^{-1}\widetilde{⊑}W$ [13].

3. Soft-Connectedness

In light of the results presented in this paper, we introduce a new definition of soft-connectedness that is more compatible with the classical notion of connectedness in topology. In general topology, the definition of disconnectedness is given in terms of a set partition by the open sets of the topology. In this study, in a similar manner, we relate the partition of a set to soft-open sets and introduce the definition of soft-disconnectedness as follows.

The following notion was previously introduced in [31] under the name “stable”; however, throughout this paper, we adopt the term “absolute soft subset”.

Definition 14.

Let X be the initial universe and $\xi$ be the set of parameters. If $A\subseteq X$, and the map $A:\xi \to \mathcal{P}\left(X\right)$ is defined as $A\left(e\right)=A$ for each $e\in \xi$, then $A_{\xi }$ is a soft set over X. The soft set $A_{\xi }$, which is a soft subset of the absolute soft set $X_{\xi }$, is called an absolute soft subset.

Definition 15.

Let ${(X,\zeta )}_{\xi }$ be a soft-topological space. If there exist two nonempty subsets A and B of X such that $A\cup B=X$ and $A\cap B=\varnothing$, with both absolute soft subsets $A_{\xi }$ and $B_{\xi }$ being soft-open, then the soft-topological space ${(X,\zeta )}_{\xi }$ is called soft-disconnected. Otherwise, ${(X,\zeta )}_{\xi }$ is called a soft-connected topological space.

Let ${(X,\zeta )}_{\xi }$ be a soft-topological space. If A and B are nonempty disjoint subsets of X such that $X=A\cup B$, then it is obvious from the definition of absolute soft subset that $X_{\xi }=A_{\xi }\tilde{\bigsqcup }{B}_{\xi }$ and $A_{\xi }\tilde{\sqcap }{B}_{\xi }={\varnothing }_{\xi }$. However, even if U and V are soft-open in $\zeta$ such that $U\tilde{\bigsqcup }V=X_{\xi }$, and $U\tilde{\sqcap }V={\varnothing }_{\xi }$, it does not necessarily follow that U and V constitute a set partition of X.

Example 3.

Let $X=\{\upsilon ,{\upsilon }^{\prime },{\upsilon }^{\prime \prime }\}$ and $\xi =\{e_1,e_2\}$, and consider the soft topology

$$\zeta =\{X_{\xi },{\varnothing }_{\xi },\{(e_1,\{\upsilon ,{\upsilon }^{\prime }\}),(e_2,\left\{{\upsilon }^{″}\right\})\},\{(e_1,\left\{{\upsilon }^{″}\right\}),(e_2,\{\upsilon ,{\upsilon }^{\prime }\})\}\}$$

on X. Here, although

$$\{(e_1,\{\upsilon ,{\upsilon }^{\prime }\}),(e_2,\left\{{\upsilon }^{″}\right\})\}\tilde{\bigsqcup }\{(e_1,\left\{{\upsilon }^{″}\right\}),(e_2,\{\upsilon ,{\upsilon }^{\prime }\})\}\}=X_{\xi }$$

and

$$\{(e_1,\{\upsilon ,{\upsilon }^{\prime }\}),(e_2,\left\{{\upsilon }^{″}\right\})\}\tilde{\sqcap }\{(e_1,\left\{{\upsilon }^{″}\right\}),(e_2,\{\upsilon ,{\upsilon }^{\prime }\})\}\}={\varnothing }_{\xi },$$

no element of X belongs to either $\{(e_1,\{\upsilon ,{\upsilon }^{\prime }\}),(e_2,\left\{{\upsilon }^{″}\right\})\}$ or $\{(e_1,\left\{{\upsilon }^{″}\right\}),(e_2,\{\upsilon ,{\upsilon }^{\prime }\})\}$. Hence, a partition of the set X cannot be obtained by these soft-open sets.

Let ${(X,\zeta )}_{\xi }$ be a soft-topological space and $A\subseteq X$. If the soft subspace ${(A,{\zeta }_A)}_{\xi }$ is soft-connected, then A is said to be soft-connected.

Theorem 1.

Let ${(X,\zeta )}_{\xi }$ be a soft-topological space. Then the following are equivalent:

1.

X is soft-connected.

2.

The only soft absolute subsets that are both soft-open and soft-closed are ${\varnothing }_{\xi }$ and $X_{\xi }$.

Proof. 

(1) ⇒ (2). Let X be soft-connected. Suppose that $A_{\xi }$ is both a soft-open and soft-closed absolute subset of $X_{\xi }$ with ${\varnothing }_{\xi }\ne A_{\xi }\ne X_{\xi }$. Since $A_{\xi }$ is soft-closed, its soft complement $A_{\xi }^c$ is also soft-open. Then, the pair A and $A^c$ are two nonempty, disjoint subsets of X whose union is X. This contradicts the soft-connectedness of X. Therefore, except for ${\varnothing }_{\xi }$ and $X_{\xi }$, there is no other soft absolute subset that is both soft-open and soft-closed.

(2) ⇒ (1). Assume that the only soft absolute subsets that are both soft-open and soft-closed in $\zeta$ are ${\varnothing }_{\xi }$ and $X_{\xi }$. If X were soft-disconnected, there would exist absolute soft-open sets $A_{\xi }$ and $B_{\xi }$ such that $X_{\xi }=A_{\xi }\tilde{\bigsqcup }{B}_{\xi }$. In this case, $A_{\xi }$ is soft-open, and its soft complement is $B_{\xi }$, which is also soft-open. Hence, $A_{\xi }$ is both soft-open and soft-closed, contradicting the assumption. Therefore, X is soft-connected. □

Remark 2.

In this paper, all results concerning soft continuity are proved without adopting the proof technique in classical topology, as the inverse image of a soft-open set under a soft-continuous mapping is not necessarily soft-open. We would like to point out that we presented our proofs by paying attention to the membership of the soft element, that is, by directly satisfying the conditions of Definitions 10 and 11. Therefore, our definition of soft-connectedness and the proof methodology we used are completely different from those in the literature.

Theorem 2.

Let ${(X,\zeta )}_{\xi }$ and ${(X^{\prime },{\zeta }^{\prime })}_{\xi }$ be soft-topological spaces, and let $({\mathbb{I}}_{\xi },\varrho ):{(X,\zeta )}_{\xi }\to {(X^{\prime },{\zeta }^{\prime })}_{\xi }$ be a soft-continuous mapping. If X is soft-connected, then $\varrho \left(X\right)$ is soft-connected.

Proof. 

Let X be soft-connected. And assume that ${(\varrho \left(X\right),{\zeta }_{\varrho \left(X\right)}^{\prime })}_{\xi }$ is a soft-disconnected space. Then there exists a soft absolute subset $A_{\xi }$ that is both soft-open and soft-closed ($A_{\xi }^c$ is soft-open), with ${\varnothing }_{\xi }\ne A_{\xi }\ne {\varrho \left(X\right)}_{\xi }$. For each $x\in X$, either $\varrho \left(x\right)\tilde{\in }{A}_{\xi }$ or $\varrho \left(x\right)\tilde{\in }{A}_{\xi }^c$. If $\varrho \left(x\right)\tilde{\in }{A}_{\xi }$ and ${\varrho }^{-1}\left(A\right)=B$, then since $({\mathbb{I}}_{\xi },\varrho )$ is soft-continuous, there exists a soft-open neighborhood $B_{\xi }$ of x such that

$$\varrho \left(x\right)\tilde{\in }({\mathbb{I}}_{\xi },\varrho )\left(B_{\xi }\right)\widetilde{⊑}{A}_{\xi }.$$

And similarly, if $\varrho \left(x\right)\tilde{\in }{A}_{\xi }^c$ and ${\varrho }^{-1}\left(A^c\right)=B^c$, then there exists a soft-open neighborhood $B_{\xi }^c$ of x such that

$$\varrho \left(x\right)\tilde{\in }({\mathbb{I}}_{\xi },\varrho )\left(B_{\xi }^c\right)\widetilde{⊑}{A}_{\xi }^c.$$

Therefore, each x belongs to only one of $B_{\xi }$ or $B_{\xi }^c$. So, we have $B_{\xi }\tilde{\bigsqcup }{B}_{\xi }^c=X_{\xi }$ and $B_{\xi }\tilde{\sqcap }{B}_{\xi }^c={\varnothing }_{\xi }$, which is a contradiction. Hence, $\varrho \left(X\right)$ is soft-connected. □

4. Soft Path and Soft-Path Connectedness

Let $\xi$ be any parameter set. For every $a,b\in \mathbb{R}$, let the set-valued functions $M_{\xi }^a:\xi \to \mathcal{P}\left(\mathbb{R}\right)$ be defined as $M_{\xi }^a\left(e\right)=(a,\infty )$ and $N_{\xi }^b:\xi \to \mathcal{P}\left(\mathbb{R}\right)$ be defined as $N_{\xi }^b\left(e\right)=(-\infty ,b)$, for all $e\in \xi$. Clearly $M_{\xi }^a$ and $N_{\xi }^b$ are soft sets over $\mathbb{R}$. Consider the class of soft sets

$${\mathcal{S}}_{\xi }=\{M_{\xi }^a\mid a\in \mathbb{R}\}\bigcup \{N_{\xi }^b\mid b\in \mathbb{R}\},$$

which forms a soft subbase for a soft topology on $\mathbb{R}$. Since each open interval $(a,b)$ in $\mathbb{R}$ can be written as $(-\infty ,b)\cap (a,\infty )$, from the soft subbase ${\mathcal{S}}_{\xi }$, we can obtain the soft base

$${\mathcal{B}}_{\xi }=\{H_{\xi }^{(a,b)}\mid a,b\in \mathbb{R},a<b\},$$

where $H_{\xi }^{(a,b)}:\xi \to \mathcal{P}\left(\mathbb{R}\right)$ is defined by $H_{\xi }^{(a,b)}\left(e\right)=(a,b)$ for all $e\in \xi$. From this soft base, we obtain the soft topology

$${\mathcal{U}}_{\xi }=\{H_{\xi }^G\mid G\subseteq \mathbb{R}\mathrm{is}\mathrm{open}\mathrm{in}\mathrm{the}\mathrm{usual}\mathrm{topology}\},$$

where $H_{\xi }^G:\xi \to \mathcal{P}\left(\mathbb{R}\right)$ is given by $H_{\xi }^G\left(e\right)=G$. The soft topology obtained in this way is called the soft usual topology, and the pair ${(\mathbb{R},{\mathcal{U}}_{\xi })}_{\xi }=(\mathbb{R},{\mathcal{U}}_{\xi })$ is called the soft usual topological space.

For each parameter $e\in \xi$, the associated topology ${\mathcal{U}}_{\xi }^e$ is the usual topology on $\mathbb{R}$.

Example 4.

Let $\xi =\{e_1,e_2,e_3\}$. Then the soft usual topology corresponding to $\xi$ is ${\mathcal{U}}_{\xi }=\{H_{\xi }^G\mid G\subseteq \mathbb{R},G\mathit{is}\mathit{open}\mathit{in}\mathit{the}\mathit{usual}\mathit{topology}\}$. If $G=(-3,2)$, then $H_{\xi }^G=H_{\xi }^{(-3,2)}=\{(e_1,(-3,2)),(e_2,(-3,2)),(e_3,(-3,2))\}$. If $G=\mathbb{R}$, then $H_{\xi }^{\mathbb{R}}=\{(e_1,\mathbb{R}),(e_2,\mathbb{R}),(e_3,\mathbb{R})\}$.

From the definition of the soft usual topology, one can easily see that a soft-open set in the soft-usual topology is also an absolute open subset.

Definition 16.

Let ${(X,\zeta )}_{\xi }$ be a soft-topological space and let $(I,{\left({\mathcal{U}}_{\xi }\right)}_I)$ be the soft subspace of $(\mathbb{R},{\mathcal{U}}_{\xi })$ on $I=[0,1]\subseteq \mathbb{R}$. Let $\gamma :I\to X$ be a function. We say that the soft mapping

$$\Gamma =({\mathbb{I}}_{\xi },\gamma ):(I,{\left({\mathcal{U}}_{\xi }\right)}_I)\to {(X,\zeta )}_{\xi }$$

is a soft path from $\gamma \left(0\right)=x$ and $\gamma \left(1\right)=y$, if $\Gamma$ is soft-continuous.

Remark 3.

Let us note that, in the definition of the soft path, the parameter sets of the soft usual topology on $\mathbb{R}$ and the soft subtopology obtained on I are the same as the parameter set of the soft space ${(X,\zeta )}_{\xi }$.

Example 5.

Let $(\mathbb{R},{\mathcal{U}}_{\xi })$ be the soft usual topology with parameter set $\xi =\{e_1,e_2\}$. Let $\gamma :I\to \mathbb{R}$ be the function defined by $\gamma \left(t\right)=2t$. Then,

$$\Gamma =({\mathbb{I}}_{\xi },\gamma ):(I,{\left({\mathcal{U}}_{\xi }\right)}_I)\to (\mathbb{R},{\mathcal{U}}_{\xi })$$

is a soft path from $\gamma \left(0\right)=0$ to $\gamma \left(1\right)=2$. To see that, it is sufficient to show that $\Gamma$ is soft-continuous. Let $t\in I$ be any point. Let us consider the open interval $G_{ϵ}=(\gamma \left(t\right)-ϵ,\gamma \left(t\right)+ϵ)=(2t-ϵ,2t+ϵ)$. Then, for each $ϵ>0$, $H_{\xi }^{G_{ϵ}}$ is a soft-open set which is a soft-open neighborhood of $\gamma \left(t\right)$. Now, let $\delta ={\displaystyle \frac{ϵ}{2}}$. By the definition of soft subtopology, for every $e\in \xi$,

$$H_{\xi }^{(t-{\displaystyle \frac{ϵ}{2}},t+{\displaystyle \frac{ϵ}{2}})}\left(e\right)\cap I=(t-{\displaystyle \frac{ϵ}{2}},t+{\displaystyle \frac{ϵ}{2}})\cap I=H_{\xi }^{(t-{\displaystyle \frac{ϵ}{2}},t+{\displaystyle \frac{ϵ}{2}})\cap I}\left(e\right),$$

and

$${\left(H_{\xi }^{G_{ϵ}}\right)}_I=H_{\xi }^{(t-{\displaystyle \frac{ϵ}{2}},t+{\displaystyle \frac{ϵ}{2}})\cap I}\in {\left({\mathcal{U}}_{\xi }\right)}_I$$

is a soft-open neighborhood of t. Since $\gamma ((t-{\displaystyle \frac{ϵ}{2}},t+{\displaystyle \frac{ϵ}{2}}))=(\gamma (t-{\displaystyle \frac{ϵ}{2}}),\gamma (t+{\displaystyle \frac{ϵ}{2}}))=(2t-2{\displaystyle \frac{ϵ}{2}},2t+2{\displaystyle \frac{ϵ}{2}})=(2t-ϵ,2t+ϵ)$, then $\Gamma (H_{\xi }^{(t-{\displaystyle \frac{ϵ}{2}},t+{\displaystyle \frac{ϵ}{2}})\cap I})\widetilde{⊑}{\left(H_{\xi }^{G_{ϵ}}\right)}_I$. Hence, $\Gamma$ is soft-continuous.

Example 6.

Let $X=\left\{v\right\}$ and $\xi =\{e_1,e_2\}$, and consider the topology $\zeta =\{X_{\xi },{\varnothing }_{\xi },\left\{(e_1,\left\{v\right\})\right\}\}$. For the function $\gamma :I\to X$ defined by $\gamma \left(t\right)=v$ for all $t\in I$, $\Gamma =({\mathbb{I}}_{\xi },\gamma ):(I,{\left({\mathcal{U}}_{\xi }\right)}_I)\to {(X,\zeta )}_{\xi }$ is a soft path. Although $U=\left\{(e_1,\left\{v\right\})\right\}\in \zeta$, ${\Gamma }^{-1}\left(U\right)=\{(e_1,I),(e_2,\varnothing )\}$ is not soft-open in ${\left({\mathcal{U}}_{\xi }\right)}_I$. Therefore, in a soft path, the inverse image of a soft-open set is not always soft-open.

Example 7.

Let $X=\{1,2,3\}$, and let $\xi =\{e_1,e_2\}$ be a set of parameters. Define a soft topology ζ over X consisting of the following soft sets:

$$\zeta =\left\{{\varnothing }_{\xi },X_{\xi },U_1,U_2\right\},$$

where the soft sets $U_1$ and $U_2$ are given as follows:

$$\begin{array}{cccc}\hfill U_1\left(e_1\right)& =\left\{1\right\},\hfill & \hfill U_1\left(e_2\right)& =\left\{2\right\},\hfill \\ \hfill U_2\left(e_1\right)& =\{1,3\},\hfill & \hfill U_2\left(e_2\right)& =\left\{2\right\}.\hfill \end{array}$$

Now define the soft maps

$${\Gamma }_{13}=({\mathbb{I}}_{\xi },{\gamma }_{13}):(I,{\left({\mathcal{U}}_{\xi }\right)}_I)\to {(X,\zeta )}_{\xi },$$

and

$${\Gamma }_{22}=({\mathbb{I}}_{\xi },{\gamma }_{22}):(I,{\left({\mathcal{U}}_{\xi }\right)}_I)\to {(X,\zeta )}_{\xi },$$

with the functions ${\gamma }_{13},{\gamma }_{22}:I\to X$,

$${\gamma }_{13}\left(t\right)=\left\{\begin{array}{cc}1,\hfill & 0\le t<{\displaystyle \frac{1}{2}},\hfill \\ 3,\hfill & {\displaystyle \frac{1}{2}}\le t\le 1,\hfill \end{array}\right.{\gamma }_{22}\left(t\right)=2\mathit{for}\mathit{all}t\in I.$$

Let us show that ${\Gamma }_{13}$ and ${\Gamma }_{22}$ are soft-continuous by using Definition 11.

If $t\in [0,{\displaystyle \frac{1}{2}})$, then ${\gamma }_{13}\left(t\right)=1$, and in ζ the only soft-open set containing 1 is $X_{\xi }$. $H_{\xi }^{[0,{\displaystyle \frac{1}{2}})}$ is a soft-open set in ${\left({\mathcal{U}}_{\xi }\right)}_I$ that consists t and $1\tilde{\in }{\Gamma }_{13}(H_{\xi }^{[0,{\displaystyle \frac{1}{2}})})\widetilde{⊑}{X}_{\xi }$. Therefore, for every element of $[0,{\displaystyle \frac{1}{2}})$, ${\Gamma }_{13}$ is soft-continuous. Similarly, it is shown that ${\Gamma }_{13}$ is soft-continuous for every $t\in [{\displaystyle \frac{1}{2}},1]$. Hence, ${\Gamma }_{13}$ is a soft path.

If $t\in [0,1]$, then ${\gamma }_{22}\left(t\right)=2$, and in ζ the only open set containing 2 is $X_{\xi }$. $H_{\xi }^{[0,1]}$ is a soft-open set in ${\left({\mathcal{U}}_{\xi }\right)}_I$ that consists of t and $2\tilde{\in }{\Gamma }_{22}\left(H_{\xi }^{[0,1]}\right)\widetilde{⊑}{X}_{\xi }$. Therefore, for every element of $[0,1]$, ${\Gamma }_{22}$ is soft-continuous. Therefore, ${\Gamma }_{22}$ is a soft path.

Now, let us consider the soft-open set $U_1$. Then, we obtain

$${\Gamma }_{13}^{-1}\left(U_1\right)=\{(e_1,{\gamma }_{13}^{-1}\left\{1\right\}),(e_2,{\gamma }_{13}^{-1}\left\{2\right\})\}=\{(e_1,[0,\frac{1}{2})),(e_2,\varnothing )\},$$

and

$${\Gamma }_{22}^{-1}\left(U_1\right)=\{(e_1,{\gamma }_{22}^{-1}\left\{1\right\}),(e_2,{\gamma }_{22}^{-1}\left\{2\right\})\}=\{(e_1,\varnothing ),(e_2,I)\}.$$

So, the inverse images of the soft-open set $U_1$ are not soft-open sets.

Definition 17.

Let ${(X,\zeta )}_{\xi }$ be a soft-topological space. If for every $x,y\in X$ there exists a soft path

$$\Gamma =({\mathbb{I}}_{\xi },\gamma ):(I,{\left({\mathcal{U}}_{\xi }\right)}_I)\to {(X,\zeta )}_{\xi },$$

from x to y, then ${(X,\zeta )}_{\xi }$ is called a soft-path-connected space.

Example 8.

According to the soft usual topology defined above, the soft usual topological space $(\mathbb{R},{\mathcal{U}}_{\xi })$ is a soft-path-connected space. Indeed, for every $a,b\in \mathbb{R}$ there exists a function

$$\gamma :I\to \mathbb{R},t\mapsto (1-t)a+tb$$

from a to b. Similarly to Example 5, it can be shown that

$$\Gamma =({\mathbb{I}}_{\xi },\gamma ):(I,{\left({\mathcal{U}}_{\xi }\right)}_I)\to (\mathbb{R},{\mathcal{U}}_{\xi })$$

is a soft path. Hence, $(\mathbb{R},{\mathcal{U}}_{\xi })$ is soft-path-connected.

Proposition 4.

Let ${(X,\zeta )}_{\xi }$ and ${(X^{\prime },{\zeta }^{\prime })}_{{\xi }^{\prime }}$ be soft-topological spaces and $(\phi ,\varrho ):{(X,\zeta )}_{\xi }\to {(X^{\prime },{\zeta }^{\prime })}_{{\xi }^{\prime }}$ be a soft-continuous mapping. If A is a non-empty subset of X endowed with the soft subspace topology ${\zeta }_A$, then the soft mapping

$$(\phi ,\varrho {|}_A):{(A,{\zeta }_A)}_{\xi }\to {(X^{\prime },{\zeta }^{\prime })}_{{\xi }^{\prime }}$$

is also soft-continuous, where ${\varrho |}_A$ denotes the restriction of ϱ to A.

Proof. 

Let $a\in A\subset X$ and $\varrho \left(a\right)\tilde{\in }{U}^{\prime }$ be an arbitrary soft-open set in ${\zeta }^{\prime }$. Since $(\phi ,\varrho )$ is soft-continuous, there is a soft-open neighborhood U of a such that $(\phi ,\varrho )\left(U\right)\widetilde{⊑}{U}^{\prime }$. By the definition of the soft subspace topology, $\left(U_A\right)\in {\zeta }_A$, and hence $(\phi ,\varrho {|}_A)\left(U_A\right)\widetilde{⊑}{U}^{\prime }$. So $(\phi ,\varrho {|}_A)$ is soft-continuous. □

Theorem 3.

Let ${(X,\zeta )}_{\xi }$ and ${(X^{\prime },{\zeta }^{\prime })}_{\xi }$ be soft-topological spaces, and $A\subseteq X$. If $({\mathbb{I}}_{\xi },\varrho ):{(X,\zeta )}_{\xi }\to {(X^{\prime },{\zeta }^{\prime })}_{\xi }$ is a soft-continuous function and the soft subspace ${(A,{\zeta }_A)}_{\xi }$ is soft-path-connected, then the soft subspace ${(\varrho \left(A\right),{\zeta }_{\varrho \left(A\right)}^{\prime })}_{\xi }$ is also soft-path-connected.

Proof. 

Let $b_1,b_2\in \varrho \left(A\right)$. Then there exist $a_1,a_2\in A$ such that $\varrho \left(a_1\right)=b_1$ and $\varrho \left(a_2\right)=b_2$. Since ${(A,{\zeta }_A)}_{\xi }$ is soft-path-connected, there exists a soft path

$$\Gamma =({\mathbb{I}}_{\xi },\gamma ):(I,{\left({\mathcal{U}}_{\xi }\right)}_I)\to {(A,{\zeta }_A)}_{\xi },$$

where

$$\gamma :I\to A,\gamma \left(0\right)=a_1,\gamma \left(1\right)=a_2.$$

Now consider the composition

$$(I,{\left({\mathcal{U}}_{\xi }\right)}_I)\stackrel{\Gamma }{\to }{(A,{\zeta }_A)}_{\xi }\stackrel{({\mathbb{I}}_{\xi },\varrho {|}_A)}{\to }{(\varrho \left(A\right),{\zeta }_{\varrho \left(A\right)}^{\prime })}_{\xi }.$$

This yields the soft path

$$({\mathbb{I}}_{\xi },\varrho {|}_A)\circ \Gamma =({\mathbb{I}}_{\xi },(\varrho {|}_A\circ \gamma )):(I,{\left({\mathcal{U}}_{\xi }\right)}_I)\to {(\varrho \left(A\right),{\zeta }_{\varrho \left(A\right)}^{\prime })}_{\xi },$$

where

$${\varrho |}_A\circ \gamma :I\to \varrho \left(A\right)$$

is a function with $(\varrho {|}_A\circ \gamma )\left(0\right)=\varrho \left(a_1\right)=b_1$ and $(\varrho {|}_A\circ \gamma )\left(1\right)=\varrho \left(a_2\right)=b_2$.

Since both $\Gamma$ and $({\mathbb{I}}_{\xi },\varrho {|}_A)$ are soft-continuous, their composition is also soft-continuous. Therefore, there exists a soft path in $\varrho \left(A\right)$ from $b_1$ to $b_2$.

Hence, ${(\varrho \left(A\right),{\zeta }_{\varrho \left(A\right)}^{\prime })}_{\xi }$ is soft-path-connected. □

Proposition 5.

Let $A\subset \mathbb{R}$ be a proper subset, and let $\xi$ be any parameter set. If A is soft-connected in the soft usual topological space $(\mathbb{R},{\mathcal{U}}_{\xi })$, then A is an interval.

Proof. 

Suppose that A is not an interval. Now, consider the soft-open sets

$$U_{\xi }=H_{\xi }^{(-\infty ,x)\cap A}\mathrm{and}{V}_{\xi }=H_{\xi }^{(x,\infty )\cap A}$$

in soft subspace ${\left({\mathcal{U}}_{\xi }\right)}_A$ where $x\in \mathbb{R}$ and $x\notin A$. Then, we obtain $U_{\xi }\tilde{\bigsqcup }{V}_{\xi }=A_{\xi }$, $U_{\xi }\tilde{\sqcap }{V}_{\xi }={\varnothing }_{\xi }$, which implies that A is soft-disconnected. This contradicts the assumption that A is not an interval, so A is an interval. □

Theorem 4.

Let ${(X,\zeta )}_{\xi }$ be a soft-connected topological space, and let

$$({\mathbb{I}}_{\xi },\varrho ):{(X,\zeta )}_{\xi }\to (\mathbb{R},{\mathcal{U}}_{\xi })$$

be a soft-continuous mapping. Let $a,b\in \varrho \left(X\right)$. Then the function ϱ takes all values between a and b.

Proof. 

Since X is soft-connected, $\varrho \left(X\right)$ is also soft-connected. If the function $\varrho$ is onto, the proof is complete. If $\varrho$ is not onto, then $\varrho \left(X\right)$ is an interval. Hence, for each $c\in [a,b]\subseteq \varrho \left(X\right)$, there exists an $x\in X$ such that $\varrho \left(x\right)=c$. □

Proposition 6.

The soft usual topological space $(\mathbb{R},{\mathcal{U}}_{\xi })$ is soft-connected.

Proof. 

Suppose that $\mathbb{R}$ is soft-disconnected. Then there exist soft-open sets $H_{\xi }^G$ and $H_{\xi }^{G^{\prime }}$ such that $H_{\xi }^G\tilde{\bigsqcup }{H}_{\xi }^{G^{\prime }}=H_{\xi }^{G\cup G^{\prime }}={\mathbb{R}}_{\xi }$ and $H_{\xi }^G\tilde{\sqcap }{H}_{\xi }^{G^{\prime }}=H_{\xi }^{G\cap G^{\prime }}={\varnothing }_{\xi }$. From this, for the open sets G and $G^{\prime }$ of the usual topology on $\mathbb{R}$, $G\cup G^{\prime }=\mathbb{R}$ and $G\cap G^{\prime }=\varnothing$, which contradicts the connectedness of $\mathbb{R}$ with respect to the usual topology. Hence, $\mathbb{R}$ is soft-connected. □

Proposition 7.

$(I,{\left({\mathcal{U}}_{\xi }\right)}_I)$ is a soft-connected space.

Proof. 

Suppose that $(I,{\left({\mathcal{U}}_{\xi }\right)}_I)$ is soft-disconnected. Then there exist soft-open sets $H_{\xi }^{G\cap I}$ and $H_{\xi }^{G^{\prime }\cap I}$ in ${\left({\mathcal{U}}_{\xi }\right)}_I$ such that

$$H_{\xi }^{G\cap I}\tilde{\bigsqcup }{H}_{\xi }^{G^{\prime }\cap I}=I_{\xi }\mathrm{and}{H}_{\xi }^{G\cap I}\tilde{\sqcap }{H}_{\xi }^{G^{\prime }\cap I}={\varnothing }_{\xi }.$$

$G\cap I$ and $G^{\prime }\cap I$ are disjoint open sets in the usual topology of I and $(G\cap I)\cup (G^{\prime }\cap I)=I$. This implies that I is disconnected with respect to the usual topology, which is a contradiction. Therefore, $(I,{\left({\mathcal{U}}_{\xi }\right)}_I)$ is a soft-connected space. □

Theorem 5.

If ${(X,\zeta )}_{\xi }$ is a soft-path-connected space, then ${(X,\zeta )}_{\xi }$ is a soft-connected space.

Proof. 

Assume that X is soft-disconnected, then there exist absolute soft-open subsets such that $A_{\xi }\tilde{\bigsqcup }{B}_{\xi }=X_{\xi }$ and $A_{\xi }\tilde{\sqcap }{B}_{\xi }={\varnothing }_{\xi }$. Choose $x\in A$ and $y\in B$ and let

$$\Gamma =({\mathbb{I}}_{\xi },\gamma ):(I,{\left({\mathcal{U}}_{\xi }\right)}_I)\to {(X,\zeta )}_{\xi }$$

be a soft path from x to y. Since $({\mathbb{I}}_{\xi },\gamma )$ is soft-continuous, $\gamma \left(I\right)$ is soft-connected; however,

$$\left(A_{\xi }\tilde{\sqcap }\gamma {\left(I\right)}_{\xi }\right)\tilde{\bigsqcup }\left(B_{\xi }\tilde{\sqcap }\gamma {\left(I\right)}_{\xi }\right)=\gamma {\left(I\right)}_{\xi }\mathrm{and}\left(A_{\xi }\tilde{\sqcap }\gamma {\left(I\right)}_{\xi }\right)\tilde{\sqcap }\left(B_{\xi }\tilde{\sqcap }\gamma {\left(I\right)}_{\xi }\right)={\varnothing }_{\xi }$$

imply that $\gamma \left(I\right)$ is soft-disconnected, which is a contradiction. Thus, ${(X,\zeta )}_{\xi }$ is a soft-connected space. □

5. Soft-Topological Groups

Now we will investigate the concepts of soft-connectedness and soft-path-connectedness in the context of soft-topological groups.

Definition 18.

Let ${(G,\zeta )}_{\xi }$ be a soft-topological group. Then ${(G,\zeta )}_{\xi }$ is called a soft-connected topological group if the underlying soft-topological space ${(G,\zeta )}_{\xi }$ is soft-connected.

Theorem 6.

Let ${(G,\zeta )}_{\xi }$ be a soft-topological group. Then, the soft group operation

$$\ast :G\times G\to G,(a,b)\mapsto a\ast b$$

is soft-continuous.

Proof. 

Suppose that ${(G,\zeta )}_{\xi }$ is a soft-topological group. Then for every $(a,b)\in G\times G$ and for every soft neighborhood W of $a\ast b$, there are soft-open neighborhoods U of a and V of b such that $U\ast V\widetilde{⊑}W$. In the soft product space ${(G\times G,{\zeta }_{\times })}_{\xi \times \xi }$, $U\tilde{\times }V$ is a soft-open neighborhood of $(a,b)$ and $\ast \left(U\tilde{\times }V\right)=U\ast V\widetilde{⊑}W$. It shows that ∗ is soft-continuous. □

Proposition 8.

Let ${(G,\zeta )}_{\xi }$ be a soft-topological group. If H is a subgroup of G, then ${(H,{\zeta }_H)}_{\xi }$ is a soft-topological group.

Proof. 

Let H be a subgroup of the soft-topological group ${(G,\zeta )}_{\xi }$ equipped with the soft subspace topology. Since H is a subgroup, for every $x,y\in H$, $x{y}^{-1}\in H\subseteq G$. And since G is a soft-topological group, for every $x,y\in G$ and for every soft-open set W containing $x{y}^{-1}$, there are soft-open sets U and V such that $x\tilde{\in }U,y\tilde{\in }V$ and $U{V}^{-1}\widetilde{⊑}W$. Then, by the definition of the soft subspace topology, for every $x,y\in H$ and for every soft-open set $W_H$ containing $x{y}^{-1}$, there are soft-open sets $U_H$ and $V_H$ such that $x\tilde{\in }{U}_H,y\tilde{\in }{V}_H$ and $\left(U_H\right){\left(V_H\right)}^{-1}\widetilde{⊑}{W}_H$. Hence, ${(H,{\zeta }_H)}_{\xi }$ is a soft-topological group. □

Proposition 9.

Let ${(G,\zeta )}_{\xi }$ be a soft-topological group and H be a subgroup of G such that $H_{\xi }$ is soft-open. Then $H_{\xi }$ is soft-closed [13].

Proposition 10.

A soft-connected soft-topological group ${(G,\zeta )}_{\xi }$ has no soft subgroup $H\subseteq G$ such that $H_{\xi }$ is soft-open.

Proof. 

Let ${(G,\zeta )}_{\xi }$ be a soft-connected soft-topological group. Assume that H is a subgroup of G such that $H_{\xi }$ is soft-open. Then, by Proposition 9, $H_{\xi }$ is soft-closed. This contradicts the soft-connectedness of G. Hence, G has no subgroup H such that $H_{\xi }$ is soft-open. □

Proposition 11.

Let ${(X,\zeta )}_{\xi }$ be a soft-topological space and ${(G,{\zeta }^{\prime })}_{\xi }$ be a soft-topological group. If $({\mathbb{I}}_{\xi },\varrho ):{(X,\zeta )}_{\xi }\to {(G,{\zeta }^{\prime })}_{\xi }$ and $({\mathbb{I}}_{\xi },{\varrho }^{\prime }):{(X,\zeta )}_{\xi }\to {(G,{\zeta }^{\prime })}_{\xi }$ are soft-continuous mappings, then

$$({\mathbb{I}}_{\xi },\varrho \ast {\varrho }^{\prime }):{(X,\zeta )}_{\xi }\to {(G,{\zeta }^{\prime })}_{\xi }$$

is also a soft-continuous mapping.

Proof. 

Let $({\mathbb{I}}_{\xi },\varrho ):{(X,\zeta )}_{\xi }\to {(G,{\zeta }^{\prime })}_{\xi }$ and $({\mathbb{I}}_{\xi },{\varrho }^{\prime }):{(X,\zeta )}_{\xi }\to {(G,{\zeta }^{\prime })}_{\xi }$ be soft-continuous mappings. Now consider the soft mapping

$$({\Delta }_{\xi },(\varrho ,{\varrho }^{\prime })):{(X,\zeta )}_{\xi }\to {(G\times G,{\zeta }_{\times }^{\prime })}_{\xi \times \xi },$$

for the functions ${\Delta }_{\xi }:\xi \to \xi \times \xi$ defined by ${\Delta }_{\xi }\left(e\right)=(e,e)$ and $(\varrho ,{\varrho }^{\prime }):X\to G\times G$ defined by $(\varrho ,{\varrho }^{\prime })\left(x\right)=(\varrho \left(x\right),{\varrho }^{\prime }\left(x\right))$,where ${\zeta }_{\times }^{\prime }$ is a soft product topology induced by ${\zeta }^{\prime }$.

Since $({\mathbb{I}}_{\xi },\varrho )$ is soft-continuous, then for every $x\in X$ and every soft-open neighborhood $U^{\prime }$ of $\varrho \left(x\right)$, there is a soft-open neighborhood U of x such that $\varrho \left(x\right)\tilde{\in }\varrho \left(U\right)\widetilde{⊑}{U}^{\prime }$. Similarly, since $({\mathbb{I}}_{\xi },{\varrho }^{\prime })$ is soft-continuous, for every soft-open set $V^{\prime }$ containing the point $\varrho \left(x\right)$, there is a soft-open set V containing x such that ${\varrho }^{\prime }\left(x\right)\tilde{\in }{\varrho }^{\prime }\left(V\right)\widetilde{⊑}{V}^{\prime }$. So for every $x\in X$ and every soft-open neighborhood $U^{\prime }\tilde{\times }{V}^{\prime }$ of $(\varrho \left(x\right),{\varrho }^{\prime }\left(x\right))$, there is a soft open neighborhood $U\tilde{\sqcap }V$ of x such that $(\varrho ,{\varrho }^{\prime })\left(x\right)\tilde{\in }(\varrho ,{\varrho }^{\prime })\left(U\tilde{\sqcap }V\right)\widetilde{⊑}{U}^{\prime }\tilde{\times }{V}^{\prime }$. Hence $({\Delta }_{\xi },(\varrho ,{\varrho }^{\prime }))$ is a soft continuous mapping.

Moreover, by Theorem 6, the group operation

$$\ast :G\times G\to G,(a,b)\mapsto a\ast b$$

is soft-continuous. Therefore, the composition

$$\begin{array}{cc}\hfill {(X,\zeta )}_{\xi }& \stackrel{({\Delta }_{\xi },(\varrho ,{\varrho }^{\prime }))}{\to }{(G\times G,{\zeta }_{\times }^{\prime })}_{\xi \times \xi }\stackrel{\ast }{\to }{(G,{\zeta }^{\prime })}_{\xi }\hfill \\ \hfill x& ⟼(\varrho \left(x\right),{\varrho }^{\prime }\left(x\right))⟼\varrho \left(x\right)\ast {\varrho }^{\prime }\left(x\right).\hfill \end{array}$$

is soft-continuous. Hence, we conclude that

$$(\ast )\circ (({\Delta }_{\xi },(\varrho ,{\varrho }^{\prime }))=({\mathbb{I}}_{\xi },\varrho \ast {\varrho }^{\prime })$$

is soft-continuous as well. □

Example 9.

Let $(G,\ast )$ be a group and let ${(G,\zeta )}_{\xi }$ be a soft-topological group. If $\Gamma :(I,{\left({\mathcal{U}}_{\xi }\right)}_I)\to {(G,\zeta )}_{\xi }$ and ${\Gamma }^{\prime }:(I,{\left({\mathcal{U}}_{\xi }\right)}_I)\to {(G,\zeta )}_{\xi }$ are soft paths, then $\Gamma \ast {\Gamma }^{\prime }:(I,{\left({\mathcal{U}}_{\xi }\right)}_I,\xi )\to {(G,\zeta )}_{\xi }$ is also a soft path.

Proposition 12.

Let ${(G,\zeta )}_{\xi }$ be a soft-topological group, and let H and K be soft-connected subsets of G. Then $H\ast K\subseteq G$ is also soft-connected.

Proof. 

Assume, to the contrary, that $H\ast K$ is soft-disconnected. Then there exist two nonempty, disjoint absolute soft-open subsets $A_{\xi }$ and $B_{\xi }$ of ${(G,\zeta )}_{\xi }$ such that $H\ast K=A\cup B,A\cap B=\varnothing$. For each $h\in H$, by ([13] Proposition 4.5), consider the left translation ${\alpha }_L\left(h\right):G\to G$ defined by ${\alpha }_L\left(h\right)\left(x\right)=h\ast x$. Since ${\alpha }_L\left(h\right)$ is soft-continuous, the image $h\ast K={\alpha }_L\left(h\right)\left(K\right)$ is soft-connected for every $h\in H$.

Since $A_{\xi }$ and $B_{\xi }$ form a soft separation of $H\ast K$, they induce a soft separation on each subset $h\ast K$. By the soft-connectedness of $h\ast K$, one must have $h\ast K\subseteq A$ or $h\ast K\subseteq B$.

Suppose that there exist $h_1,h_2\in H$ with $h_1\ast K\subseteq A$ and $h_2\ast K\subseteq B$. Choose $k^{\prime }\in K$ and note that since the right translation is soft-continuous, $H\ast k^{\prime }$ is also soft-connected. Then $h_1\ast k^{\prime }\in A$ and $h_2\ast k^{\prime }\in B$. Therefore, $h_1\ast k^{\prime }\tilde{\in }{A}_{\xi }$ and $h_2\ast k^{\prime }\tilde{\in }{B}_{\xi }$, contradicting the soft-connectedness of $H\ast k^{\prime }$. Hence, all $h\ast K$ subsets must lie in the same soft-open set, say $A_{\xi }$, implying

$$H\ast K=\bigcup _{h\in H}h\ast K\subseteq A.$$

Thus, $B_{\xi }={\varnothing }_{\xi }$, contradicting the assumption that both A and B are nonempty. Therefore, $H\ast K$ is soft-connected. □

Theorem 7.

Let ${(X,\zeta )}_{\xi }$ and ${(X^{\prime },{\zeta }^{\prime })}_{{\xi }^{\prime }}$ be soft-path-connected spaces. Then the soft product space ${(X\times X^{\prime },{\zeta }_{\times })}_{\xi \times {\xi }^{\prime }}$ is also soft-path-connected.

Proof. 

Let $(x_1,y_1),(x_2,y_2)\in X\times X^{\prime }$. Since ${(X,\zeta )}_{\xi }$ and ${(X^{\prime },{\zeta }^{\prime })}_{{\xi }^{\prime }}$ are soft-path-connected, there exist soft paths

$${\Gamma }_X=({\mathbb{I}}_{\xi },{\gamma }_X):(I,{\left({\mathcal{U}}_{\xi }\right)}_I)\to {(X,\zeta )}_{\xi },{\Gamma }_{X^{\prime }}=({\mathbb{I}}_{\xi }^{\prime },{\gamma }_{X^{\prime }})(I,{\left({\mathcal{U}}_{\xi }\right)}_I)\to {(X^{\prime },{\zeta }^{\prime })}_{{\xi }^{\prime }},$$

such that ${\gamma }_X\left(0\right)=x_1$, ${\gamma }_X\left(1\right)=x_2$ and ${\gamma }_{X^{\prime }}\left(0\right)=y_1$, ${\gamma }_{X^{\prime }}\left(1\right)=y_2$.

Define a soft mapping

$$({\mathbb{I}}_{\xi \times {\xi }^{\prime }},\gamma ):(I,{\left({\mathcal{U}}_{\xi \times {\xi }^{\prime }}\right)}_I)\to {(X\times X^{\prime },{\zeta }_{\times })}_{\xi \times {\xi }^{\prime }}$$

by

$$\Gamma ={\mathbb{I}}_{\xi }\times {\mathbb{I}}_{{\xi }^{\prime }}={\mathbb{I}}_{\xi \times {\xi }^{\prime }}:\xi \times {\xi }^{\prime }\to \xi \times {\xi }^{\prime }\mathrm{and}\gamma \left(t\right)=({\gamma }_X\left(t\right),{\gamma }_{X^{\prime }}\left(t\right)).$$

Since both $({\mathbb{I}}_{\xi },{\gamma }_X)$ and $({\mathbb{I}}_{\xi }^{\prime },{\gamma }_{X^{\prime }})$ are soft-continuous, the soft continuity of $({\mathbb{I}}_{\xi \times {\xi }^{\prime }},\gamma )$ can be established in a manner similar to Theorem 11.

Moreover, $\gamma \left(0\right)=(x_1,y_1)$ and $\gamma \left(1\right)=(x_2,y_2)$. Hence, $\Gamma$ is a soft path in $X\times X^{\prime }$ connecting $(x_1,y_1)$ and $(x_2,y_2)$.

Therefore, ${(X\times X^{\prime },{\zeta }_{\times })}_{\xi \times {\xi }^{\prime }}$ is soft-path-connected. □

Theorem 8.

Let ${(G,\zeta )}_{\xi }$ be a soft-topological group. If U and V are soft-path-connected subsets of G, then $U\ast V$ is also soft-path-connected.

Proof. 

Since U and V are soft-path-connected, by Theorem 7, their soft product $U\tilde{\times }V$ is also soft-path-connected. On the other hand, by Theorems 3 and 6, since

$$\ast :G\times G\to G,(a,b)\mapsto a\ast b$$

is soft-continuous, it follows that the image of $U\tilde{\times }V$ under this operation, $U\ast V$ is also soft-path-connected. □

Theorem 9.

Let ${(G,\zeta )}_{\xi }$ and ${(G^{\prime },{\zeta }^{\prime })}_{\xi }$ be soft-topological groups, and let

$$\varrho :G\to G^{\prime }$$

be a group homomorphism. Then $({\mathbb{I}}_{\xi },\varrho )$ is soft-continuous if and only if it is soft-continuous at the identity element $e_G$.

Proof. 

(⇒) If $({\mathbb{I}}_{\xi },\varrho )$ is soft-continuous on G, then it is certainly soft-continuous at $e_G$.

(⇐) Assume that $({\mathbb{I}}_{\xi },\varrho )$ is soft-continuous at $e_G$. We need to show that $({\mathbb{I}}_{\xi },\varrho )$ is soft-continuous at an arbitrary $a\in G$. Let $W^{\prime }$ be a soft-open neighborhood of $\varrho \left(a\right)$ in ${\zeta }^{\prime }$. By ([13] Proposition 4.5),

$${\alpha }_L\left(a\right):G\to G,x\mapsto a\ast x,\mathrm{and}{\beta }_L\left(\varrho \left(a\right)\right):G^{\prime }\to G^{\prime },x\mapsto \varrho \left(a\right)\ast x.$$

are soft homeomorphisms, then the soft set ${\beta }_L\left(\varrho {\left(a\right)}^{-1}\right)(W^{\prime }{)=\varrho \left(a\right)}^{-1}\ast W^{\prime }$ is a soft-open neighborhood of the identity element $e_{G^{\prime }}$ of the group $G^{\prime }$. So, there exists a soft neighborhood W of $e_G$ in ${(G,\zeta )}_{\xi }$ such that $({\mathbb{I}}_{\xi },\varrho )\left(W\right)\widetilde{⊏}{W}^{\prime }$. Thus, ${\alpha }_L\left(a\right)\left(W\right)=a\ast W$ is a soft-open neighborhood of a. For any $a\in G$ we have

$$\varrho \left(a\right)\tilde{\in }({\mathbb{I}}_{\xi },\varrho )(a\ast W)\widetilde{⊏}{W}^{\prime }.$$

This proves the soft continuity of $\varrho$ at a. Since a was arbitrary, $({\mathbb{I}}_{\xi },\varrho )$ is soft-continuous on G. □

6. Category of Soft-Topological Groups

In this section, we first define the morphisms of the category whose objects are soft-topological groups, as well as the composition operation for these morphisms.

Definition 19.

Let ${(G,\zeta )}_{\xi }$ and ${(G^{\prime },{\zeta }^{\prime })}_{{\xi }^{\prime }}$ be soft-topological groups. If $(\phi ,\varrho ):{(G,\zeta )}_{\xi }\to {(G^{\prime },{\zeta }^{\prime })}_{{\xi }^{\prime }}$ is a soft-continuous mapping such that $\phi :\xi \to {\xi }^{\prime }$ is a function, and $\varrho :G\to G^{\prime }$ is a group homomorphism, then $(\phi ,\varrho )$ is called a morphism of soft-topological groups.

Example 10.

Let ${(G,\zeta )}_{\xi }$ be a soft-topological group. Let ${\mathbb{I}}_{\xi }:\xi \to \xi$ be the identity function on the parameter set, and ${\mathbb{I}}_G:G\to G$ be the identity group homomorphism. Then the soft mapping

$$({\mathbb{I}}_{\xi },{\mathbb{I}}_G):{(G,\zeta )}_{\xi }\to {(G,\zeta )}_{\xi }$$

is soft-continuous. This morphism of soft-topological groups is called the identity morphism of soft-topological groups and is denoted by ${\mathbb{I}}_{{(G,\zeta )}_{\xi }}.$

Definition 20.

Let $({\phi }_1,{\varrho }_1):{(G_1,{\zeta }_1)}_{{\xi }_1}\to {(G_2,{\zeta }_2)}_{{\xi }_2}$ and $({\phi }_2,{\varrho }_2):{(G_2,{\zeta }_2)}_{{\xi }_2}\to {(G_3,{\zeta }_3)}_{{\xi }_3}$ be morphisms of soft-topological groups. Then $({\phi }_1,{\varrho }_1)=({\phi }_2,{\varrho }_2)$ if and only if ${\phi }_1={\phi }_2$ and ${\varrho }_1={\varrho }_2$.

Proposition 13.

Let ${(G_1,{\zeta }_1)}_{{\xi }_1}$, ${(G_2,{\zeta }_2)}_{{\xi }_2}$, and ${(G_3,{\zeta }_3)}_{{\xi }_3}$ be soft-topological groups. Suppose that

$$({\phi }_1,{\varrho }_1):{(G_1,{\zeta }_1)}_{{\xi }_1}\to {(G_2,{\zeta }_2)}_{{\xi }_2}\mathrm{and}({\phi }_2,{\varrho }_2):{(G_2,{\zeta }_2)}_{{\xi }_2}\to {(G_3,{\zeta }_3)}_{{\xi }_3}$$

are morphisms of soft-topological groups. Then the composition

$$({\phi }_1,{\varrho }_1)\circ ({\phi }_2,{\varrho }_2)=({\phi }_2\circ {\phi }_1,{\varrho }_2\circ {\varrho }_1)$$

is also a morphism of soft-topological groups.

Proof. 

Since ${\phi }_2\circ {\phi }_1$ is a function and ${\varrho }_2\circ {\varrho }_1$ is a group homomorphism, it is sufficient to show that $({\phi }_2\circ {\phi }_1,{\varrho }_2\circ {\varrho }_1)$ is a soft-continuous mapping. And since $({\phi }_1,{\varrho }_1)$ is a soft-continuous mapping, for every $a\in G_1$ and for every soft-open neighborhood $U^{\prime }$ of ${\varrho }_1\left(a\right)$, there is a soft-open neighborhood U of a such that ${\varrho }_1\left(a\right)\tilde{\in }({\phi }_1,{\varrho }_1)\left(U\right)\widetilde{⊑}{U}^{\prime }$.

Let $b={\varrho }_1\left(a\right)$. Because $({\phi }_2,{\varrho }_2)$ is a soft-continuous mapping, for every $b\in G_2$ and for every soft-open neighborhood $U^{\prime \prime }$ of ${\varrho }_2\left(b\right)={\varrho }_2\left({\varrho }_1\left(a\right)\right)$ there is a soft-open neighborhood $U^{\prime }$ of b such that ${\varrho }_2\left({\varrho }_1\left(x\right)\right)\tilde{\in }({\phi }_2\circ {\phi }_1,{\varrho }_2\circ {\varrho }_1)\left(U^{\prime }\right)\widetilde{⊑}{U}^{″}.$

Hence, the composition $({\phi }_1,{\varrho }_1)\circ ({\phi }_2,{\varrho }_2)$ is also soft-continuous. □

Theorem 10.

The soft-topological groups and the morphisms between the soft-topological groups together form a category.

Proof. 

The objects of the category are all soft-topological groups, and the morphisms are all soft-topological group morphisms between objects.

The partial composition of the category is defined as the composition of morphisms of soft-topological groups.

For every object ${(G,\zeta )}_{\xi }$, there exists an identity morphism

$${\mathbb{I}}_{{(G,\zeta )}_{\xi }}=({\mathbb{I}}_{\xi },{\mathbb{I}}_G):{(G,\zeta )}_{\xi }⟶{(G,\zeta )}_{\xi }$$

as given in Example 10. For any morphism

$$(\phi ,\varrho ):{(G,\zeta )}_{\xi }\to {(G^{\prime },{\zeta }^{\prime })}_{{\xi }^{\prime }},$$

the composition satisfies

$$(\phi ,\varrho )\circ {\mathbb{I}}_{{(G,\zeta )}_{\xi }}=(\phi ,\varrho )\circ ({\mathbb{I}}_{\xi },{\mathbb{I}}_G)=(\phi \circ {\mathbb{I}}_{\xi },\varrho \circ {\mathbb{I}}_G)=(\phi ,\varrho ).$$

Similarly,

$${\mathbb{I}}_{{(G^{\prime },{\zeta }^{\prime })}_{{\xi }^{\prime }}}\circ (\phi ,\varrho )=(\phi ,\varrho ).$$

Let ${(G_1,{\zeta }_1)}_{{\xi }_1}$, ${(G_2,{\zeta }_2)}_{{\xi }_2}$, ${(G_3,{\zeta }_3)}_{{\xi }_3}$, and ${(G_4,{\zeta }_4)}_{{\xi }_4}$ be soft-topological groups. And let $({\phi }_1,{\varrho }_1):{(G_1,{\zeta }_1)}_{{\xi }_1}\to {(G_2,{\zeta }_2)}_{{\xi }_2}$, $({\phi }_2,{\varrho }_2):{(G_2,{\zeta }_2)}_{{\xi }_2}\to {(G_3,{\zeta }_3)}_{{\xi }_3}$ and $({\phi }_3,v_3):{(G_3,{\zeta }_3)}_{{\xi }_3}\to {(G_4,{\zeta }_4)}_{{\xi }_4}$ be morphisms of soft-topological groups. Since the composition of functions and the composition of group homomorphisms are associative, we also have

$$\begin{array}{cc}\hfill ({\phi }_3,{\varrho }_3)\circ \left(({\phi }_2,{\varrho }_2)\circ ({\phi }_1,{\varrho }_1)\right)& =({\phi }_3,{\varrho }_3)\circ ({\phi }_2\circ {\phi }_1,{\varrho }_2\circ {\varrho }_1)\hfill \\ \hfill & =({\phi }_3\circ ({\phi }_2\circ {\phi }_1),{\varrho }_3\circ ({\varrho }_2\circ {\varrho }_1))\hfill \\ \hfill & =(({\phi }_3\circ {\phi }_2)\circ {\phi }_1,({\varrho }_3\circ {\varrho }_2)\circ {\varrho }_1)\hfill \\ \hfill & =({\phi }_3\circ {\phi }_2,{\varrho }_3\circ {\varrho }_2)\circ ({\phi }_1,{\varrho }_1)\hfill \\ \hfill & =(({\phi }_3,{\varrho }_3)\circ ({\phi }_2,{\varrho }_2))\circ ({\phi }_1,{\varrho }_1).\hfill \end{array}$$

Thus, the composition of the category is associative. □

The category we have obtained here is called the category of soft-topological groups and is denoted by $\mathbf{STGrp}$.

Let $\mathbf{Grp}$ the category of groups. We can define a functor

$$\mathcal{U}:\mathbf{STGrp}⟶\mathbf{Grp}$$

by assigning to each object ${(G,\zeta )}_{\xi }$ of $\mathbf{STGrp}$ the underlying group G, and to each morphism $(\phi ,\varrho ):{(G,\zeta )}_{\xi }\to {(G^{\prime },{\zeta }^{\prime })}_{{\xi }^{\prime }}$ the group homomorphism $\varrho :G\to G^{\prime }$. Then $\mathcal{U}$ is a forgetful functor.

Remark 4.

While the functor $\mathcal{U}:\mathbf{STGrp}\to \mathbf{Grp}$ is naturally defined, there is no natural functor from $\mathbf{STGrp}$ to the category $\mathbf{SGp}$, whose objects are soft groups and whose morphisms are soft group homomorphisms [19]. Indeed, when the objects of these two categories are compared, it becomes clear that there is no natural connection between them. The category $\mathbf{STGrp}$ consists of groups equipped with a soft topology, whereas $\mathbf{SGp}$ consists of algebraic structures defined by subgroups parameterized by a given set. Since these two categories are based on different notions of softness, a natural relationship between them cannot be established.

Proposition 14.

Let ${(G,\zeta )}_{\xi }$ and ${(G^{\prime },{\zeta }^{\prime })}_{{\xi }^{\prime }}$ be soft-topological groups. Then for the morphism of soft-topological groups $(\phi ,\varrho ):{(G,\zeta )}_{\xi }\to {(G^{\prime },{\zeta }^{\prime })}_{{\xi }^{\prime }}$:

(i) 

If φ is an injective function and ϱ is an injective group homomorphism, then $(\phi ,\varrho )$ is a monomorphism.

(ii) 

If φ is a surjective function and ϱ is a surjective group homomorphism, then $(\phi ,\varrho )$ is an epimorphism.

Proof. 

(i)

Let $({\phi }_1,{\varrho }_1),({\phi }_2,{\varrho }_2):{(G^{″},{\zeta }^{″})}_{{\xi }^{″}}\to {(G,\zeta )}_{\xi }$ be morphisms of soft-topological groups such that

$$(\phi ,\varrho )\circ ({\phi }_1,{\varrho }_1)=(\phi ,\varrho )\circ ({\phi }_2,{\varrho }_2).$$

In this case,

$$\phi \circ {\phi }_1=\phi \circ {\phi }_2\mathrm{and}\varrho \circ {\varrho }_1=\varrho \circ {\varrho }_2.$$

Since $\phi$ is an injective function and $\varrho$ is an injective group homomorphism, it follows that ${\phi }_1={\phi }_2\mathrm{and}{\varrho }_1={\varrho }_2.$ Therefore, $({\phi }_1,{\varrho }_1)=({\phi }_2,{\varrho }_2)$, which shows that $(\phi ,\varrho )$ is a monomorphism.

(ii)

Let $({\phi }_3,{\varrho }_3),({\phi }_4,{\varrho }_4):{(G^{\prime },{\zeta }^{\prime })}_{{\xi }^{\prime }}\to {(G^{‴},{\zeta }^{‴})}_{{\xi }^{‴}}$ be morphisms of soft-topological groups such that

$$({\phi }_3,{\varrho }_3)\circ (\phi ,\varrho )=({\phi }_4,{\varrho }_4)\circ (\phi ,\varrho ).$$

Then,

$$({\phi }_3\circ \phi ,{\varrho }_3\circ \varrho )=({\phi }_4\circ \phi ,{\varrho }_4\circ \varrho ),$$

which implies ${\phi }_3\circ \phi ={\phi }_4\circ \phi \mathrm{and}{\varrho }_3\circ \varrho ={\varrho }_4\circ \varrho .$ Since $\phi$ is a surjective function and $\varrho$ is a surjective group homomorphism, it follows that ${\phi }_3={\phi }_4\mathrm{and}{\varrho }_3={\varrho }_4.$ Therefore, $({\phi }_3,{\varrho }_3)=({\phi }_4,{\varrho }_4)$, which shows that $(\phi ,\varrho )$ is an epimorphism. □

Theorem 11.

Let ${(G,\zeta )}_{\xi }$ and ${(G^{\prime },{\zeta }^{\prime })}_{{\xi }^{\prime }}$ be soft-topological groups, and let

$$(\phi ,\varrho ):{(G,\zeta )}_{\xi }\to {(G^{\prime },{\zeta }^{\prime })}_{{\xi }^{\prime }}$$

be a morphism of soft-topological groups. If ${(K,{\zeta }_K)}_{\xi }$ is a soft-topological subgroup of ${(G,\zeta )}_{\xi }$, then ${(\varrho \left(K\right),{\zeta }_{\varrho \left(K\right)})}_{{\xi }^{\prime }}$ is a soft-topological subgroup of ${(G^{\prime },{\zeta }^{\prime })}_{{\xi }^{\prime }}$.

Proof. 

Since $\varrho$ is a homomorphism and K is a subgroup of G, the image $\varrho \left(K\right)$ is a subgroup of $G^{\prime }$.

By Proposition 8, since $\varrho \left(K\right)$ is a subgroup of $G^{\prime }$, ${(\varrho \left(K\right),{\zeta }_{\varrho \left(K\right)})}_{{\xi }^{\prime }}$ is a soft-topological subgroup. □

Proposition 15.

Let $(\phi ,\varrho ):{(G,\zeta )}_{\xi }\to {(G^{\prime },{\zeta }^{\prime })}_{{\xi }^{\prime }}$ be a morphism of soft-topological groups. If $(\phi ,\varrho )$ is a monomorphism, then ϱ is injective.

Proof. 

Let ${\mathbb{I}}_{\xi }:\xi \to \xi$ be the identity function, $K=ker\left(\varrho \right)$, and $i:K\to G$ be the inclusion group homomorphism. And assume that ${\zeta }_K$ is the soft subtopology on K induced from $\zeta$. Then,

$$({\mathbb{I}}_{\xi },i):{(K,{\zeta }_K)}_{\xi }\to {(G,\zeta )}_{\xi }$$

is a morphism of soft-topological groups. Here, it is clear from Proposition 4 that $({\mathbb{I}}_{\xi },i)$ is soft-continuous.

Let ${\mathbb{I}}_{\xi }:\xi \to \xi$ be the identity function and ${\varrho }_1:K\to G$ be the trivial group homomorphism. Now, let us show that

$$({\mathbb{I}}_{\xi },{\varrho }_1):{(K,{\zeta }_K)}_{\xi }\to {(G,\zeta )}_{\xi }$$

is soft-continuous. Let $x\in K$, and $U^{\prime }$ be any soft-open neighborhood of ${\varrho }_1\left(x\right)=1_G$. Then $U_K^{\prime }$ is a soft-open neighborhood of x in ${\zeta }_K$ such that $({\mathbb{I}}_{\xi },{\varrho }_1)\left(U_K^{\prime }\right)\widetilde{⊑}{U}^{\prime }$. Therefore,$({\mathbb{I}}_{\xi },{\varrho }_1)$ is soft-continuous. Thus,

$$\begin{array}{ccc}\hfill {(K,{\zeta }_K)}_{\xi }\underset{({\mathbb{I}}_{\xi },{\varrho }_1)}{\overset{({\mathbb{I}}_{\xi },i)}{{\displaystyle \overrightarrow{\to }}}}& {(G,\zeta )}_{\xi }\stackrel{(\phi ,\varrho )}{\to }& {(G^{\prime },{\zeta }^{\prime })}_{{\xi }^{\prime }}.\hfill \end{array}$$

$$(\phi ,\varrho )\circ ({\mathbb{I}}_{\xi },i)=(\phi ,\varrho )\circ ({\mathbb{I}}_{\xi },{\varrho }_1)$$

$$(\phi \circ {\mathbb{I}}_{\xi },\varrho \circ i)=(\phi \circ {\mathbb{I}}_{\xi },\varrho \circ {\varrho }_1)$$

because both sides of the equality $\varrho \circ i=\varrho \circ {\varrho }_1$ are trivial maps from K to $G^{\prime }$. Since $(\phi ,\varrho )$ is monic, it follows that

$$({\mathbb{I}}_{\xi },i)=({\mathbb{I}}_{\xi },{\varrho }_1),$$

and, hence,

$$i={\varrho }_1.$$

This means that the inclusion map $i:K\to G$ is trivial. Consequently, the kernel $K=ker\left(\varrho \right)=\left\{1_G\right\}$ is a singleton. Therefore, $\varrho$ is injective. □

Proposition 16.

Let $(\phi ,\varrho ):{(G,\zeta )}_{\xi }\to {(G^{\prime },{\zeta }^{\prime })}_{{\xi }^{\prime }}$ be a morphism of soft-topological groups. If $(\phi ,\varrho )$ is an epimorphism, then ϱ is surjective.

Proof. 

Let $K=Im\left(\varrho \right)$. Let ${\varrho }_1:H\to H/K$ be the canonical epimorphism, and let ${\varrho }_2:H\to H/K$ be the trivial group homomorphism. Define the soft indiscrete topology on $H/K$ as ${\zeta }^{″}=\{{\varnothing }_{{\xi }^{\prime }},{H/K}_{{\xi }^{\prime }}\}.$ Then,

$$({\mathbb{I}}_{\xi },{\varrho }_1),({\mathbb{I}}_{\xi },{\varrho }_2):{(G^{\prime },{\zeta }^{\prime })}_{{\xi }^{\prime }}\to {(H/K,{\zeta }^{\prime \prime })}_{{\xi }^{\prime }}$$

are morphisms of soft-topological groups, and

$$\begin{array}{ccc}{(G,\zeta )}_{\xi }\stackrel{(\phi ,\varrho )}{\to }& {(G^{\prime },{\zeta }^{\prime })}_{{\xi }^{\prime }}\underset{({\mathbb{I}}_{\xi },{\varrho }_2)}{\overset{({\mathbb{I}}_{\xi },{\varrho }_1)}{{\displaystyle \overrightarrow{\to }}}}& {(H/K,{\zeta }^{″})}_{{\xi }^{\prime }}.\end{array}$$

If $({\mathbb{I}}_{\xi },{\varrho }_1)\circ (\phi ,\varrho )=({\mathbb{I}}_{\xi },{\varrho }_2)\circ (\phi ,\varrho )$, then

$$({\mathbb{I}}_{\xi }\circ \phi ,{\varrho }_1\circ \varrho )=({\mathbb{I}}_{\xi }\circ \phi ,{\varrho }_2\circ \varrho ),$$

and thus

$${\varrho }_1\circ \varrho ={\varrho }_2\circ \varrho .$$

(1)

Moreover, since

$$({\mathbb{I}}_{\xi },{\varrho }_1)\circ (\phi ,\varrho )=({\mathbb{I}}_{\xi },{\varrho }_2)\circ (\phi ,\varrho )$$

and $(\phi ,\varrho )$ is epic, it follows that

$$({\mathbb{I}}_{\xi },{\varrho }_1)=({\mathbb{I}}_{\xi },{\varrho }_2).$$

and hence

$${\varrho }_1={\varrho }_2.$$

(2)

From (1) and (2), it follows that $\varrho$ is a group homomorphism which is an epimorphism. Since every group epimorphism is surjective, $\varrho$ is surjective. □

Proposition 17.

Let ${(G_1,{\zeta }_1)}_{{\xi }_1}$ and ${(G_2,{\zeta }_2)}_{{\xi }_2}$ be soft-topological groups. Let ${\zeta }_{\times }$ be the soft product topology induced from the collection $\{W_1\tilde{\times }{W}_2\mid W_i\in {\zeta }_i,i=1,2\}.$ Then

$${(G_1\times G_2,{\zeta }_{\times })}_{{\xi }_1\times {\xi }_2}$$

is a soft-topological group where $G_1\times G_2$ is the direct product group, and ${\xi }_1\times {\xi }_2$ is the Cartesian product of the parameter sets.

Proof. 

Since ${(G_1,{\zeta }_1)}_{{\xi }_1}$ is a soft-topological group, for any $a_1,b_1\in G_1$ and any soft-open set $W_1\in {\zeta }_1$ such that $a_1{b}_1^{-1}\tilde{\in }{W}_1$, there are soft-open sets $U_1,V_1\in {\zeta }_1$ such that

$$a_1\tilde{\in }{U}_1,b_1\tilde{\in }{V}_1,\mathrm{and}{U}_1{\left(V_1\right)}^{-1}\widetilde{⊑}{W}_1.$$

Similarly, since ${(G_2,{\zeta }_2)}_{{\xi }_2}$ is a soft-topological group, for any $a_2,b_2\in G_2$ and any soft-open set $W_2\in {\zeta }_2$ such that $a_2{b}_2^{-1}\tilde{\in }{W}_2$, there exist soft-open sets $U_2,V_2\in {\zeta }_2$ such that

$$a_2\tilde{\in }{U}_2,b_2\tilde{\in }{V}_2,\mathrm{and}{U}_2{\left(V_2\right)}^{-1}\widetilde{⊑}{W}_2.$$

$(a_1,a_2),(b_1,b_2)\in G_1\times G_2$. Then $(a_1{b}_1^{-1},a_2{b}_2^{-1})\in G_1\times G_2$. Moreover, since $a_1\tilde{\in }{U}_1,b_1\tilde{\in }{V}_1,a_2\tilde{\in }{U}_2,b_2\tilde{\in }{V}_2,$ it follows that

$$\left(U_1{\left(V_1\right)}^{-1}\right)\tilde{\times }\left(U_2{\left(V_2\right)}^{-1}\right)\widetilde{⊑}{W}_1\tilde{\times }{W}_2.$$

This shows that

$${(G_1\times G_2,{\zeta }_{\times })}_{{\xi }_1\times {\xi }_2}$$

is a soft-topological group. □

The soft-topological group ${(G_1\times G_2,{\zeta }_{\times })}_{{\xi }_1\times {\xi }_2}$ is called the soft product topological group.

Theorem 12.

Let ${(G_1,{\zeta }_1)}_{{\xi }_1}$ and ${(G_2,{\zeta }_2)}_{{\xi }_2}$ be two soft-topological groups, and let ${(G_1\times G_2,{\zeta }_{\times })}_{{\xi }_1\times {\xi }_2}$ be the soft product topological group. Define projection maps:

$$P_{{\xi }_1}:{\xi }_1\times {\xi }_2\to {\xi }_1,q_{G_1}:G_1\times G_2\to G_1,$$

$$P_{{\xi }_2}:{\xi }_1\times {\xi }_2\to {\xi }_2,q_{G_2}:G_1\times G_2\to G_2.$$

Then the pairs

$$(P_{{\xi }_1},q_{G_1}):{(G_1\times G_2,{\zeta }_{\times })}_{{\xi }_1\times {\xi }_2}\to {(G_1,{\zeta }_1)}_{{\xi }_1}$$

and

$$(P_{{\xi }_2},q_{G_2}):{(G_1\times G_2,{\zeta }_{\times })}_{{\xi }_1\times {\xi }_2}\to {(G_2,{\zeta }_2)}_{{\xi }_2}$$

are morphisms of soft-topological groups.

Proof. 

It is known that the maps $q_{G_1}:G_1\times G_2\to G_1\mathrm{and}{q}_{G_2}:G_1\times G_2\to G_2$ are group homomorphisms. Let us show that the morphism

$$(P_{{\xi }_1},q_{G_1}):{(G_1\times G_2,{\zeta }_{\times })}_{{\xi }_1\times {\xi }_2}\to {(G_1,{\zeta }_1)}_{{\xi }_1}$$

is soft-continuous.

Let $(x,y)\in G_1\times G_2$. Since $q_{G_1}(x,y)=x,$ for every soft neighborhood $U_1$ of $q_{G_1}(x,y)=x$, we have

$$x=q_{G_1}(x,y)\tilde{\in }(P_{{\xi }_1},q_{G_1})\left(U_1\tilde{\times }{V}_1\right)=U_1\widetilde{⊑}{U}_1.$$

So, $(P_{{\xi }_1},q_{G_1})$ is soft-continuous. Similarly, one can show that

$$(P_{{\xi }_2},q_{G_2}):{(G_1\times G_2,{\zeta }_{\times })}_{{\xi }_1\times {\xi }_2}\to {(G_2,{\zeta }_2)}_{{\xi }_2}$$

is also soft-continuous. □

Theorem 13.

Let ${(G_1,{\zeta }_1)}_{{\xi }_1}$ and ${(G_2,{\zeta }_2)}_{{\xi }_2}$ be any two objects in $\mathbf{STGrp}$. Then, the product of ${(G_1,{\zeta }_1)}_{{\xi }_1}$ and ${(G_2,{\zeta }_2)}_{{\xi }_2}$ in $\mathbf{STGrp}$ is

$$({(G_1\times G_2,{\zeta }_{\times })}_{{\xi }_1\times {\xi }_2},(P_{{\xi }_i},q_{G_i})),$$

for $i=1,2$.

Proof. 

Let ${(H,\sigma )}_{\xi }$ be an object in $\mathbf{STGrp}$, and let $({\phi }_1,{\varrho }_1):{(H,\sigma )}_{\xi }\to {(G_1,{\zeta }_1)}_{{\xi }_1}$ and $({\phi }_2,{\varrho }_2):{(H,\sigma )}_{\xi }\to {(G_2,{\zeta }_2)}_{{\xi }_2}$ be morphisms of soft-topological groups. Consider the maps $({\phi }_1,{\phi }_2):\xi \to {\xi }_1\times {\xi }_2,$ defined by $({\phi }_1,{\phi }_2)\left(e\right)=({\phi }_1\left(e\right),{\phi }_2\left(e\right))$, which is a function, and $({\varrho }_1,{\varrho }_2):H\to G_1\times G_2,$ defined by $({\varrho }_1,{\varrho }_2)\left(h\right)=({\varrho }_1\left(h\right),{\varrho }_2\left(h\right))$, which is a group homomorphism. Now we can define the soft mapping

$$(\vartheta ,\chi )=\left(({\phi }_1,{\phi }_2),({\varrho }_1,{\varrho }_2)\right):{(H,\sigma )}_{\xi }\to {(G_1\times G_2,{\zeta }_{\times })}_{{\xi }_1\times {\xi }_2}$$

by using the maps $({\phi }_1,{\phi }_2)$ and $({\varrho }_1,{\varrho }_2)$. Since $({\phi }_1,{\varrho }_1)$ and $({\phi }_2,{\varrho }_2)$ are soft-continuous, for every $h\in H$ and for every soft neighborhood $W_1$ of ${\varrho }_1\left(h\right)$, there is a soft neighborhood $U_1$ such that

$${\varrho }_1\left(h\right)\tilde{\in }({\phi }_1,{\varrho }_1)\left(U_1\right)\widetilde{⊑}{W}_1.$$

Similarly, for every $h\in H$ and every soft neighborhood $W_2$ of ${\varrho }_2\left(h\right)$, there is a soft neighborhood $U_2$ such that

$${\varrho }_2\left(h\right)\tilde{\in }({\phi }_2,{\varrho }_2)\left(U_2\right)\widetilde{⊑}{W}_2.$$

Hence, for every $h\in H$ and every soft neighborhood $\left(U_1\tilde{\times }{U}_2\right)$ of $\chi \left(h\right)=({\varrho }_1,{\varrho }_2)\left(h\right)=({\varrho }_1\left(h\right),{\varrho }_2\left(h\right)),$ we have

$$({\varrho }_1,{\varrho }_2)\left(h\right)\tilde{\in }(({\phi }_1,{\varrho }_1),({\phi }_2,{\varrho }_2))\left(U_1\tilde{\times }{U}_2\right)\widetilde{⊑}{W}_1\tilde{\times }{W}_2.$$

Therefore, the map $(\vartheta ,\chi )$ is soft-continuous, which shows the existence of the soft-topological group morphism $(\vartheta ,\chi )$.

Let $(\gamma ,\theta ):{(H,\sigma )}_{\xi }\to {(G_1\times G_2,{\zeta }_{\times })}_{{\xi }_1\times {\xi }_2}$ be another soft-topological group morphism such that

$$(P_{{\xi }_1},q_{G_1})\circ (\gamma ,\theta )=({\phi }_1,{\varrho }_1)\mathrm{and}(P_{{\xi }_2},q_{G_2})\circ (\gamma ,\theta )=({\phi }_2,{\varrho }_2).$$

Since $P_{{\xi }_1}\circ \gamma =P_{{\xi }_1}\circ \vartheta ,$ it follows that $\gamma =\vartheta$, and since $q_{G_1}\circ \theta =q_{G_1}\circ \chi ,$ it follows that $\theta =\chi$. Therefore, $(\vartheta ,\chi )$ is unique.

□

Theorem 14.

Let $\left\{1_G\right\}$ be the trivial group and $\left\{e\right\}$ be a singleton set. Consider the soft indiscrete topology ζ on $\left\{1_G\right\}$. Then the triple ${(\left\{1_G\right\},\zeta )}_{\left\{e\right\}}$ is the terminal object in the category of soft-topological groups STGrp.

Proof. 

Let ${(H,\sigma )}_{\xi }$ be an arbitrary soft-topological group. And let $\varrho :H\to \left\{1_G\right\}$ be the trivial group homomorphism and $\phi :\xi \to \left\{e_1\right\}$ be the constant function. Then there exists a unique morphism

$${(H,\sigma )}_{\xi }\stackrel{(\phi ,\varrho )}{\to }{(\left\{1_G\right\},\zeta )}_{\left\{e\right\}}$$

in the category STGrp. Here, since $\zeta$ is the soft indiscrete topology, the morphism $(\phi ,\varrho )$ is soft-continuous. Therefore, ${(\left\{1_G\right\},\zeta )}_{\left\{e\right\}}$ is the terminal object in the category STGrp. □

Corollary 1.

The category STGrp is a symmetric monoidal category.

7. Conclusions

This study establishes the structural and continuity foundations necessary for defining the soft fundamental group and the soft covering concepts corresponding to soft-topological groups. Although these notions are not explicitly defined here, the framework developed provides a theoretical basis for extending classical homotopy and covering space theory to the setting of soft-topological groups. The next phase of this research will focus on the formal definition of the soft fundamental group and soft covering mappings, as well as a detailed investigation of their soft-homotopic properties within soft-topological groups.