Connectedness of Soft-Ideal Topological Spaces

Abstract

Despite its apparent simplicity, the idea of connectedness has significant effects on topology and its applications. An essential part of the intermediate-value theorem is the idea of connectedness. In many applications, such as population modeling, robotics motion planning, and geographic information systems, connectedness is significant, and it is a critical factor in differentiating between various topological spaces. This study uses soft open sets and the concept of soft ideals as a new class of soft sets to present and explore the ideas of soft connected spaces and strongly soft connected spaces with soft ideals. Also, under certain assumptions regarding the subsequent concepts—soft-ideal connectedness and stronglysoft-ideal connectedness in soft-ideal topological spaces—we characterize this new class of sets by employing soft open sets and soft ideals to examine its fundamental features. Furthermore, we look at a symmetry between our new notions and other existing ones, and this study examines the relationships between these concepts.

1. Introduction

Scientific processes, as opposed to tried-and-true methods, must be utilized to answer most empirical issues in technology domains—such as engineering, computer science, and social sciences—that deal with uncertainty and unreliability. In [1], Molodtsov outlined a novel mathematical method known as soft set theory to solve some issues. Molodtsov effectively utilized the soft set theory in [1,2] in several domains, including probability, measurement theory, Riemann integration, game theory, operation research, and smoothness of functions. Considerable advancements in theory and technology have resulted from an exponential growth in the study of and applications of this soft set (see [3,4,5,6,7,8,9]). Next, a well-focused investigation of an abstract soft set operator theory with applications to decision-making problems was conducted by Maji et al. [10]. The work of Shabir and Naz focused on the theoretical studies that introduced the idea of soft topological spaces [11]. These spaces are defined over an initial universe that has a predetermined set of soft topological space features. Kandil et al. [12] presented the concept of soft ideals initially. In addition, they established the concept of the soft local function. Several symmetric concepts have been proposed for identifying new soft topologies from old ones, commonly known as soft topological spaces with soft ideals $(X_{\epsilon },\tau ,{\mathbb{I}}_{\epsilon })$. Kandil et al. [13] explored some applications of soft sets in several sectors. Lin [14] developed the concept of connectedness in soft topological spaces. In [13], the authors explored the concept of (∗-soft separated, ∗-soft connected, and ${\ast }_s$-soft connected) sets in soft topological spaces with soft ideals. Al-shami et al. [15] examined notions of connectedness via the class of soft somewhat-open sets. Additionally, Al-shami et al. [16] introduced new types of symmetrical soft connected spaces and infra-soft locally connected spaces and components. Local connection and connectedness are some essential concepts in soft topology. In 1983, Atanassov introduced intuitionistic fuzzy sets [17]. Basic results on intuitionistic fuzzy sets are reported in [18,19]. Yager [20,21] suggested a new form of fuzzy sets called Pythagorean fuzzy sets. Yager [22] introduced the q-rung orthopair fuzzy set for dual universes. Saber Y. et al. [23] discussed the connectedness and stratification of single-valued neutral topological spaces. Li et al. [24] introduced the notion of q-rung picture fuzzy sets. Ajmal et al. [25] investigated the idea of connectedness and local connectedness in fuzzy topological spaces. Refs. [26,27] provide further information about soft connectedness. Research in soft topology and related topics remains active (see, for example, [28,29]), with opportunities for significant contributions. The principal characteristics of these notions are defined and examined by the researchers of [6,15]. Thus, our objectives in this article are to introduce the concept of soft-ideal connected spaces and soft-strongly-ideal connected spaces using ideals, and to extend some significant results on strong connectedness. Also, we aim to ascertain under which conditions soft connectedness, soft-ideal connectedness, and soft -strongly-ideal connectedness are equivalent. Finally, we provide some more examples to illustrate some of the characteristics.

This manuscript is a continuation of the previous works based on symmetric soft connected spaces. The structure of this paper is as follows: Section 2 reviews the main definitions and findings of soft theory. After defining “soft connected” and “soft ideal connected spaces”, some of their many extensions are shown in Section 3. With the aid of strong examples, we examine their salient features and make clear the relationships between them. We present the idea of strongly-soft -${\mathbb{I}}_{\epsilon }$-connected spaces in Section 4 and explore deeply its main characteristics. We show that this concept is comparable to that of “soft connected”. Lastly, we provide some insights and pave the way for further research in Section 5.

In this study, topological space and soft topological space, respectively, shall be denoted by the terms used in publications: TS and STS. This essay aims to introduce and explore the notion of soft connected spaces and strongly soft connected spaces with soft ideals.

2. Preliminary

In order to keep this publication self-contained, we will discuss the ideas and conclusions from previous research that are necessary to understand the results that we have found here.

Definition 1 

([1]). Let ε be a set of parameters and X be an initial universe. Let B be a non-null subset of parameters ε and $2^X$ be the power set of X. A pair $(\lambda ,B)$ indicated by ${\lambda }_B$ is a soft set over $X_{\epsilon }$, such that λ is a mapping provided by $\lambda :B\to 2^X$. Alternatively stated, a soft set over $X_{\epsilon }$ is a parameterized collection of subsets of the universe $X_{\epsilon }$. For a certain $e\in \epsilon$, $\lambda \left(e\right)$ may be thought of as the set of e-approximate elements of the soft set $(\lambda ,\epsilon )={\lambda }_{\epsilon }$; also, if $e\notin \epsilon$, then $\lambda \left(e\right)=\varphi$, i.e., ${\lambda }_{\epsilon }=\{\lambda \left(e\right):e\in \epsilon , \lambda :\epsilon \to 2^X\}$. The collection of all soft sets is represented by $SS{\left(X\right)}_{\epsilon }$.

Definition 2 

([10]). Assume $A_{\epsilon }, B_{\epsilon }\in SS{\left(X\right)}_{\epsilon }$. We call $A_{\epsilon }$:

(a) 

A soft subset of $B_{\epsilon }$, represented by $A_{\epsilon }⊑B_{\epsilon }$, if $A\left(e\right)\subseteq B\left(e\right)$, for all $e\in \epsilon$.

(b) 

An absolute, represented by $X_{\epsilon }$, if $A\left(e\right)=X$, for all $e\in \epsilon$.

(c) 

A null, represented by ${\varphi }_{\epsilon }$, if $A\left(e\right)=\varphi$, for all $e\in \epsilon .$

In this instance, $A_{\epsilon }$ is a soft subset of $B_{\epsilon }$ and $B_{\epsilon }$ is a soft superset of $A_{\epsilon }$, $A_{\epsilon }⊑B_{\epsilon }$.

Definition 3 

([30]).

(a) 

Asume Δ to be an arbitrary index set and $\Omega =\{{\left({\lambda }_{\alpha }\right)}_{\epsilon }:\alpha \in \Delta \}$ to be a subfamily of $SS{\left(X\right)}_{\epsilon }$. Then:

(i) 

The union of all ${\left({\lambda }_{\alpha }\right)}_{\epsilon }$ is the soft set $A_{\epsilon }$, where $A\left(e\right)={\cup }_{\alpha \in \Delta }{\left({\lambda }_{\alpha }\right)}_{\epsilon }\left(e\right)$ for all $e\in \epsilon$. We compose ${\bigsqcup }_{\alpha \in \Delta }{\left({\lambda }_{\alpha }\right)}_{\epsilon }=A_{\epsilon }$.

(ii) 

The intersection of all ${\left({\lambda }_{\alpha }\right)}_{\epsilon }$ is the soft set $N_{\epsilon }$, where $N\left(e\right)={\cap }_{\alpha \in \Delta }{\left({\lambda }_{\alpha }\right)}_{\epsilon }\left(e\right)$ for all $e\in \epsilon$. We compose ${\sqcap }_{\alpha \in \Delta }{\left({\lambda }_{\alpha }\right)}_{\epsilon }=N_{\epsilon }$.

(b) 

A soft set $A_{\epsilon }$ in $(X_{\epsilon },\tau )$ is a soft neighborhood of the soft point $x_{\epsilon }\in X_{\epsilon }$ if there exists a set $B_{\epsilon }\in \tau$ such that $x_{\epsilon }\in B_{\epsilon }⊑A_{\epsilon }$.

(c) 

A soft set ${\lambda }_{\epsilon }\in SS{\left(X\right)}_{\epsilon }$ is a soft point in $X_{\epsilon }$ if there exist $x\in X$ and $e\in \epsilon$ such that $\lambda \left(e\right)=\left\{x\right\}$ and $\lambda \left(e^c\right)=\varphi$ for all $e^c\in \epsilon -\left\{e\right\}$. This soft point ${\lambda }_{\epsilon }$ is denoted by $x_{\epsilon }$.

Definition 4 

([11]). Let $(X_{\epsilon },\tau )$ be a STS and ${\lambda }_{\epsilon }\in SS{\left(X\right)}_{\epsilon }$.

(a) 

A complement of a soft set ${\lambda }_{\epsilon }$, indicated by ${\lambda }_{\epsilon }^c$, is described as follows: ${\lambda }^c:\epsilon \to 2^X$ is a mapping given by ${\lambda }^c\left(e\right)=X_{\epsilon }\left(e\right)-\lambda \left(e\right)$, for all $e\in \epsilon$, and ${\lambda }^c$ is called a soft complement function of ${\lambda }_{\epsilon }$.

(b) 

A difference of two soft sets ${\lambda }_{\epsilon }$ and $A_{\epsilon }$ over the common universe $X_{\epsilon }$, denoted by ${\lambda }_{\epsilon }-A_{\epsilon }$, is the soft set $B_{\epsilon }$, for all $e\in \epsilon$, such that $B\left(e\right)=\lambda \left(e\right)-A\left(e\right)$.

(c) 

The soft interior of ${\lambda }_{\epsilon }$ is $Int\left({\lambda }_{\epsilon }\right)=\bigsqcup \{A_{\epsilon }:A_{\epsilon }\in \tau$ and $A_{\epsilon }⊑{\lambda }_{\epsilon }\}$.

(d) 

The soft closure of ${\lambda }_{\epsilon }$ is $cl\left({\lambda }_{\epsilon }\right)=\sqcap \{A_{\epsilon }:A_{\epsilon }\in {\tau }^c$ and ${\lambda }_{\epsilon }⊑A_{\epsilon }\}$.

(e) 

Let ${\lambda }_{\epsilon }$ be a soft set over $X_{\epsilon }$ and $x_{\epsilon }\in X_{\epsilon }$. We say that $x_{\epsilon }\in {\lambda }_{\epsilon }$ indicates that $x_{\epsilon }$ belongs to the soft set ${\lambda }_{\epsilon }$ whenever $x_{\epsilon }\left(e\right)\in \lambda \left(e\right)$, for all $e\in \epsilon$.

More details about soft set theory and its applications in various mathematical structures may be found in [8,31,32].

Definition 5 

([12]). Let ${\mathbb{I}}_{\epsilon }$ be a non-null collection of soft sets over a universe $X_{\epsilon }$. Then, ${\mathbb{I}}_{\epsilon }\subseteq SS{\left(X\right)}_{\epsilon }$ is called a soft ideal on $X_{\epsilon }$ if:

1. 

$A_{\epsilon }\in {\mathbb{I}}_{\epsilon }$ and $B_{\epsilon }\in {\mathbb{I}}_{\epsilon }$, then $A_{\epsilon }\bigsqcup B_{\epsilon }\in {\mathbb{I}}_{\epsilon }$.

2. 

$A_{\epsilon }\in {\mathbb{I}}_{\epsilon }$ and $B_{\epsilon }⊑A_{\epsilon }$, then $B_{\epsilon }\in {\mathbb{I}}_{\epsilon }$.

Definition 6 

([12]). Let $(X_{\epsilon },\tau )$ be an STS and ${\mathbb{I}}_{\epsilon }$ be a soft ideal over $X_{\epsilon }$ with the same set of parameters ε. Then, ${\overline{D_{\epsilon }}}^{\ast }({\mathbb{I}}_{\epsilon },\tau )\left(\mathit{or}{\overline{D}}_{\epsilon }^{\ast }\right)=\bigsqcup \{x_{\epsilon }\in X_{\epsilon }:O_{x_{\epsilon }}\sqcap D_{\epsilon }\notin {\mathbb{I}}_{\epsilon }\mathit{for}\mathit{all}\mathit{soft}\mathit{open}\mathit{set}{O}_{x_{\epsilon }}\}$ is the soft local function of $D_{\epsilon }$ with respect to ${\mathbb{I}}_{\epsilon }$ and soft topology τ, where $O_{x_{\epsilon }}$ is a soft open set containing $x_{\epsilon }$.

Theorem 1 

([33]). Let $(X_{\epsilon },\tau )$ be an STS and ${\mathbb{I}}_{\epsilon }$ be a soft ideal over X with the same set of parameters ε. Then, the following statements are equivalent:

1. 

$\tau \sqcap {\mathbb{I}}_{\epsilon }={\varphi }_{\epsilon }$;

2. 

$A_{\epsilon }\in {\mathbb{I}}_{\epsilon }$ implies $Int\left(A_{\epsilon }\right)={\varphi }_{\epsilon }$;

3. 

For all $B_{\epsilon }\in \tau$, $B_{\epsilon }⊑{\overline{B}}_{\epsilon }^{\ast }$;

4. 

$X_{\epsilon }={\overline{X}}_{\epsilon }^{\ast }$.

Definition 7 

([1]). An STS $(X_{\epsilon },\tau )$ is called soft connected if X cannot be articulated as the soft union of two soft separated sets in $(X_{\epsilon },\tau )$. If it is not soft connected, $(X_{\epsilon },\tau )$ is said to be soft disconnected.

3. Soft-Ideal Connected Spaces

In this section, we introduce and explain the concept of soft-ideal connected spaces. Additionally, we demonstrate that, subject to specific limitations, the product of soft-ideal connected spaces is also a soft-ideal connected space.

Definition 8. 

Let $(X_{\epsilon },\tau )$ be am STS with a soft ideal ${\mathbb{I}}_{\epsilon }$ on X. A soft subset $Y_{\epsilon }$ of $X_{\epsilon }$ is soft ${\mathbb{I}}_{\epsilon }$-connected if $Y_{\epsilon }$ cannot be expressed as a union of two non-soft-ideal sets $A_{\epsilon }$ and $B_{\epsilon }$ such that $Y_{\epsilon }\sqcap {\overline{A}}_{\epsilon }\sqcap B_{\epsilon }={\varphi }_{\epsilon }=A_{\epsilon }\sqcap {\overline{B}}_{\epsilon }\sqcap Y_{\epsilon }$. A non-soft-ideal set $A_{\epsilon }$ in $X_{\epsilon }$ is a subset of $X_{\epsilon }$, such that $A_{\epsilon }\notin {\mathbb{I}}_{\epsilon }$, and ${\overline{A}}_{\epsilon }$ is a soft closure of $A_{\epsilon }$ in τ.

Any soft connected set is soft ${\mathbb{I}}_{\epsilon }$-connected, but the opposite is not always true, as shown below.

Example 1. 

Let $(X,{\tau }_u)=(\mathbb{R},{\tau }_u)$ denote the reals with usual topology and $Y=[0,2]\bigsqcup \{3,4,5\}\subseteq X$. Let ${\mathbb{I}}_{\epsilon }$ denote the ideal of all finite subsets of $X_{\epsilon }$ and $E=\left\{e\right\}$. Let $\tau =\{F\in SS(X,E):F\left(e\right)\in {\tau }_u$, for all $e\in E\}$, be a soft topological space. Then, $Y_{\epsilon }$ is soft ${\mathbb{I}}_{\epsilon }$-connected but not soft connected.

Example 2. 

Let $X=\{h_1,h_2,h_3\},$ε={${\epsilon }_1$, ${\epsilon }_2$}, and $\tau =\{X_{\epsilon },{\varphi }_{\epsilon },A_{\epsilon },B_{\epsilon }\}$, where $A_{\epsilon }$ and $B_{\epsilon }$ are soft sets over X defined as follows: $A\left({\epsilon }_1\right)=\left\{h_2\right\}$, $A\left({\epsilon }_2\right)=\{h_1,h_3\}$, and $B\left({\epsilon }_1\right)=\{h_1,h_3\}$, $B\left({\epsilon }_2\right)=\left\{h_2\right\}$. Then, τ is a soft topology on X. Let ${\mathbb{I}}_{\epsilon }=\{{\varphi }_{\epsilon },I_{\epsilon },J_{\epsilon },K_{\epsilon },L_{\epsilon }\}$, where $I_{\epsilon },J_{\epsilon },K_{\epsilon },L_{\epsilon }$ are soft sets over X defined by $I\left({\epsilon }_1\right)=\left\{h_2\right\}$, $I\left({\epsilon }_2\right)=\left\{h_1\right\}$, $J\left({\epsilon }_1\right)=\{h_1,h_2\}$, $J\left({\epsilon }_2\right)=\left\{h_3\right\}$, $K\left({\epsilon }_1\right)=\left\{h_2\right\}$, $K\left({\epsilon }_2\right)=\{h_1,h_3\}$, and $L\left({\epsilon }_1\right)=\left\{h_1\right\}$, $L\left({\epsilon }_2\right)=\left\{h_3\right\}$. Then, ${\mathbb{I}}_{\epsilon }$ is a soft ideal on X. Hence, $X_{\epsilon }$ is not soft connected but it is soft ${\mathbb{I}}_{\epsilon }$-connected.

Next, we characterize the soft-ideal connected spaces.

Theorem 2. 

Let $(X_{\epsilon },\tau )$ be an STS with a soft ideal ${\mathbb{I}}_{\epsilon }$ on X. Then, the subsequent statements are equivalent:

1. 

$X_{\epsilon }$ is soft ${\mathbb{I}}_{\epsilon }$-connected.

2. 

$X_{\epsilon }$ cannot be expressed as a union of two disjoint non-soft-ideal soft open sets.

3. 

$X_{\epsilon }$ cannot be expressed as a union of two disjoint non-soft-ideal soft closed sets.

Proof. 

(1) ⇒ (2): Suppose (2) is not true. Then, $X_{\epsilon }=A_{\epsilon }\bigsqcup B_{\epsilon }$, for some soft subset $A_{\epsilon },B_{\epsilon }\notin {\mathbb{I}}_{\epsilon }$ such that $A_{\epsilon },B_{\epsilon }$ are disjoint soft open sets. Then, $A_{\epsilon }={\overline{A}}_{\epsilon }$ and $B_{\epsilon }={\overline{B}}_{\epsilon }$, implying ${\overline{A}}_{\epsilon }\sqcap B_{\epsilon }={\varphi }_{\epsilon }=A_{\epsilon }\sqcap {\overline{B}}_{\epsilon }$. This contradicts (1). Therefore, $X_{\epsilon }$ is not characterized as a union of two disjoint non-soft-ideal soft open sets.

(2) ⇒ (3): Suppose (3) is not true. Then, $X_{\epsilon }=A_{\epsilon }\bigsqcup B_{\epsilon }$ for some soft subsets $A_{\epsilon },B_{\epsilon }\notin {\mathbb{I}}_{\epsilon }$ such that $A_{\epsilon },B_{\epsilon }$ are disjoint soft closed sets. Then, $X_{\epsilon }=A_{\epsilon }\cup B_{\epsilon }$, where $A_{\epsilon },B_{\epsilon }\notin {\mathbb{I}}_{\epsilon }$, $A_{\epsilon }\sqcap B_{\epsilon }={\varphi }_{\epsilon }$, and $A_{\epsilon }=X_{\epsilon }-B_{\epsilon }$, $B_{\epsilon }=X_{\epsilon }-A_{\epsilon }$ are soft open sets, which contradicts (2). Therefore, $X_{\epsilon }$ not characterized as a union of two disjoint non-soft-ideal soft closed sets.

(3) ⇒ (1): Suppose $X_{\epsilon }$ is not soft ${\mathbb{I}}_{\epsilon }$-connected. Then, $X_{\epsilon }=A_{\epsilon }\cup B_{\epsilon }$, for some soft subsets $A_{\epsilon },B_{\epsilon }\notin {\mathbb{I}}_{\epsilon }$ such that ${\overline{A}}_{\epsilon }\sqcap B_{\epsilon }={\varphi }_{\epsilon }=A_{\epsilon }\sqcap {\overline{B}}_{\epsilon }$. This implies that ${\overline{A}}_{\epsilon }⊑A_{\epsilon }$ and ${\overline{B}}_{\epsilon }⊑B_{\epsilon }$. Hence, $X_{\epsilon }=A_{\epsilon }\bigsqcup B_{\epsilon }$, where $A_{\epsilon },B_{\epsilon }\notin {\mathbb{I}}_{\epsilon }$, and $A_{\epsilon },B_{\epsilon }$ are disjoint soft closed sets; which is a contradiction to (3). So, $X_{\epsilon }$ is soft ${\mathbb{I}}_{\epsilon }$-connected. □

It is well known that the union of two connected sets is connected if $A_{\epsilon }\sqcap B_{\epsilon }\ne {\varphi }_{\epsilon }$. This result can be generalized as follows.

Theorem 3. 

Let $(X_{\epsilon },\tau )$ be an STS with a soft ideal ${\mathbb{I}}_{\epsilon }$ on X and $A_{\epsilon }$, $B_{\epsilon }$ be two soft ${\mathbb{I}}_{\epsilon }$-connected sets with $A_{\epsilon }\sqcap B_{\epsilon }\notin {\mathbb{I}}_{\epsilon }$. Then, $A_{\epsilon }\bigsqcup B_{\epsilon }$ is soft ${\mathbb{I}}_{\epsilon }$-connected.

Proof. 

Suppose $A_{\epsilon }\bigsqcup B_{\epsilon }$ is not soft ${\mathbb{I}}_{\epsilon }$-connected. Then, $A_{\epsilon }\bigsqcup B_{\epsilon }=C_{\epsilon }\bigsqcup D_{\epsilon }$, where $C_{\epsilon },D_{\epsilon }\notin {\mathbb{I}}_{\epsilon }$ and $(A_{\epsilon }\bigsqcup B_{\epsilon })\sqcap {\overline{C}}_{\epsilon }\sqcap D_{\epsilon }={\varphi }_{\epsilon }$ and $(A_{\epsilon }\bigsqcup B_{\epsilon })\sqcap C_{\epsilon }\sqcap {\overline{D}}_{\epsilon }={\varphi }_{\epsilon }$. We have $(A_{\epsilon }\sqcap B_{\epsilon })=(C_{\epsilon }\sqcap A_{\epsilon }\sqcap B_{\epsilon })\bigsqcup (D_{\epsilon }\sqcap A_{\epsilon }\sqcap B_{\epsilon })\notin {\mathbb{I}}_{\epsilon }$. Then, either $C_{\epsilon }\sqcap A_{\epsilon }\sqcap B_{\epsilon }\notin {\mathbb{I}}_{\epsilon }$ or $D_{\epsilon }\sqcap A_{\epsilon }\sqcap B_{\epsilon }\notin {\mathbb{I}}_{\epsilon }$. Suppose $C_{\epsilon }\sqcap A_{\epsilon }\sqcap B_{\epsilon }\notin {\mathbb{I}}_{\epsilon }$, then $C_{\epsilon }\sqcap A_{\epsilon }\notin {\mathbb{I}}_{\epsilon }$ and $C_{\epsilon }\sqcap B_{\epsilon }\notin {\mathbb{I}}_{\epsilon }$. Since $A_{\epsilon }=(C_{\epsilon }\sqcap A_{\epsilon })\bigsqcup (D_{\epsilon }\sqcap A_{\epsilon })$ is a soft ${\mathbb{I}}_{\epsilon }$-connected, either $C_{\epsilon }\sqcap A_{\epsilon }\in {\mathbb{I}}_{\epsilon }$ or $D_{\epsilon }\sqcap A_{\epsilon }\in {\mathbb{I}}_{\epsilon }$. As $C_{\epsilon }\sqcap A_{\epsilon }\notin {\mathbb{I}}_{\epsilon }$, we have $D_{\epsilon }\sqcap A_{\epsilon }\in {\mathbb{I}}_{\epsilon }$. Similarly, we have $D_{\epsilon }\sqcap B_{\epsilon }\in {\mathbb{I}}_{\epsilon }$. So, $D_{\epsilon }=(D_{\epsilon }\sqcap A_{\epsilon })\bigsqcup (D_{\epsilon }\sqcap B_{\epsilon })\in {\mathbb{I}}_{\epsilon }$ and hence $D\in {\mathbb{I}}_{\epsilon }$, which is a contradiction. Hence, $A_{\epsilon }\bigsqcup B_{\epsilon }$ is soft ${\mathbb{I}}_{\epsilon }$-connected. □

Theorem 4. 

Let $(X_{\epsilon },\tau )$ be an STS with soft ideal ${\mathbb{I}}_{\epsilon }$ on $X_{\epsilon }$ with $\tau \sqcap {\mathbb{I}}_{\epsilon }={\varphi }_{\epsilon }$. Then, $X_{\epsilon }$ is soft ${\mathbb{I}}_{\epsilon }$-connected if $X_{\epsilon }$ is soft connected.

Proof. 

It is enough to prove that if $X_{\epsilon }$ is soft ${\mathbb{I}}_{\epsilon }$-connected, then it is soft connected. Suppose $X_{\epsilon }$ is not soft connected. Then, $X_{\epsilon }=A_{\epsilon }\bigsqcup B_{\epsilon }$, where $A_{\epsilon }$, $B_{\epsilon }$ are non-null soft open sets and ${\overline{A}}_{\epsilon }\sqcap B_{\epsilon }={\varphi }_{\epsilon }=A_{\epsilon }\sqcap {\overline{B}}_{\epsilon }$. Since $\tau \sqcap {\mathbb{I}}_{\epsilon }={\varphi }_{\epsilon }$, we have $A_{\epsilon },B_{\epsilon }\notin {\mathbb{I}}_{\epsilon }$. So, $X_{\epsilon }$ is not soft ${\mathbb{I}}_{\epsilon }$-connected, which produces a contradiction. Consequently, $X_{\epsilon }$ is soft connected. □

Theorem 5. 

Let $(X_{\epsilon },\tau )$ be an STS with soft ideal ${\mathbb{I}}_{\epsilon }$ on $X_{\epsilon }$. If $A_{\epsilon }⊑X_{\epsilon }$ is soft ${\mathbb{I}}_{\epsilon }$-connected and $A_{\epsilon }⊑B_{\epsilon }⊑{\overline{A}}_{\epsilon }^{\ast })$ (closure of $A_{\epsilon }$ in ${\tau }^{\ast }$), then $B_{\epsilon }$ is soft ${\mathbb{I}}_{\epsilon }$-connected.

Proof. 

Suppose that $B_{\epsilon }$ is not soft ${\mathbb{I}}_{\epsilon }$-connected. Then, $B_{\epsilon }=C_{\epsilon }\bigsqcup D_{\epsilon }$, where $C_{\epsilon },D_{\epsilon }\notin {\mathbb{I}}_{\epsilon }$ and $B_{\epsilon }\sqcap {\overline{C}}_{\epsilon }\sqcap D_{\epsilon }={\varphi }_{\epsilon }=B_{\epsilon }\sqcap C_{\epsilon }\sqcap {\overline{D}}_{\epsilon }$. Now, we have $A_{\epsilon }=(A_{\epsilon }\sqcap C_{\epsilon })\bigsqcup (A_{\epsilon }\sqcap D_{\epsilon })$. Since $A_{\epsilon }$ is soft ${\mathbb{I}}_{\epsilon }$-connected, either $A_{\epsilon }\sqcap C_{\epsilon }\in {\mathbb{I}}_{\epsilon }$ or $A_{\epsilon }\sqcap D_{\epsilon }\in {\mathbb{I}}_{\epsilon }$. Suppose $A_{\epsilon }\sqcap D_{\epsilon }\in {\mathbb{I}}_{\epsilon }$ and let $x_{\epsilon }\in D_{\epsilon }-A_{\epsilon }$. Then, for every neighborhood $V_{\epsilon }$ of $x_{\epsilon }$, $V_{\epsilon }\sqcap A_{\epsilon }\notin {\mathbb{I}}_{\epsilon }$. As $V_{\epsilon }\sqcap A_{\epsilon }=(V_{\epsilon }\sqcap A_{\epsilon }\sqcap C_{\epsilon })\bigsqcup (V_{\epsilon }\sqcap A_{\epsilon }\sqcap D_{\epsilon })\notin {\mathbb{I}}_{\epsilon }$, we have $V_{\epsilon }\sqcap A_{\epsilon }\sqcap C_{\epsilon }\notin {\mathbb{I}}_{\epsilon }$. In particular, $V_{\epsilon }\sqcap A_{\epsilon }\sqcap C_{\epsilon }\ne {\varphi }_{\epsilon }$. Then, $V_{\epsilon }\sqcap C_{\epsilon }\ne {\varphi }_{\epsilon }$ and hence $x_{\epsilon }\in C_{\epsilon }$. Therefore, $x_{\epsilon }\in D_{\epsilon }-A_{\epsilon }$, implying that $x_{\epsilon }\in C_{\epsilon }$, which is a contradiction to $B_{\epsilon }\sqcap {\overline{C}}_{\epsilon }\sqcap D_{\epsilon }={\varphi }_{\epsilon }$. Hence, $D_{\epsilon }-A_{\epsilon }={\varphi }_{\epsilon }$ and $D_{\epsilon }⊑A_{\epsilon }$. Therefore, $D_{\epsilon }=D_{\epsilon }\sqcap A_{\epsilon }\in {\mathbb{I}}_{\epsilon }$, which is a contradiction. Consequently, $B_{\epsilon }$ is soft ${\mathbb{I}}_{\epsilon }$-connected. □

The above theorem is not true if we replace ∗-closure with closure. We give the following example.

Example 3. 

Let $(X,{\tau }_u)=(\mathbb{R},{\tau }_u)$ be the real line with usual topology. Let $A=[0,1]\cup \{x:x$ be rational, $4<x<5\}$ and let ${\mathbb{I}}_{\epsilon }$ be the ideal of measure zero sets $X_{\epsilon }$ and $E=\left\{\mathbb{Z}\right\}$. Let $\tau =\{F\in SS(X,E):F\left(e\right)\in {\tau }_u$ for all $e\in E\}$ be soft topological. Then, $A_{\epsilon }$ is soft ${\mathbb{I}}_{\epsilon }$-connected, but $\overline{A}=[0,1]\cup [4,5]$ is not soft ${\mathbb{I}}_{\epsilon }$-connected.

Theorem 6. 

Let $(X_{\epsilon },\tau )$ and $(Y_{\epsilon },\sigma )$ be STSs and let ${\mathbb{I}}_{\epsilon }$ be a soft ideal on $X_{\epsilon }$. Let $f_{pu}:(X_{\epsilon },\tau )\to (Y_{\epsilon },\sigma )$ be soft continuous surjective. If $(X_{\epsilon },\tau )$ is soft ${\mathbb{I}}_{\epsilon }$-connected, then $(Y_{\epsilon },\sigma )$ is soft $f_{pu}\left({\mathbb{I}}_{\epsilon }\right)$-connected.

Proof. 

Let $f_{pu}:(X_{\epsilon },\tau )\to (Y_{\epsilon },\sigma )$ be a soft continuous surjective map and $(X_{\epsilon },\tau )$ be soft ${\mathbb{I}}_{\epsilon }$-connected. Assume that $(Y_{\epsilon },\sigma )$ is not soft $f_{pu}\left({\mathbb{I}}_{\epsilon }\right)$-connected; then, $Y_{\epsilon }=B_{\epsilon }\bigsqcup C_{\epsilon }$, for some $B_{\epsilon },C_{\epsilon }\notin f_{pu}\left({\mathbb{I}}_{\epsilon }\right)$, $B_{\epsilon }\sqcap C_{\epsilon }={\varphi }_{\epsilon }$ and $B_{\epsilon },C_{\epsilon }$ are soft open sets. Since $f_{pu}$ is soft continuous, $f_{pu}^{-1}\left(B_{\epsilon }\right)$, $f_{pu}^{-1}\left(C_{\epsilon }\right)$ are soft open and $f_{pu}^{-1}\left(B_{\epsilon }\right)\sqcap f_{pu}^{-1}\left(C_{\epsilon }\right)=f_{pu}^{-1}(B_{\epsilon }\sqcap C_{\epsilon })=f_{pu}^{-1}\left({\varphi }_{\epsilon }\right)={\varphi }_{\epsilon }$. Also, $f_{pu}^{-1}\left(B_{\epsilon }\right),f_{pu}^{-1}\left(C_{\epsilon }\right)\notin {\mathbb{I}}_{\epsilon }$ (if $f_{pu}^{-1}\left(B_{\epsilon }\right)\in {\mathbb{I}}_{\epsilon }$, then $B_{\epsilon }\in f_{pu}\left({\mathbb{I}}_{\epsilon }\right)$). Now, $X_{\epsilon }=f_{pu}^{-1}\left(B_{\epsilon }\right)\bigsqcup f_{pu}^{-1}\left(C_{\epsilon }\right)$, where $f_{pu}^{-1}\left(B_{\epsilon }\right)$, $f_{pu}^{-1}\left(C_{\epsilon }\right)$ are soft open sets such that $f_{pu}^{-1}\left(B_{\epsilon }\right)\sqcap f_{pu}^{-1}\left(C_{\epsilon }\right)={\varphi }_{\epsilon }$ and $f_{pu}^{-1}\left(B_{\epsilon }\right),f_{pu}^{-1}\left(C_{\epsilon }\right)\notin {\mathbb{I}}_{\epsilon }$. Hence, $(X_{\epsilon },\tau )$ is not soft ${\mathbb{I}}_{\epsilon }$-connected, which is a contradiction to our assumption. Thus, $(Y_{\epsilon },\sigma )$ is soft $f_{pu}\left({\mathbb{I}}_{\epsilon }\right)$-connected. □

Theorem 7. 

Let $(X_{\epsilon },\tau )$ be an STS and ${\mathbb{I}}_{\epsilon }$ be a soft ideal on $X_{\epsilon }$ and let $A_{\epsilon },B_{\epsilon }⊑X_{\epsilon }$. If $A_{\epsilon }$ is soft ${\mathbb{I}}_{\epsilon }$-connected and $B_{\epsilon }\in {\mathbb{I}}_{\epsilon }$, then $A_{\epsilon }\bigsqcup B_{\epsilon }$ is soft ${\mathbb{I}}_{\epsilon }$-connected.

Proof. 

If $A_{\epsilon }\bigsqcup B_{\epsilon }$ is not soft ${\mathbb{I}}_{\epsilon }$-connected, then there exist soft open sets $C_{\epsilon }$ and $D_{\epsilon }$ in $X_{\epsilon }$ such that:

• $(A_{\epsilon }\bigsqcup B_{\epsilon })\sqcap {\overline{C}}_{\epsilon }\sqcap D_{\epsilon }={\varphi }_{\epsilon }$ and $(A_{\epsilon }\bigsqcup B_{\epsilon })\sqcap C_{\epsilon }\sqcap {\overline{D}}_{\epsilon }={\varphi }_{\epsilon }$.
• $(A_{\epsilon }\bigsqcup B_{\epsilon })\sqcap C_{\epsilon }\notin {\mathbb{I}}_{\epsilon }$ and $(A_{\epsilon }\bigsqcup B_{\epsilon })\sqcap D_{\epsilon }\notin {\mathbb{I}}_{\epsilon }$.

Since $B_{\epsilon }\in {\mathbb{I}}_{\epsilon }$ and $B_{\epsilon }\sqcap C_{\epsilon }\in {\mathbb{I}}_{\epsilon }$, we have $(A_{\epsilon }\sqcap C_{\epsilon })\notin {\mathbb{I}}_{\epsilon }$. Similarly, we have $(A_{\epsilon }\sqcap D_{\epsilon })\notin {\mathbb{I}}_{\epsilon }$. Now, $A_{\epsilon }=(A_{\epsilon }\sqcap C_{\epsilon })\bigsqcup (A_{\epsilon }\sqcap D_{\epsilon })$, which is a contradiction to $A_{\epsilon }$ being soft ${\mathbb{I}}_{\epsilon }$-connected. Hence, $A_{\epsilon }\bigsqcup B_{\epsilon }$ is soft ${\mathbb{I}}_{\epsilon }$-connected. □

Corollary 1. 

Let $(X_{\epsilon },\tau )$ be a STS and let ${\mathbb{I}}_{\epsilon }$ be a soft ideal on $X_{\epsilon }$. If $A_{\epsilon }$ is a soft ${\mathbb{I}}_{\epsilon }$-connected and $X_{\epsilon }-A_{\epsilon }\in {\mathbb{I}}_{\epsilon }$, then $X_{\epsilon }$ is a soft ${\mathbb{I}}_{\epsilon }$-connected.

Now, we show that the product of soft ${\mathbb{I}}_{\epsilon }$-connected spaces is soft ${\mathbb{I}}_{\epsilon }$-connected under some conditions.

Theorem 8. 

Let $(X_{\epsilon },\tau )$ be soft ${\mathcal{J}}_{\epsilon }$-connected and $(Y_{\epsilon },\sigma )$ be soft ${\mathcal{H}}_{\epsilon }$-connected. Assume that ${\mathcal{J}}_{\epsilon }\sqcap \tau$ is closed under arbitrary unions. If ${\mathbb{I}}_{\epsilon }$ is a soft ideal in $X_{\epsilon }\times Y_{\epsilon }$ containing $p_1^{-1}\left({\mathcal{J}}_{\epsilon }\right)$ and $p_2^{-1}\left({\mathcal{H}}_{\epsilon }\right)$, then $X_{\epsilon }\times Y_{\epsilon }$ is soft ${\mathbb{I}}_{\epsilon }$-connected.

Proof. 

If $X_{\epsilon }\in {\mathcal{J}}_{\epsilon }$, then $X_{\epsilon }\times Y_{\epsilon }$ is in the soft ideal ${\mathbb{I}}_{\epsilon }$ and hence $X_{\epsilon }\times Y_{\epsilon }$ is soft ${\mathbb{I}}_{\epsilon }$-connected. Now suppose that $X_{\epsilon }\notin {\mathcal{J}}_{\epsilon }$ and $X_{\epsilon }\times Y_{\epsilon }$ is not soft ${\mathbb{I}}_{\epsilon }$-connected. Then, $X_{\epsilon }\times Y_{\epsilon }=A_{\epsilon }\bigsqcup B_{\epsilon }$, where $A_{\epsilon },B_{\epsilon }\notin {\mathbb{I}}_{\epsilon }$, $A_{\epsilon }\sqcap B_{\epsilon }={\varphi }_{\epsilon }$ such that $A_{\epsilon },B_{\epsilon }$ are soft open sets in $X_{\epsilon }\times Y_{\epsilon }$. For each $y_{\epsilon }\in Y_{\epsilon }$, define $A_{\epsilon }\left(y_{\epsilon }\right)=\{x_{\epsilon }\in X_{\epsilon }:(x_{\epsilon },y_{\epsilon })\in A_{\epsilon }\}$ and $B_{\epsilon }\left(y_{\epsilon }\right)=\{x_{\epsilon }\in X_{\epsilon }:(x_{\epsilon },y_{\epsilon })\in B_{\epsilon }\}$.

Let $C_{\epsilon }=\{y_{\epsilon }\in Y_{\epsilon }:A_{\epsilon }\left(y_{\epsilon }\right)\in {\mathcal{J}}_{\epsilon }\}$ and $D_{\epsilon }=\{y_{\epsilon }\in Y_{\epsilon }:B_{\epsilon }\left(y_{\epsilon }\right)\in {\mathcal{J}}_{\epsilon }\}$.

Then, $X_{\epsilon }=A_{\epsilon }\left(y_{\epsilon }\right)\bigsqcup B_{\epsilon }\left(y_{\epsilon }\right)$. For each $y_{\epsilon }$, both $A_{\epsilon }\left(y_{\epsilon }\right)$ and $B_{\epsilon }\left(y_{\epsilon }\right)$ are soft open sets, and $A_{\epsilon }\left(y_{\epsilon }\right)\sqcap B_{\epsilon }\left(y_{\epsilon }\right)={\varphi }_{\epsilon }$. As $X_{\epsilon }$ is soft ${\mathcal{J}}_{\epsilon }$-connected, either $A_{\epsilon }\left(y_{\epsilon }\right)\in {\mathcal{J}}_{\epsilon }$ or $B_{\epsilon }\left(y_{\epsilon }\right)\in {\mathcal{J}}_{\epsilon }$. In fact, to each $y_{\epsilon }\in Y_{\epsilon }$, exactly one of $A_{\epsilon }\left(y_{\epsilon }\right)$ and $B_{\epsilon }\left(y_{\epsilon }\right)\in {\mathcal{J}}_{\epsilon }$. $Y_{\epsilon }=C_{\epsilon }\bigsqcup D_{\epsilon }$ and $C_{\epsilon }\sqcap D_{\epsilon }={\varphi }_{\epsilon }$. Now, we claim that $C_{\epsilon }$ is a soft closed set. Fix $y_{\epsilon }\in {\overline{C}}_{\epsilon }$. If $A_{\epsilon }\left(y_{\epsilon }\right)\notin {\mathcal{J}}_{\epsilon }$, then $A_{\epsilon }\left(y_{\epsilon }\right)\ne {\varphi }_{\epsilon }$. Since $A_{\epsilon }$ is a soft open set, for each $x_{\epsilon }\in A_{\epsilon }\left(y_{\epsilon }\right)$, there exist soft neighborhoods $U_{\epsilon }\left(x_{\epsilon }\right)$ of $x_{\epsilon }$ and $V_{\epsilon }\left(y_{\epsilon }\right)$ of $y_{\epsilon }$ such that $(x_{\epsilon },y_{\epsilon })\in U_{\epsilon }\left(x_{\epsilon }\right)\times V_{\epsilon }\left(y_{\epsilon }\right)⊑A_{\epsilon }$. As $y_{\epsilon }\in {\overline{C}}_{\epsilon }$, there is one $y_{\epsilon }^{\prime }\in V_{\epsilon }\left(y_{\epsilon }\right)\sqcap C_{\epsilon }$ so $U_{\epsilon }\left(x_{\epsilon }\right)\times \left\{y_{\epsilon }^{\prime }\right\}⊑A_{\epsilon }$ and hence $U_{\epsilon }\left(x_{\epsilon }\right)⊑A_{\epsilon }\left(y_{\epsilon }^{\prime }\right)$. As $A_{\epsilon }\left(y_{\epsilon }^{\prime }\right)\in {\mathcal{J}}_{\epsilon }$, we have $U_{\epsilon }\left(x_{\epsilon }\right)\in {\mathcal{J}}_{\epsilon }$. Therefore, ${\displaystyle A_{\epsilon }\left(y_{\epsilon }\right)⊑{\cup }_{x_{\epsilon }\in A_{\epsilon }\left(y_{\epsilon }\right)}{U}_{\epsilon }\left(x_{\epsilon }\right)\in {\mathcal{J}}_{\epsilon }}$ (by assumption).

Hence, $A_{\epsilon }\left(y_{\epsilon }\right)\in {\mathcal{J}}_{\epsilon }$ and hence $y_{\epsilon }=C_{\epsilon }$. Thus, $C_{\epsilon }$ is a soft closed set. Similarly $D_{\epsilon }$ is a soft closed set. Since $Y_{\epsilon }$ is soft ${\mathcal{H}}_{\epsilon }$-connected, we have $C_{\epsilon }\in {\mathcal{H}}_{\epsilon }$ or $D_{\epsilon }\in {\mathcal{H}}_{\epsilon }$.

Case (1): If $C_{\epsilon }\in {\mathcal{H}}_{\epsilon }$, then $X_{\epsilon }\times C_{\epsilon }\in {\mathbb{I}}_{\epsilon }$. Take $M_{\epsilon }=\bigsqcup \{B_{\epsilon }\left(y_{\epsilon }\right):y_{\epsilon }\in D_{\epsilon }\}\in {\mathcal{J}}_{\epsilon }\sqcap \tau$ (assumption). So, $M_{\epsilon }\times Y_{\epsilon }\in {\mathbb{I}}_{\epsilon }$ and $(X_{\epsilon }\times C_{\epsilon })\bigsqcup (M_{\epsilon }\times Y_{\epsilon })\in {\mathbb{I}}_{\epsilon }$. Fix $(x_{\epsilon },y_{\epsilon })\in B_{\epsilon }$. If $y_{\epsilon }\in C_{\epsilon }$, then $(x_{\epsilon },y_{\epsilon })\in X_{\epsilon }\times C_{\epsilon }$. If $y_{\epsilon }\notin C_{\epsilon }$, then $y_{\epsilon }\in D_{\epsilon }$ and $x_{\epsilon }\in B_{\epsilon }\left(y_{\epsilon }\right)⊑M_{\epsilon }$. Therefore, $(x_{\epsilon },y_{\epsilon })\in M_{\epsilon }\times Y_{\epsilon }$. Hence, $B_{\epsilon }⊑(X_{\epsilon }\times C_{\epsilon })\bigsqcup (M_{\epsilon }\times Y_{\epsilon })$. Hence, $B_{\epsilon }\in {\mathbb{I}}_{\epsilon }$, which contradicts the fact that $B_{\epsilon }\notin {\mathbb{I}}_{\epsilon }$.

Case (2): If $D_{\epsilon }\in {\mathcal{H}}_{\epsilon }$, then $X_{\epsilon }\times D_{\epsilon }\in {\mathbb{I}}_{\epsilon }$. As in case (1), we obtain a contradiction. Thus, $X_{\epsilon }\times Y_{\epsilon }$ is soft ${\mathbb{I}}_{\epsilon }$-connected. □

Now, we introduce the definition of soft connected component $C_{\epsilon }$ and discuss the product of soft ${\mathbb{I}}_{\epsilon }$-connected spaces.

Definition 9. 

Let $(X_{\epsilon },\tau )$ be an STS and ${\mathbb{I}}_{\epsilon }$ be a soft ideal on $X_{\epsilon }$. A soft connected component $C_{\epsilon }$ of $X_{\epsilon }$ with respect to τ is said to be a soft-ideal component in $X_{\epsilon }$ if $C_{\epsilon }\in {\mathbb{I}}_{\epsilon }$.

Theorem 9. 

Let $(X_{\epsilon },\tau )$ be soft ${\mathcal{J}}_{\epsilon }$-connected and $(Y_{\epsilon },\sigma )$ be soft ${\mathcal{H}}_{\epsilon }$-connected. Assume that any union of soft-ideal components is a member of ${\mathcal{J}}_{\epsilon }$. If ${\mathbb{I}}_{\epsilon }$ is a soft ideal in $X_{\epsilon }\times Y_{\epsilon }$ containing $p_1^{-1}\left({\mathcal{J}}_{\epsilon }\right)$ and $p_2^{-1}\left({\mathcal{H}}_{\epsilon }\right)$, then $X_{\epsilon }\times Y_{\epsilon }$ is soft ${\mathbb{I}}_{\epsilon }$-connected.

Proof. 

If $X_{\epsilon }\in {\mathcal{J}}_{\epsilon }$, then $X_{\epsilon }\times Y_{\epsilon }$ is in the soft ideal ${\mathbb{I}}_{\epsilon }$. Hence, $X_{\epsilon }\times Y_{\epsilon }$ is soft ${\mathbb{I}}_{\epsilon }$-connected. Suppose that $X_{\epsilon }\notin {\mathcal{J}}_{\epsilon }$, and $X_{\epsilon }\times Y_{\epsilon }$ is not soft ${\mathbb{I}}_{\epsilon }$-connected. Then, $X_{\epsilon }\times Y_{\epsilon }=A_{\epsilon }\bigsqcup B_{\epsilon }$, where $A_{\epsilon },B_{\epsilon }\notin {\mathbb{I}}_{\epsilon }$, $A_{\epsilon }\sqcap B_{\epsilon }={\varphi }_{\epsilon }$ and $A_{\epsilon },B_{\epsilon }$ are soft open sets. For every soft component $C_{\epsilon }$ of $X_{\epsilon }$ and $D_{\epsilon }$ of $Y_{\epsilon }$, $C_{\epsilon }\times D_{\epsilon }$ is a soft connected subset of $X_{\epsilon }\times Y_{\epsilon }$ and hence $C_{\epsilon }\times D_{\epsilon }⊑A_{\epsilon }$ or $C_{\epsilon }\times D_{\epsilon }⊑B_{\epsilon }$$\dots \dots$(1). For every component $D_{\epsilon }$ of $Y_{\epsilon }$, write

$$\begin{array}{cc}\hfill {\tilde{A}}_{\epsilon }=& \cup \{C_{\epsilon }:C_{\epsilon }\mathrm{is}\mathrm{a}\mathrm{soft}\mathrm{component}\mathrm{of}{X}_{\epsilon }\mathrm{and}{C}_{\epsilon }\times D_{\epsilon }⊑A_{\epsilon }\},\hfill \\ \hfill {\tilde{B}}_{\epsilon }=& \cup \{C_{\epsilon }:C_{\epsilon }\mathrm{is}\mathrm{a}\mathrm{soft}\mathrm{component}\mathrm{of}{X}_{\epsilon }\mathrm{and}{C}_{\epsilon }\times D_{\epsilon }⊑B_{\epsilon }\}.\hfill \end{array}$$

Now, we claim that ${\tilde{A}}_{\epsilon }$ is a soft open set. Let $x_{\epsilon }\in {\tilde{A}}_{\epsilon }$. Then, there exists a soft component $C_{\epsilon }$ of $X_{\epsilon }$ such that $x_{\epsilon }\in C_{\epsilon }$ and $C_{\epsilon }\times D_{\epsilon }⊑A_{\epsilon }$. Fix $y_{\epsilon }\in D_{\epsilon }$. Therefore, $(x_{\epsilon },y_{\epsilon })\in C_{\epsilon }\times D_{\epsilon }⊑A_{\epsilon }$. Since $A_{\epsilon }$ is soft open, there exist soft neighborhoods $U_{\epsilon }\left(x_{\epsilon }\right)$ and $V_{\epsilon }\left(y_{\epsilon }\right)$ of $x_{\epsilon }$ and $y_{\epsilon }$, respectively, such that $U_{\epsilon }\left(x_{\epsilon }\right)\times V_{\epsilon }\left(y_{\epsilon }\right)⊑A_{\epsilon }$. If $x_{\epsilon }\in {\overline{\tilde{B}}}_{\epsilon }$, then $U_{\epsilon }\left(x_{\epsilon }\right)\sqcap {\tilde{B}}_{\epsilon }\ne {\varphi }_{\epsilon }$. Let $x_{\epsilon }^{\prime }\in U_{\epsilon }\left(x_{\epsilon }\right)\sqcap {\tilde{B}}_{\epsilon }$, i.e., $x_{\epsilon }^{\prime }\in U_{\epsilon }\left(x_{\epsilon }\right)\sqcap C_{\epsilon }^{\prime }$, for some soft component $C_{\epsilon }^{\prime }$, where $C_{\epsilon }^{\prime }\times D_{\epsilon }⊑B_{\epsilon }$. Let $(x_{\epsilon }^{\prime },y_{\epsilon })\in U_{\epsilon }\left(x_{\epsilon }^{\prime }\right)\times V_{\epsilon }^{\prime }\left(y_{\epsilon }\right)⊑B_{\epsilon }$, where $U_{\epsilon }\left(x_{\epsilon }^{\prime }\right),V_{\epsilon }^{\prime }\left(y_{\epsilon }\right)$ are some soft neighborhoods of $x_{\epsilon }^{\prime }$ and $y_{\epsilon }$, respectively. Then, $(x_{\epsilon }^{\prime },y_{\epsilon })\in (U_{\epsilon }\left(x_{\epsilon }\right)\sqcap U_{\epsilon }\left(x_{\epsilon }^{\prime }\right))\times (V_{\epsilon }\left(y_{\epsilon }\right)\sqcap V_{\epsilon }^{\prime }\left(y_{\epsilon }\right))⊑A_{\epsilon }\sqcap B_{\epsilon }$, which is a contradiction to $A_{\epsilon }\sqcap B_{\epsilon }={\varphi }_{\epsilon }$. Therefore, $x_{\epsilon }\in {\tilde{A}}_{\epsilon }$ implies that $x_{\epsilon }$ is not a limit of the soft point of ${\tilde{B}}_{\epsilon }$. That is, ${\tilde{A}}_{\epsilon }$ is a soft open set. Similarly, ${\tilde{B}}_{\epsilon }$ is a soft open set. Thus, $X_{\epsilon }={\tilde{A}}_{\epsilon }\bigsqcup {\tilde{B}}_{\epsilon }$, ${\tilde{A}}_{\epsilon }$, and ${\tilde{B}}_{\epsilon }$ are soft open sets, so exactly one of ${\tilde{A}}_{\epsilon }$ or ${\tilde{B}}_{\epsilon }$ is in ${\mathcal{J}}_{\epsilon }$, because $X_{\epsilon }\notin {\mathcal{J}}_{\epsilon }$.

$$\begin{array}{cc}\hfill \mathrm{Let}{D}_{\epsilon }^{\prime }=& \{D_{\epsilon }⊑Y_{\epsilon }:D_{\epsilon }\mathrm{is}\mathrm{a}\mathrm{soft}\mathrm{component}\mathrm{of}{Y}_{\epsilon }\mathrm{and}{\tilde{A}}_{\epsilon }\in {\mathcal{J}}_{\epsilon }\},\hfill \\ \hfill D_{\epsilon }^{″}=& \{D_{\epsilon }⊑Y_{\epsilon }:D_{\epsilon }\mathrm{is}\mathrm{a}\mathrm{soft}\mathrm{component}\mathrm{of}{Y}_{\epsilon }\mathrm{and}{\tilde{B}}_{\epsilon }\in {\mathcal{J}}_{\epsilon }\}.\hfill \end{array}$$

Write $H_{\epsilon }={\bigsqcup }_{D_{\epsilon }\in D_{\epsilon }^{\prime }}{D}_{\epsilon }$ and $K_{\epsilon }={\bigsqcup }_{D_{\epsilon }\in D_{\epsilon }^{″}}{D}_{\epsilon }$. Then, $Y_{\epsilon }=H_{\epsilon }\bigsqcup K_{\epsilon }$ and $H_{\epsilon }\sqcap K_{\epsilon }={\varphi }_{\epsilon }$.

We claim that $H_{\epsilon }$ is a soft closed set. Fix $h_{\epsilon }\in {\overline{H}}_{\epsilon }$. Let $D_{\epsilon }$ be the soft component of $Y_{\epsilon }$ such that $h_{\epsilon }\in D_{\epsilon }$. Suppose $h_{\epsilon }\notin H_{\epsilon }$. Then, $D_{\epsilon }\notin D_{\epsilon }^{\prime }$ implies that ${\tilde{A}}_{\epsilon }\notin {\mathcal{J}}_{\epsilon }$, so ${\tilde{B}}_{\epsilon }\in {\mathcal{J}}_{\epsilon }$. By (1) and our assumption, there is a soft component $C_{\epsilon }$ of $X_{\epsilon }$ such that $C_{\epsilon }\notin {\mathcal{J}}_{\epsilon }$, $C_{\epsilon }\times D_{\epsilon }⊑A_{\epsilon }$. Fix a member $c_{\epsilon }\in C_{\epsilon }$. Then, $(c_{\epsilon },h_{\epsilon })\in A_{\epsilon }$. Since $A_{\epsilon }$ is a soft open set, there exist soft neighborhoods $U_{\epsilon }\left(c_{\epsilon }\right)$ and $V_{\epsilon }\left(h_{\epsilon }\right)$ of $c_{\epsilon }$, $h_{\epsilon }$ in $X_{\epsilon }$, $Y_{\epsilon }$, respectively, such that $(c_{\epsilon },h_{\epsilon })\in U_{\epsilon }\left(c_{\epsilon }\right)\times V_{\epsilon }\left(h_{\epsilon }\right)⊑A_{\epsilon }$. So there is a member $h_{\epsilon }^{\prime }\in V_{\epsilon }\left(h_{\epsilon }\right)\sqcap H_{\epsilon }$ and there is a component $H_{\epsilon }^{\prime }$ of $Y_{\epsilon }$ such that $h_{\epsilon }^{\prime }\in H_{\epsilon }^{\prime }$ and ${\widetilde{A^{\prime }}}_{\epsilon }=\cup \{C_{\epsilon }:C_{\epsilon }\mathrm{is}\mathrm{a}\mathrm{soft}\mathrm{component}\mathrm{of}{X}_{\epsilon }\mathrm{and}{C}_{\epsilon }\times H_{\epsilon }^{\prime }⊑A_{\epsilon }\}\in {\mathcal{J}}_{\epsilon }$. Then, $(c_{\epsilon },h_{\epsilon }^{\prime })\in U_{\epsilon }\left(c_{\epsilon }\right)\times V_{\epsilon }\left(h_{\epsilon }\right)⊑A_{\epsilon }$. Therefore, $C_{\epsilon }\times H_{\epsilon }^{\prime }⊑A_{\epsilon }$ and $C_{\epsilon }\in {\mathcal{J}}_{\epsilon }$, because $C_{\epsilon }⊑{\widetilde{A^{\prime }}}_{\epsilon }\in {\mathcal{J}}_{\epsilon }$. This contradicts $C_{\epsilon }\notin {\mathcal{J}}_{\epsilon }$. Therefore, $h_{\epsilon }\in H_{\epsilon }$, i.e., $H_{\epsilon }={\overline{H}}_{\epsilon }$ and $H_{\epsilon }$ is a soft closed set. Similarly $K_{\epsilon }$ is a soft closed set. Thus, $Y_{\epsilon }=H_{\epsilon }\bigsqcup K_{\epsilon }$, where $H_{\epsilon },K_{\epsilon }$ are soft open sets and $H_{\epsilon }\sqcap K_{\epsilon }={\varphi }_{\epsilon }$. Since $Y_{\epsilon }$ is soft ${\mathcal{H}}_{\epsilon }$-connected, either $H_{\epsilon }\in {\mathcal{H}}_{\epsilon }$ or $K_{\epsilon }\in {\mathcal{H}}_{\epsilon }$. Without sacrificing generality, we may assume $H_{\epsilon }\in {\mathcal{H}}_{\epsilon }$. Then, $X_{\epsilon }\times H_{\epsilon }\in {\mathbb{I}}_{\epsilon }$.

Take $M_{\epsilon }={\bigsqcup }_{D_{\epsilon }\in D_{\epsilon }^{″}}{\tilde{B}}_{\epsilon }\in {\mathcal{J}}_{\epsilon }$ (by assumption). So, $M_{\epsilon }\times Y_{\epsilon }\in {\mathbb{I}}_{\epsilon }$ and hence $(X_{\epsilon }\times H_{\epsilon })\bigsqcup (M_{\epsilon }\times Y_{\epsilon })\in {\mathbb{I}}_{\epsilon }$. It is enough to prove that $B_{\epsilon }⊑(X_{\epsilon }\times H_{\epsilon })\bigsqcup (M_{\epsilon }\times Y_{\epsilon })$. Fix $(x_{\epsilon },y_{\epsilon })\in B_{\epsilon }$. Then, there exist soft components $C_{\epsilon }$ and $D_{\epsilon }$ such that $(x_{\epsilon },y_{\epsilon })\in C_{\epsilon }\times D_{\epsilon }⊑B_{\epsilon }$. If $y_{\epsilon }\in H_{\epsilon }$, then $(x_{\epsilon },y_{\epsilon })\in X_{\epsilon }\times H_{\epsilon }$. If $y_{\epsilon }\notin H_{\epsilon }$, then $y_{\epsilon }\in K_{\epsilon }={\bigsqcup }_{D_{\epsilon }\in D_{\epsilon }^{″}}{D}_{\epsilon }$ and hence $x_{\epsilon }\in C_{\epsilon }⊑{\tilde{B}}_{\epsilon }⊑M_{\epsilon }$ for some $D_{\epsilon }\in D_{\epsilon }^{″}$. Therefore, $(x_{\epsilon },y_{\epsilon })\in M_{\epsilon }\times Y_{\epsilon }$. Hence, $B_{\epsilon }⊑(X_{\epsilon }\times H_{\epsilon })\bigsqcup (M_{\epsilon }\times Y_{\epsilon })\in {\mathbb{I}}_{\epsilon }$. This is a contradiction to $B_{\epsilon }\notin {\mathbb{I}}_{\epsilon }$. Hence, $X_{\epsilon }\times Y_{\epsilon }$ is soft ${\mathbb{I}}_{\epsilon }$-connected. □

4. Strongly Soft ${\mathbb{I}}_{\mathit{\epsilon }}$-Connected

We present and define the idea of strongly soft ${\mathbb{I}}_{\epsilon }$-connected spaces and characterize the concept of strongly soft ${\mathbb{I}}_{\epsilon }$-connected spaces in this part. Additionally, we show that the image of the strongly soft ${\mathbb{I}}_{\epsilon }$-connected set is preserved by the soft continuous surjective maps.

Definition 10. 

Let $(X_{\epsilon },\tau )$ be an STS with a soft ideal ${\mathbb{I}}_{\epsilon }$ on $X_{\epsilon }$. A subset $A_{\epsilon }$ of $X_{\epsilon }$ is called strongly soft ${\mathbb{I}}_{\epsilon }$-connected if there is a soft connected subset $B_{\epsilon }$ of $X_{\epsilon }$ such that $A_{\epsilon }=B_{\epsilon }\bigsqcup C_{\epsilon }$, where $C_{\epsilon }\in {\mathbb{I}}_{\epsilon }$.

Now, we show that every strongly soft ${\mathbb{I}}_{\epsilon }$-connected set is soft ${\mathbb{I}}_{\epsilon }$-connected.

Theorem 10. 

Assume that $(X_{\epsilon },\tau )$ is an STS with a soft ideal ${\mathbb{I}}_{\epsilon }$ on $X_{\epsilon }$. Let $(X_{\epsilon },\tau ,{\mathbb{I}}_{\epsilon })$ be strongly soft ${\mathbb{I}}_{\epsilon }$-connected. Then, $(X_{\epsilon },\tau ,{\mathbb{I}}_{\epsilon })$ is soft ${\mathbb{I}}_{\epsilon }$-connected.

Proof. 

Suppose that $(X_{\epsilon },\tau ,{\mathbb{I}}_{\epsilon })$ is strongly soft ${\mathbb{I}}_{\epsilon }$-connected and $X_{\epsilon }=B_{\epsilon }\bigsqcup C_{\epsilon }$, where $B_{\epsilon }$ is soft connected and $C_{\epsilon }\in {\mathbb{I}}_{\epsilon }$. Let $X_{\epsilon }=D_{\epsilon }^{\prime }\bigsqcup D_{\epsilon }^{″}$, where $D_{\epsilon }^{\prime }$ and $D_{\epsilon }^{″}$ are soft open sets and $D_{\epsilon }^{\prime }\sqcap D_{\epsilon }^{″}={\varphi }_{\epsilon }$. Then, $B_{\epsilon }=(D_{\epsilon }^{\prime }\sqcap B_{\epsilon })\bigsqcup (D_{\epsilon }^{″}\sqcap B_{\epsilon })$ and $D_{\epsilon }^{\prime }\sqcap B_{\epsilon }={\varphi }_{\epsilon }$ or $D_{\epsilon }^{″}\sqcap B_{\epsilon }={\varphi }_{\epsilon }$. Then, $D_{\epsilon }^{\prime }⊑X_{\epsilon }-B_{\epsilon }$ or $D_{\epsilon }^{″}⊑X_{\epsilon }-B_{\epsilon }$. Therefore, $D_{\epsilon }^{\prime }⊑C_{\epsilon }$ or $D_{\epsilon }^{″}⊑C_{\epsilon }$ and $D_{\epsilon }^{\prime }\in {\mathbb{I}}_{\epsilon }$ or $D_{\epsilon }^{″}\in {\mathbb{I}}_{\epsilon }$. Hence, $X_{\epsilon }$ is soft ${\mathbb{I}}_{\epsilon }$-connected. □

The converse of the above theorem is not true.

Example 4. 

Let $X=\{h_1,h_2,h_3,h_4,h_5\}$, $\epsilon =\left\{{\epsilon }_1\right\}$, and $\tau =\{X_{\epsilon },{\varphi }_{\epsilon },A_{\epsilon },B_{\epsilon },C_{\epsilon },D_{\epsilon }\}$, where $A_{\epsilon }$, $B_{\epsilon }$, $C_{\epsilon }$, and $D_{\epsilon }$ are soft sets over X defined as follows: $A\left({\epsilon }_1\right)=\left\{h_2\right\}$, $B\left({\epsilon }_1\right)=\{h_3,h_4\}$, $C\left({\epsilon }_1\right)=\{h_2,h_3,h_4\}$, and $D\left({\epsilon }_1\right)=\{h_1,h_3,h_4,h_5\}$. Then, τ defines a soft topology on X. Let ${\mathbb{I}}_{\epsilon }=\{{\varphi }_{\epsilon },I_{\epsilon }\}$, where $I_{\epsilon }$ is a soft set over X defined by $I\left({\epsilon }_1\right)=\left\{h_2\right\}$. Then, ${\mathbb{I}}_{\epsilon }$ defines a soft ideal on X. Clearly, $X_{\epsilon }$ is not soft connected and not soft ${\mathbb{I}}_{\epsilon }$-connected. Now, if $Y=\{h_1,h_3,h_4,h_5\}⊑X$, $\epsilon =\left\{{\epsilon }_1\right\}$ and ${\tau }_Y=\{Y_{\epsilon },{\varphi }_{\epsilon },E_{\epsilon }\}$, where $E\left({\epsilon }_1\right)=\{h_3,h_4\}$, then ${\tau }_Y$ defines a soft topology on Y. Then, $Y_{\epsilon }$ is soft connected and $X_{\epsilon }=Y_{\epsilon }\bigsqcup E_{\epsilon }$ and $E_{\epsilon }\in {\mathbb{I}}_{\epsilon }$. Therefore, $X_{\epsilon }$ is strongly soft ${\mathbb{I}}_{\epsilon }$-connected and not soft ${\mathbb{I}}_{\epsilon }$-connected.

Remark 1. 

Let $(X_{\epsilon },\tau )$ be an STS and let ${\mathbb{I}}_{\epsilon }$ be a soft ideal on $X_{\epsilon }$. If $A_{\epsilon }$ is a strongly soft ${\mathbb{I}}_{\epsilon }$-connected set and $B_{\epsilon }\in {\mathbb{I}}_{\epsilon }$, then $A_{\epsilon }\bigsqcup B_{\epsilon }$ is strongly soft ${\mathbb{I}}_{\epsilon }$-connected.

We obtain equivalent conditions for connectedness in ideal soft spaces in the following corollary.

Corollary 2. 

Let $(X_{\epsilon },\tau )$ be an STS and let ${\mathbb{I}}_{\epsilon }$ be a soft ideal on $X_{\epsilon }$ with $\tau \sqcap {\mathbb{I}}_{\epsilon }={\varphi }_{\epsilon }$. Then, the subsequent statements are equivalent. $X_{\epsilon }$ is:

1. 

Soft connected.

2. 

Soft ${\mathbb{I}}_{\epsilon }$-connected.

3. 

Strongly soft ${\mathbb{I}}_{\epsilon }$-connected.

Example 5. 

Let $X=\{h_1,h_2,h_3\}$, $\epsilon =\{{\epsilon }_1,{\epsilon }_2\}$, and $\tau =\{X_{\epsilon },{\varphi }_{\epsilon },A_{\epsilon },B_{\epsilon }\}$, where $A_{\epsilon }$ and $B_{\epsilon }$ are soft sets over X defined as follows: $A\left({\epsilon }_1\right)=\left\{h_2\right\}$, $A\left({\epsilon }_2\right)=\{h_1,h_3\}$, and $B\left({\epsilon }_1\right)=\{h_1,h_3\}$, $B\left({\epsilon }_2\right)=\left\{h_2\right\}$. Then, τ defines a soft topology on X. Let ${\mathbb{I}}_{\epsilon }=\{{\varphi }_{\epsilon },I_{\epsilon }\}$ be a soft set over X defined by $I\left({\epsilon }_1\right)=\left\{h_2\right\}$, $I\left({\epsilon }_2\right)=\left\{h_1\right\}$. Then, ${\mathbb{I}}_{\epsilon }$ defines a soft ideal on X. Clearly, $X_{\epsilon }$ is not soft connected since $\tau \sqcap {\mathbb{I}}_{\epsilon }={\varphi }_{\epsilon }$. Therefore, $X_{\epsilon }$ is not strongly soft ${\mathbb{I}}_{\epsilon }$-connected.

It is well known that the union of two connected sets is connected if $A_{\epsilon }\sqcap B_{\epsilon }\ne {\varphi }_{\epsilon }$. This result can be generalized as follows.

Theorem 11. 

Let $(X_{\epsilon },\tau )$ be an STS and let ${\mathbb{I}}_{\epsilon }$ be a soft ideal on $X_{\epsilon }$. If $B_{\epsilon }$ and $A_{\epsilon }$ are strongly soft ${\mathbb{I}}_{\epsilon }$-connected sets such that $A_{\epsilon }\sqcap B_{\epsilon }\notin {\mathbb{I}}_{\epsilon }$, then $A_{\epsilon }\bigsqcup B_{\epsilon }$ is strongly soft ${\mathbb{I}}_{\epsilon }$-connected.

Proof. 

Let $B_{\epsilon }$ and $A_{\epsilon }$ be strongly soft ${\mathbb{I}}_{\epsilon }$-connected sets. Then, $A_{\epsilon }=H_{\epsilon }\bigsqcup K_{\epsilon }$ and $B_{\epsilon }=M_{\epsilon }\bigsqcup N_{\epsilon }$, where $H_{\epsilon }$ and $M_{\epsilon }$ are soft connected sets and $K_{\epsilon },N_{\epsilon }\in {\mathbb{I}}_{\epsilon }$. Since $A_{\epsilon }\sqcap B_{\epsilon }\notin {\mathbb{I}}_{\epsilon }$, the $A_{\epsilon },B_{\epsilon },H_{\epsilon },M_{\epsilon }\notin {\mathbb{I}}_{\epsilon }$. Let $D_{\epsilon }^{\prime }=(A_{\epsilon }\sqcap B_{\epsilon })\sqcap K_{\epsilon }$ and $D_{\epsilon }^{″}=(A_{\epsilon }\sqcap B_{\epsilon })\sqcap N_{\epsilon }$. Then, $D_{\epsilon }^{\prime },D_{\epsilon }^{″}\in {\mathbb{I}}_{\epsilon }$. Therefore, $D_{\epsilon }^{\prime }\bigsqcup D_{\epsilon }^{″}\in {\mathbb{I}}_{\epsilon }$. Now, put $D_{\epsilon }=(A_{\epsilon }\sqcap B_{\epsilon })\setminus (D_{\epsilon }^{\prime }\bigsqcup D_{\epsilon }^{″})$. Then, $D_{\epsilon }\notin {\mathbb{I}}_{\epsilon }$ because $A_{\epsilon }\sqcap B_{\epsilon }\notin {\mathbb{I}}_{\epsilon }$ and $D_{\epsilon }^{\prime },D_{\epsilon }^{″}\in {\mathbb{I}}_{\epsilon }$. Since $D_{\epsilon }⊑H_{\epsilon }$ and $D_{\epsilon }⊑M_{\epsilon }$, then $D_{\epsilon }⊑M_{\epsilon }\sqcap H_{\epsilon }\notin {\mathbb{I}}_{\epsilon }$, which implies that $M_{\epsilon }\sqcap H_{\epsilon }\ne {\varphi }_{\epsilon }$. Hence, $M_{\epsilon }\bigsqcup H_{\epsilon }$ is soft connected and $A_{\epsilon }\bigsqcup B_{\epsilon }=(M_{\epsilon }\bigsqcup H_{\epsilon })\bigsqcup C_{\epsilon }$, where $C_{\epsilon }⊑(A_{\epsilon }\setminus H_{\epsilon })\bigsqcup (B_{\epsilon }\setminus M_{\epsilon })⊑(K_{\epsilon }\bigsqcup N_{\epsilon })\in {\mathbb{I}}_{\epsilon }$. Thus, $C_{\epsilon }\in {\mathbb{I}}_{\epsilon }$. Hence, $A_{\epsilon }\bigsqcup B_{\epsilon }$ is strongly soft ${\mathbb{I}}_{\epsilon }$-connected. □

Theorem 12. 

Let $(X_{\epsilon },\tau )$ be an STS and let ${\mathbb{I}}_{\epsilon }$ be a soft ideal on $X_{\epsilon }$. Let $\overline{I}\in {\mathbb{I}}_{\epsilon }$ for all $I\in {\mathbb{I}}_{\epsilon }$. If $A_{\epsilon }⊑B_{\epsilon }⊑{\overline{A}}_{\epsilon }$ such that $A_{\epsilon }$ is strongly soft ${\mathbb{I}}_{\epsilon }$-connected, then $B_{\epsilon }$ is also strongly soft ${\mathbb{I}}_{\epsilon }$-connected for all $B_{\epsilon }$. In particular, ${\overline{A}}_{\epsilon }$ is strongly soft ${\mathbb{I}}_{\epsilon }$-connected.

Proof. 

Suppose $A_{\epsilon }$ is strongly soft ${\mathbb{I}}_{\epsilon }$-connected. Then, $A_{\epsilon }=C_{\epsilon }\bigsqcup D_{\epsilon }$, where $C_{\epsilon }$ is soft connected and $D_{\epsilon }\in {\mathbb{I}}_{\epsilon }$. Since $A_{\epsilon }⊑B_{\epsilon }⊑{\overline{A}}_{\epsilon }$ and $A_{\epsilon }=C_{\epsilon }\bigsqcup D_{\epsilon }⊑B_{\epsilon }$, we have $B_{\epsilon }=(C_{\epsilon }\sqcap B_{\epsilon })\sqcap (D_{\epsilon }\sqcap B_{\epsilon })$, where $C_{\epsilon }\sqcap B_{\epsilon }$ is soft connected as $C_{\epsilon }⊑({\overline{C}}_{\epsilon }\sqcap B_{\epsilon })⊑{\overline{C}}_{\epsilon }$ and ${\overline{D}}_{\epsilon }\sqcap B_{\epsilon }\in {\mathbb{I}}_{\epsilon }$. Hence, $B_{\epsilon }$ is also strongly soft ${\mathbb{I}}_{\epsilon }$-connected. As a particular case when $A_{\epsilon }$ is strongly soft ${\mathbb{I}}_{\epsilon }$-connected, $A_{\epsilon }$ is strongly soft ${\mathbb{I}}_{\epsilon }$-connected for all $\overline{I}\in {\mathbb{I}}_{\epsilon }$. □

It is well known that the continuous image of a connected set is connected. This result can be generalized as follows.

Theorem 13. 

Let $(X_{\epsilon },\tau )$ and $(Y_{\epsilon },\sigma )$ be a STSs and let ${\mathbb{I}}_{\epsilon }$ be a soft ideal on $X_{\epsilon }$. Let $f_{pu}:(X_{\epsilon },\tau )\to (Y_{\epsilon },\sigma )$ be a soft continuous surjection. If $(X_{\epsilon },\tau )$ is a strongly soft ${\mathbb{I}}_{\epsilon }$-connected, then $(Y_{\epsilon },\sigma )$ is a strongly soft $f_{pu}\left({\mathbb{I}}_{\epsilon }\right)$-connected.

Proof. 

Let $f_{pu}:(X_{\epsilon },\tau )\to (Y_{\epsilon },\sigma )$ be a soft continuous surjective map and $(X_{\epsilon },\tau )$ be strongly soft ${\mathbb{I}}_{\epsilon }$-connected. $X_{\epsilon }=B_{\epsilon }\bigsqcup C_{\epsilon }$, where $B_{\epsilon }$ is soft connected and $C_{\epsilon }\in {\mathbb{I}}_{\epsilon }$. Therefore, $Y_{\epsilon }=f_{pu}\left(X_{\epsilon }\right)=f_{pu}(B_{\epsilon }\bigsqcup C_{\epsilon })=f_{pu}\left(B_{\epsilon }\right)\bigsqcup f_{pu}\left(C_{\epsilon }\right)$, where $f_{pu}\left(B_{\epsilon }\right)$ is soft connected and $f_{pu}\left(C_{\epsilon }\right)\in f_{pu}\left({\mathbb{I}}_{\epsilon }\right)$. Thus, $(Y_{\epsilon },\sigma )$ is strongly soft $f_{pu}\left({\mathbb{I}}_{\epsilon }\right)$-connected. □

Theorem 14. 

Let $(X_{\epsilon },\tau )$ be strongly soft ${\mathcal{J}}_{\epsilon }$-connected and $(Y_{\epsilon },\sigma )$ be strongly soft ${\mathcal{H}}_{\epsilon }$-connected. If ${\mathbb{I}}_{\epsilon }$ is a soft ideal in $X_{\epsilon }\times Y_{\epsilon }$ containing $p_1^{-1}\left({\mathcal{J}}_{\epsilon }\right)$ and $p_2^{-1}\left({\mathcal{H}}_{\epsilon }\right)$, then $X_{\epsilon }\times Y_{\epsilon }$ is strongly soft ${\mathbb{I}}_{\epsilon }$-connected.

Proof. 

Suppose that $X_{\epsilon }$ is strongly soft ${\mathcal{J}}_{\epsilon }$-connected and $Y_{\epsilon }$ is strongly soft ${\mathcal{H}}_{\epsilon }$-connected. Then, $X_{\epsilon }=A_{\epsilon }\bigsqcup H_{\epsilon }$ and $X_{\epsilon }=B_{\epsilon }\bigsqcup K_{\epsilon }$, such that $A_{\epsilon }$ and $B_{\epsilon }$ are soft connected subsets of $X_{\epsilon }$ and $Y_{\epsilon }$, respectively, and $H_{\epsilon },K_{\epsilon }\in {\mathbb{I}}_{\epsilon }$. Therefore, $X_{\epsilon }\times Y_{\epsilon }=(A_{\epsilon }\times B_{\epsilon })\bigsqcup [(H_{\epsilon }\times Y_{\epsilon })\bigsqcup (X_{\epsilon }\times K_{\epsilon })]$. Since $A_{\epsilon }\times B_{\epsilon }$ is soft connected with $\tau \times \sigma$ and $H_{\epsilon }\times Y_{\epsilon }\in {\mathbb{I}}_{\epsilon }$ and $X_{\epsilon }\times K_{\epsilon }\in {\mathbb{I}}_{\epsilon }$, then $(H_{\epsilon }\times Y_{\epsilon })\bigsqcup (X_{\epsilon }\times K_{\epsilon })\in {\mathbb{I}}_{\epsilon }$. Thus, $X_{\epsilon }\times Y_{\epsilon }$ is strongly soft ${\mathbb{I}}_{\epsilon }$-connected. □

5. Conclusions and Future Work

Shabir and Naz [11] and Çaǧman et al. [34] have individually shown how a soft topology on a universal set expands the classical (crisp) topology. Investigating this topological generalization is starting to seem intriguing. There are a lot of approaches for generating soft topologies available in the literature that can be used. We broadened our understanding of soft topology by exploring the concepts of soft connected, soft-ideal connected, and strongly soft-ideal connected spaces. The connection of soft-ideal spaces is the foundation of this investigation. We discussed some basic activities in pliable ideal places. A description of a soft-ideal connected space is given, along with an outline of its attributes. These are preliminary results, and additional investigations will examine more aspects of soft-ideal connected spaces. Prospects for further contributions to this trend are created by our study in terms of soft primal topologies and fuzzy soft topologies in soft and classical environments, as well as soft connectedness along with idea structures using generalized rough approximation spaces. This is accomplished by combining these two strategies.