On soft weak structures

Abstract

This paper takes some investigations on soft weak structures which are a generalization of weak structures and soft topological spaces with some separation axioms and compactness. Also, we give a systematic discussion on the relationship among these notions.

1 Introduction

In the real world, the exact solution cannot be found for many problems. As a solution to this problem, scientists have resorted to use approximate solutions. Molodtsov (1999) introduced a new approach for the real world problems. He proposed the soft sets as a tool for dealing with uncertainty.

Shabir and Naz (2011) introduced the notion of soft topological spaces and Min corrected some of their results (Min 2011). Zorlutuna et al. (2012) continued to study the properties of soft topological spaces by defining the concepts of soft interior point, soft interior, soft neighborhood, soft continuity and soft compactness. Later on, Nazmul and Samanta (2013) characterized the soft neighborhoods in soft topological space. Husain and Ahmad (2012) strengthened the theory of soft topological spaces by defining it on a fixed initial universe. Varol and Aygün (2013) presented soft Hausdorff spaces and introduced some new concepts such as convergence of sequences.

The attempts to develop the soft topology did not stop. Cagman et al. (2011) defined soft topological spaces by modifying the soft set. Also, Roy and Samanta (2011) strengthen the definition of the soft topological spaces presented in Cagman et al. (2011) and they used the base and the subbase to characterize its properties.

Soft sets theory received the attention of the topologists who always seeking to generalize and apply the topological notions. One of the most important characteristics of qualitative properties of spatial data and probably the most essential aspect of space is topology and topological relationships. Topological relations between spatial objects such as meet and overlap are the relationships that are invariant with respect to particular transformations due to homeomorphism. Hassan and Ghareeb (2015) gave the fundamental concepts and properties of a soft spatial region. They provided a theoretical framework for both dominant ontologies used in GIS.

Császár (2011) introduced the concept of generalized topology as a generalization of the concepts of general topology. Császár (2002, 2005, 2006, 2008) and many other authors investigated the properties of the generalized topology.

To refresh the fundamental concepts of weak structures, Császár (2011) defined weak structures and showed that these structures can replace general topologies, generalized topologies or minimal structures. Let X be a non-empty set and P(X) its power set of X. A collection w is said to be weak structure (WS, for short) on X if and only if $\emptyset \in w$.

In 2012, a new kind of sets called generalized w-closed (briefly gw-closed) sets is introduced in the concept of weak structures by Al-Omari and Noiri (2012). The class of all gw-closed sets is a generalization of the class of all w-closed sets. Furthermore, g-closed sets (Császár 2011) is a special case of gw-closed sets (since every general topology is a weak structure). Some of their properties are discussed. Also, they presented the notions of w-regular and w-normal spaces.

In the same year, Navaneethakrishnan and Thamaraiselvi (2013) defined and studied the properties of some subsets of X with respect to a weak structure on X and generalized some already established results. Further, they extended the study of weak structures and m-structures defined on a set X and prove that an m-structure generates a finer topology (see Navaneethakrishnan and Thamaraiselvi 2012).

Zahran et al. (2012) modified the weak structure by the hereditary classes and proved that the structures resulting from a generalized topology and a hereditary class introduced by Császár (2007) still valid. Recently, Güldürdek (2012) studied the p-stacks in weak structures. Ghareeb and Khalaf (2015) presented the $\phi (w)$-separation axioms in weak structures and also they defined the product of such structures.

The purpose of this paper is to define the soft weak structures and discuss some of its properties. Also, we will verify some new separation axioms and compactness in it. The results which we will get in this paper are a generalization to all the corresponding notions in weak structures, general topology and soft topology and therefore we think it will be more applicable for GIS modeling.

2 Preliminaries

Throughout this paper, let X be a non-empty set, E be a set of parameters, P(X) denote the power set of X and A be a non-empty subset of E. The pair (F, A) is called a soft set over X, where F is the mapping given by $F:A\rightarrow P(X)$. For $e\in A$, F(e) may be considered as the set of e-approximate elements of the soft set (F, A). A soft set (F, A) is called a finite (resp. countable) soft set if F(e) is finite (resp. countable) set for each $e\in A$. For any two soft sets (F, A) and (G, B) defined over a common universe X, we have:

• 1. $(F,A)\,\tilde{\subset }\,(G,B)$ iff $A\subset B$ and $F(e)\subset G(e)$ for all $e\in A$.

• 2. $(F,A)\,\tilde{=}\,(G,B)$ iff $(F,A)\,\tilde{\subset }\,(G,B)$ and (G, B) $\,\tilde{\subset }\,$ (F, A).

• 3. $(F,A)\,\tilde{\cup }\, (G,B)\,\tilde{=}\,(H,C)$ where $C=A\cup B$ and

  $$\begin{aligned} H(e)=\left\{ \begin{array}{ll} F(e), &{} \quad \hbox {if}\ e\in A-B, \\ G(e), &{} \quad \hbox {if}\ e\in B-A, \\ F(e)\cup G(e), &{} \quad \hbox {if}\ e\in A\cap B. \end{array} \right. \end{aligned}$$

  for all $e\in C$.

• 4. $(F,A)\,\tilde{\cap }\, (G,B)\,\tilde{=}\,(K,D)$ where $D=A\cap B$ and $K(e)=F(e)\cap G(e)$ for all $e\in C$.

• 5. $x\in (F,A)$ where $x\in X$ iff $x\in F(e)$ for all $e\in A$ and $x\not \in (F,A)$ whenever $x\not \in F(e)$ for some $e\in A$.

• 6. $(F,E)\tilde{-}(G,E)\,\tilde{=}\,(M,E)$ where $M(e)=F(e)-G(e)$ for all $e\in E$.

A soft set (F, A) is called a null soft set (denoted by $\tilde{\emptyset }$) if $F(e)=\emptyset$ for all $e\in A$ and called an absolute soft set (denoted by $\tilde{X}$) if $F(e)=X$ for all $e\in A$. If (F, E) is a soft set over X and $Y\subset X$, then the soft set $(^YF,E)$ is defined by $^YF(e)=Y\cap F(e)$ for each $e\in E$, i.e., $(^YF,E)\,\tilde{=}\,{\tilde{Y}}\cap (F,E)$. The relative complement of soft set (F, A) (denoted by $(F,A)'$) is defined by $(F,A)'\,\tilde{=}\,(F',A)$, where $F':A\rightarrow P(X)$ is given by $F'(e)=X-F(e)$ for all $e\in A$. It is clear that $\tilde{\emptyset }'\,\tilde{=}\,\tilde{X}$ and $\tilde{X}'\,\tilde{=}\,\tilde{\emptyset }$. The soft singleton set (x, A) is defined by $x(e)=\{x\}$ for all $e\in A$. Additional soft sets theory terminology can be found in Molodtsov (1999), Shabir and Naz (2011), Min (2011), Zorlutuna et al. (2012), Nazmul and Samanta (2013), Husain and Ahmad (2012), Varol and Aygün (2013), Cagman et al. (2011) and Roy and Samanta (2011).

3 Soft weak structure

Definition 1

Let X be a non-empty set and E be a set of parameters. A collection ${{\mathcal {SW}}}$ of soft sets defined over X with respect to E is called a soft weak structure iff $\tilde{\emptyset }\in {{\mathcal {SW}}}$. A soft set (F, A) is called ${\mathcal {W}}$-open soft set iff $(F,A)\in {\mathcal {SW}}$ and called ${\mathcal {W}}$-closed soft set iff $(F,A)'\in {\mathcal {SW}}$. A soft weak structure ${\mathcal {SW}}$ is called strong soft weak structure or soft minimal structure if ${\tilde{X}}\in {\mathcal {SW}}$.

Clearly, every soft topology is a soft weak structure.

Proposition 1

If ${\mathcal {SW}}$ is a soft weak structure defined over X with respect to the set of parameters E,  then ${\mathcal {SW}}_e=\{F(e):(F,E)\in {\mathcal {SW}}\}$ is weak structure on X for each $e\in E$.

Proof

Let ${\mathcal {SW}}$ be a soft weak structure, then $\tilde{\emptyset }\in {\mathcal {SW}}$, i.e., there exists a soft set $(F,E)\in {\mathcal {SW}}$ such that $F(e)=\emptyset$ for all $e\in E$. Hence $\emptyset \in {\mathcal {SW}}_e$ for all $e\in E$. Thus ${\mathcal {SW}}_e$ is a weak structure on X.

Definition 2

Let ${\mathcal {SW}}$ be a soft weak structure on X with respect to parameters set E and $Y\subset X$. Then ${\mathcal {SW}}_Y=\{(^YF,E): (F,E)\in {\mathcal {SW}}\}$ is a soft weak structure on Y and is called a relative soft weak structure of X.

Proposition 1 remains valid for the relative soft weak structure. The following example shows that the converse of Proposition 1 is not true in general.

Example 1

Let $X=\{h_1, h_2\}$, $E=\{e_1,e_2\}$ and ${\mathcal {SW}}=\{(F_1,E),(F_2,E)\}$ where $(F_1,E)$ and $(F_2,E)$ are soft sets over X, defined as follows:

$$\begin{aligned} F_1(e_1)= & {} \emptyset , \quad F_1(e_2)=\{h_1\},\\ F_2(e_1)= & {} \{h_2\},\quad F_2(e_2)=\emptyset . \end{aligned}$$

Then ${\mathcal {SW}}_{e_1}$ and ${\mathcal {SW}}_{e_2}$ are weak structures on X while ${\mathcal {SW}}$ is not soft weak structure where $\tilde{\emptyset }\not \in {\mathcal {SW}}$.

Contrary to the soft topology, the union and the intersection of two soft weak structures will be a soft weak structure. Let ${\mathcal {SW}}$ be a soft weak structure over X with respect to E and (F, E) is a soft set over X, we define $\widehat{(F,E)}$ as the soft intersection of all ${\mathcal {W}}$-closed soft sets containing (F, E) and $\widetilde{(F,E)}$ is the soft union of all ${\mathcal {W}}$-open soft subsets of (F, E).

Theorem 1

Let ${\mathcal {SW}}$ be a soft weak structure defined over X with respect to E and (F, E),  (G, E) are two soft sets over X,  then : 

• 1. $\widetilde{(F,E)}\,\tilde{\subset }\, (F,E)\,\tilde{\subset }\,\widehat{(F,E)},$

• 2. If $(F,E)\,\tilde{\subset }\, (G,E),$ then $\widetilde{(F,E)}\,\tilde{\subset }\, \widetilde{(G,E)}$ and $\widehat{(F,E)}\,\tilde{\subset }\,\widehat{(G,E)},$

• 3. $\widetilde{\widetilde{(F,E)}}=\widetilde{(F,E)}$ and $\widehat{\widehat{(F,E)}}=\widehat{(F,E)},$

• 4. $\widetilde{ (F,E)' }=(\widehat{(F,E)})'$ and $\widehat{(F,E)'}=(\widetilde{(F,E)})'$.

Proof

• 1. Obvious.

• 2. If (H, E) is ${\mathcal {W}}$-open soft set and (K, E) is ${\mathcal {W}}$-closed soft set such that $(H,E)\,\tilde{\subset }\, (F,E)$ and $(G,E)\,\tilde{\subset }\, (K,E),$ then $(H,E)\,\tilde{\subset }\,(G,E)$ and $(F,E)\,\tilde{\subset }\, (K,E)$. Thus $\widetilde{(F,E)}\,\tilde{\subset }\, \widetilde{(G,E)}$ and $\widehat{(F,E)}\,\tilde{\subset }\,\widehat{(G,E)}$.

• 3. Let (G, E) be ${\mathcal {W}}$-open soft set. It is clear that $(G,E)\,\tilde{\subset }\, (F,E)$ iff $(G,E)\,\tilde{\subset }\,\widetilde{(F,E)},$ i.e.,  $(G,E)\,\tilde{\subset }\,\widetilde{(F,E)}$ iff $(G,E)\,\tilde{\subset }\,\widetilde{\widetilde{(F,E)}}$. Thus $\widetilde{\widetilde{(F,E)}}=\widetilde{(F,E)}$. Similarly,  we can prove the second part.

• 4. Obvious.

Let ${\mathcal {SW}}$ be a soft weak structure over X with respect to E and (F, E) be a soft set. If (F, E) is ${\mathcal {W}}$-closed (resp. ${\mathcal {W}}$-open) soft set, then $(F,E)=\widehat{(F,E)}$ (resp. $(F,E)=\widetilde{(F,E)}$). The following example shows that the converse need not be true in general.

Example 2

Let $X=\{h_1, h_2, h_3\}$, $E=\{e_1, e_2\}$ and ${\mathcal {SW}}=\{\tilde{\emptyset }, (F_1,E), (F_2,E), (F_3,E)\}$, where

$$\begin{aligned} F_1(e_1)= & {} X \quad F_1(e_2)=\{h_1\} \\ F_2(e_1)= & {} \{h_1,h_2\}\quad F_2(e_2)=\{h_1,h_3\} \\ F_3(e_1)= & {} \{h_2,h_3\}\quad F_3(e_2)=\{h_3\} \end{aligned}$$

and let (G, E) be a soft set defined by $G(e_1)=\emptyset$, $G(e_2)=\{h_2\}$, then $\widehat{(G,E)}=(G,E)$ but (G, E) is not ${\mathcal {W}}$-closed soft set. Similarly, $\widetilde{\,{\tilde{X}}\,}=(H,E)$ but (H, E) is not ${\mathcal {W}}$-open soft set since $(H,E)\not \in {\mathcal {SW}}$ and

$$\begin{aligned} H(e_1)=X, \quad H(e_2)=\{h_1,h_3\}. \end{aligned}$$

Proposition 2

Let ${\mathcal {SW}}$ be a soft weak structure over X with respect to the parameters set E and (F, E) be a soft set,  then : 

• 1. If there exists a ${\mathcal {W}}$-open soft set (G, E) such that $x\in (G,E)\,\tilde{\subset }\, (F,E),$ then $x\in \widetilde{(F,E)}$.

• 2. $x\in \widehat{(F,E)}$ if and only if $(G,E)\,\tilde{\cap }\, (F,E)\,\tilde{\ne }\,\tilde{\emptyset }$ for all $(G,E)\in {\mathcal {SW}}$ such that $x\in (G,E)$.

Proof

• 1. It follows from the definition of $\widetilde{(F,E)}$.

• 2. Suppose that $x\in (G,E)\in {\mathcal {SW}}$ such that $(G,E)\,\tilde{\cap }\, (F,E)\,\tilde{=}\,\tilde{\emptyset }$. Then $(F,E)\,\tilde{\subset }\,(G,E)'$ and thus $x\not \in \widehat{(F,E)}$. Conversely, let $x\not \in \widehat{(F,E)}$. Then there exists a ${\mathcal {W}}$-closed soft set (H, E) with $(F,E)\,\tilde{\subset }\, (H,E)$ and $x\not \in (H,E)$. Therefore $x\in (H,E)'\in {\mathcal {SW}}$ and $(H,E)'\,\tilde{\cap }\, (F,E)\,\tilde{=}\,\tilde{\emptyset }$.

The following example shows that the converse of (1) is not true in general:

Example 3

Let $X=\{h_1,h_2,h_3\}$, $E=\{e_1,e_2\}$ and ${\mathcal {SW}}=\{\tilde{\emptyset }, (F_1,E), (F_2,E), (F_3,E)\}$ such that

$$\begin{aligned} F_1(e_1)= & {} \{h_1\}\quad F_1(e_2)=\{h_2\},\\ F_2(e_1)= & {} \{h_3\}\quad F_2(e_2)=\{h_1,h_3\},\\ F_3(e_1)= & {} \{h_2\}\quad F_3(e_2)=\{h_3\}. \end{aligned}$$

Let (H, E) be a soft set defined by

$$\begin{aligned} H(e_1)=\{h_1,h_2\},\quad H(e_2)=X. \end{aligned}$$

Since $\tilde{\emptyset }$, $(F_1,E)$ and $(F_3,E)$ are the only ${\mathcal {W}}$-open soft subsets of (H, E), then $\widetilde{(H,E)}=(K,E)$ such that

$$\begin{aligned} K(e_1)=\{h_1,h_2\},\quad K(e_2)=\{h_2,h_3\}. \end{aligned}$$

It is clear that $h_2\in (K,E)$ but there isn’t ${\mathcal {W}}$-open soft set (G, E) such that $h_2\in (G,E)\, \tilde{\subseteq }\, (H,E)$.

Proposition 3

Let ${\mathcal {SW}}_Y$ be a relative soft weak structure of ${\mathcal {SW}}$ and (F, E) be a soft set in Y. Then (F, E) is ${\mathcal {W}}_Y$-closed soft set if and only if $(F,E)\,\tilde{=}\,{\tilde{Y}}\,\tilde{\cap }\,(G,E)$ for some ${\mathcal {W}}$-closed soft set (G, E).

Proof

Straightforward.

Proposition 4

Let ${\mathcal {SW}}_Y$ be a relative soft weak structure of ${\mathcal {SW}},$ (F, E) be a soft set on Y. The following statements hold : 

• 1. $\widehat{(G,E)}^Y\,\tilde{=}\,\widehat{(F,E)}\,\tilde{\cap }\, {\tilde{Y}},$

• 2. $\widetilde{(F,E)}\,\tilde{\cap }\,{\tilde{Y}}\, \tilde{\subseteq }\,\widetilde{(F,E)}^Y,$

where $\widehat{(G,E)}^Y$ and $\widetilde{(F,E)}^Y$ denote relative closure and relative interior,  respectively.

Proof

Straightforward.

Definition 3

Let ${\mathcal {SW}}$ be a soft weak structure over X with respect to the parameters set E. The soft set (F, E) is called ${\mathcal {W}}$-dense soft set if $\widehat{(F,E)}\,\tilde{=}\,{\tilde{X}}$.

Example 4

Let $X=\{h_1,h_2,h_3\}$, $E=\{e_1, e_2\}$. If:

• 1. ${\mathcal {SW}}=\{\tilde{\emptyset }\}$, then all non-empty soft subsets are ${\mathcal {W}}$-dense soft sets.

• 2. ${\mathcal {SW}}=\{\tilde{\emptyset }, (F_1,E), (F_2,E)\}$ is a soft weak structure, where $(F_1,E)$ and $(F_2,E)$ are soft sets defined by

  $$\begin{aligned}&F_1(e_1)= \{h_1\},\quad F_1(e_2)=\{h_2\},\\&F_2(e_1)= \{h_1,h_2\},\quad F_2(e_2)=\{h_2,h_3\}. \end{aligned}$$

  Then $(F_1,E)$, $(F_2,E)$, and $\tilde{X}$ are ${\mathcal {W}}$-dense soft sets.

Theorem 2

Let ${\mathcal {SW}}$ be a soft weak structure defined over X with respect to the set of parameters E. A soft set (F, E) is ${\mathcal {W}}$-dense soft set if and only if $\widetilde{~(F,E)'~}\,\tilde{=}\,~\tilde{\emptyset }$.

Proof

Follows directly from Theorem 1(4).

Theorem 3

Let ${\mathcal {SW}}$ be soft weak structure defined over X with respect to the parameters set E and (F, E) be a soft set. Then (F, E) is a ${\mathcal {W}}$-dense soft set if and only if $(F,E)\,\tilde{\cap }\, (G,E)\,\tilde{\ne }\,\tilde{\emptyset }$ whenever $\tilde{\emptyset }\,\tilde{\ne }\,(G,E)\in {\mathcal {SW}}$.

Proof

Suppose that (F, E) is ${\mathcal {W}}$-dense soft set, then $\widehat{(F,E)}\,\tilde{=}\,{\tilde{X}}$. Also, suppose that $(G,E)\in {\mathcal {SW}}$ such that $(G,E)\,\tilde{\cap }\, (F,E)\,\tilde{=}\,\tilde{\emptyset }$. Then for each $x\in (G,E)$, we have $x\not \in \widehat{(F,E)}\,\tilde{=}\,{\tilde{X}}$, which is a contradiction.

Conversely, let $(G,E)\,\tilde{\cap }\, (F,E)\,\tilde{\ne }\,\tilde{\emptyset }$ for all $(G,E)\in {\mathcal {SW}}$ and $\widehat{(F,E)}\,\tilde{\ne }\,{\tilde{X}}$. Then there exists $x\in {\tilde{X}}$ such that $x\not \in \widehat{(F,E)}$. Hence, then there exists ${\mathcal {W}}$-open soft set (G, E) such that $x\in (G,E)$ and $(F,E)\,\tilde{\cap }\, (G,E)\,\tilde{=}\,\tilde{\emptyset }$ which is a contradiction.

Theorem 4

The soft union of two ${\mathcal {W}}$-dense soft sets is ${\mathcal {W}}$-dense soft set.

Proof

Let $(F_1, E)$ and $(F_2,E)$ be two ${\mathcal {W}}$-dense soft sets, then $\widehat{(F_1,E)}\,\tilde{=}\,{\tilde{X}}$ and $\widehat{(F_2,E)}\,\tilde{=}\,{\tilde{X}}$. Since ${\tilde{X}}\,\tilde{=}\,\widehat{(F_1,E)}\,\tilde{\subset }$ 

. Thus $(F_1,E)\tilde{\cup }(F_2,E)$ is ${\mathcal {W}}$-dense soft set.

The soft intersection of two ${\mathcal {W}}$-dense soft sets need not be a ${\mathcal {W}}$-dense soft set as shown by the following example:

Example 5

Let $X=\{h_1,h_2,h_3\}$, $E=\{e_1,e_2\}$ and ${\mathcal {SW}}=\{\tilde{\emptyset },(F_1,E), (F_2,E) \}$ where $(F_1,E)$ and $(F_2,E)$ are soft sets defined by

$$\begin{aligned} F_1(e_1)= & {} \{h_1,h_2\},\quad F_1(e_2)=\{h_3\}, \\ F_2(e_1)= & {} \{h_2,h_3\},\quad F_2(e_2)=\{h_1\}. \end{aligned}$$

Also, let (G, E) and (H, E) are soft sets defined by

$$\begin{aligned} G(e_1)= & {} \{h_1,h_3\},\quad G(e_2)=\{h_1,h_2\}, \\ H(e_1)= & {} \{h_3\},\quad H(e_2)=X. \end{aligned}$$

Then $\widehat{(G,E)}\,\tilde{=}\,\widehat{(H,E)}\,\tilde{=}\,{\tilde{X}}$ and

$\,\tilde{=}\,(F_1,E)'$.

Definition 4

Let ${\mathcal {SW}}$ be a soft weak structure on X with respect to E. A soft set (F, E) is called ${\mathcal {W}}$-nowhere dense soft set if $\widetilde{(\widehat{(F,E)})}\,\tilde{=}\,\tilde{\emptyset }$.

The collection of all ${\mathcal {W}}$-dense soft sets is not a soft weak structure where $\tilde{\emptyset }$ does not belong to this collection.

Theorem 5

The collection of all ${\mathcal {W}}$-nowhere dense soft sets establishes a soft weak structure.

Proof

If

$$\begin{aligned} \widehat{\tilde{\emptyset }}{\,\tilde{=}\,}{\tilde{\bigcap }}\{(F,E)'|~(F,E)~\text{ is }~{\mathcal {W}}\text{-open }\}{\,\tilde{=}\,}(G,E){\,\tilde{\ne }\ }{\tilde{\emptyset },} \end{aligned}$$

then for each $e\in E$ and $h\in G(e)$, we have $h\in F'(e)$ for all (F, E). Thus $h\not \in F(e)$ for all (F, E). So, $(F,E){\,\tilde{\nsubseteq }\,} (G,E)$ for all (F, E). Therefore, $\widetilde{\widehat{\tilde{\emptyset }}}{\,\tilde{=}\,}{\tilde{\emptyset }}$. This means that ${\tilde{\emptyset }}$ is ${\mathcal {W}}$-nowhere dense.

The following example shows that the soft union of two ${\mathcal {W}}$-nowhere dense soft sets need not be ${\mathcal {W}}$-nowhere dense soft set.

Example 6

In Example 5, let (M, E) and (N, E) be two soft sets defined by

$$\begin{aligned} M(e_1)= & {} \{h_3\},\quad M(e_2)=\{h_1,h_2\}, \\ N(e_1)= & {} \{h_1\},\quad N(e_2)=\{h_2,h_3\}. \end{aligned}$$

Then (M, E) and (N, E) are ${\mathcal {W}}$-nowhere dense soft sets but $(M,E)\tilde{\cup }(N,E)=(P,E)$, where

$$\begin{aligned} P(e_1)=\{h_1,h_3\}, \quad P(e_2)=X \end{aligned}$$

is not ${\mathcal {W}}$-nowhere dense soft set.

Theorem 6

Let ${\mathcal {W}}$ be a soft weak structure over X with respect to E. A soft set (F, E) is ${\mathcal {W}}$-nowhere dense soft set if and only if $\widehat{(F,E)}'$ is a ${\mathcal {W}}$-dense soft set.

Proof

Let (F, E) be ${\mathcal {W}}$-nowhere dense soft set. Then $\widetilde{(\widehat{(F,E)})}\,\tilde{=}\,\tilde{\emptyset }$ and hence $(\widetilde{(\widehat{(F,E)})})'\,\tilde{=}\,{\tilde{X}}$. So $\widehat{(\widehat{(F,E)}')}\,\tilde{=}\,{\tilde{X}}$. Then $\widehat{(F,E)}'$ is a ${\mathcal {W}}$-dense soft set.

Conversely, let $\widehat{(F,E)}'$ be ${\mathcal {W}}$-dense soft set. Then $\widehat{(\widehat{(F,E)}')}\,\tilde{=}\,{\tilde{X}}$ and hence $(\widetilde{(\widehat{(F,E)})})'\,\tilde{=}\,{\tilde{X}}$. Thus $\widetilde{(\widehat{(F,E)})}\,\tilde{=}\,\tilde{\emptyset }$ and hence (F, E) is ${\mathcal {W}}$-nowhere dense soft set.

Definition 5

Let ${\mathcal {SW}}$ be a soft weak structure on X with respect to E. Then ${\mathcal {SW}}$ is called separable if and only if there exists a countable ${\mathcal {W}}$-dense soft set on X, i.e., $\widehat{(F,E)}\,\tilde{=}\,\tilde{X}$ where (F, E) is countable soft set.

Example 7

Let $\mathcal {SR}$ be the family of all soft real sets. Then $\mathcal {SR}$ is a separable soft weak structure over the universe R (R is the set of real numbers) since (Q, E) is countable soft set (where Q is the set of rational numbers) and admits the condition $\widehat{(Q,E)}\,\tilde{=}\,\tilde{R}$.

Theorem 7

The relative soft weak structure of separable soft weak structure is separable.

Proof

From Proposition 1, the proof is clear.

4 Separation axioms

Definition 6

Let ${\mathcal {SW}}$ be a soft weak structure on X with respect to E. A soft set (F, E) is called $D_{\mathcal {W}}$-soft set if there exist two ${\mathcal {W}}$-open soft sets (G, E) and (H, E) such that $(G,E)\,\tilde{\ne }\,{\tilde{X}}$ and $(F,E)\,\tilde{=}\,(G,E)\tilde{-}(H,E)$.

Remark 1

It is clear that each ${\mathcal {W}}$-open soft set that is different from the absolute soft set is $D_{\mathcal {W}}$-soft set (By putting $(F,E)\,\tilde{=}\,(G,E)$ and $(H,E)\,\tilde{=}\,\tilde{\emptyset }$).

Definition 7

A soft weak structure ${\mathcal {SW}}$ is called:

• 1. ${\mathcal {W}}$-$D_0$ (resp. ${\mathcal {W}}$-$T_0$) if for each $x,y\in X$ such that $x\ne y$, there exists a $D_{\mathcal {W}}$-soft set (resp. ${\mathcal {W}}$-open soft set) (F, E) such that $x\in (F,E)$ and $y\not \in (F,E)$ or $x\not \in (F,E)$ and $y\in (F,E)$.

• 2. ${\mathcal {W}}$-$D_1$ (resp. ${\mathcal {W}}$-$T_1$) if for each $x,y\in X$ such that $x\ne y$, there exist $D_{\mathcal {W}}$-soft set (resp. ${\mathcal {W}}$-open soft sets) (F, E) and (G, E) such that $x\in (F,E)$, $y\not \in (F,E)$ and $x\not \in (G,E)$, $y\in (G,E)$.

• 3. ${\mathcal {W}}$-$D_2$ (resp. ${\mathcal {W}}$-$T_2$) if for each $x,y\in X$ such that $x\ne y$, there exist $D_{\mathcal {W}}$-soft sets (resp. ${\mathcal {W}}$-open soft sets) (F, E) and (G, E) such that $x\in (F,E)$, $y\in (G,E)$ and $(F,E)\,\tilde{\cap }\,(G,E)\,\tilde{=}\,\tilde{\emptyset }$.

Theorem 8

A soft weak structure ${\mathcal {SW}}$ is ${\mathcal {W}}$-$D_0$ if and only if ${\mathcal {W}}$-$T_0$.

Proof

Let ${\mathcal {W}}$ be a ${\mathcal {W}}$-$D_0$, then for any $x,y\in X$ such that $x\ne y$ there exist a $D_{\mathcal {W}}$-soft set (F, E) such that $x\in (F,E)$ and $y\not \in (F,E)$. Suppose that $(F,E)\,\tilde{=}\,(G,E)\tilde{-}(H,E)$ such that $(G,E)\,\tilde{\ne }\,{\tilde{X}}$ and (G, E), (H, E) are ${\mathcal {W}}$-open soft sets, then $x\in (G,E)$ and the cases:

• (i) $x\in (G,E)$ and $y\not \in (G,E)$.

• (ii) $y\in (H,E)$ and $x\not \in (H,E)$.

Hence ${\mathcal {SW}}$ is a ${\mathcal {W}}$-$T_0$.

The second part of the proof follows from the fact that each ${\mathcal {W}}$-open soft set that is different from the absolute soft set is $D_{\mathcal {W}}$-soft set.

Corollary 1

If a soft weak structure ${\mathcal {SW}}$ is ${\mathcal {W}}$-$D_1,$ then it is ${\mathcal {W}}$-$T_0$.

The following diagram summarizes the relationships between the introduced separation axioms in soft weak structures.

Example 8

Let $X=\{h_1,h_2,h_3\}$, $E=\{e_1,e_2\}$. If:

• 1. ${\mathcal {SW}}=\{\tilde{\emptyset }, (F_1,E), (F_2,E), (F_3,E)\}$ is a soft weak structure on X where $(F_1,E)$, $(F_2,E)$ and $(F_3,E)$ are soft sets defined as follows:

  $$\begin{aligned} F_1(e_1)= & {} \{h_1, h_2\},\quad F_1(e_2)=\{h_2\}, \\ F_2(e_1)= & {} \{h_1,h_3\},\quad F_2(e_2)=\{h_1\},\\ F_3(e_1)= & {} \{h_2,h_3\},\quad F_3(e_2)=\{h_3\}. \end{aligned}$$

  Then ${\mathcal {SW}}$ is ${\mathcal {W}}$-$T_1$ and ${\mathcal {W}}$-$D_2$ but it is not ${\mathcal {W}}$-$T_2$.

• 2. ${\mathcal {SW}}=\{\tilde{\emptyset }, (F_1,E), (F_2,E)\}$ is a soft weak structure on X where $(F_1,E)$ and $(F_2,E)$ are soft sets defined as follows:

  $$\begin{aligned} F_1(e_1)= & {} \{h_1\},\quad F_1(e_2)=\{h_1\}, \\ F_2(e_1)= & {} \{h_1,h_2\},\quad F_2(e_2)=X. \end{aligned}$$

  Then ${\mathcal {SW}}$ is ${\mathcal {W}}$-$T_0$ (resp. ${\mathcal {W}}$-$D_0$) but it is not ${\mathcal {W}}$-$T_1$ (resp. ${\mathcal {W}}$-$D_1$).

• 3. ${\mathcal {SW}}=\{\tilde{\emptyset }, (F_1,E), (F_2,E), (F_3,E)\}$ is a soft weak structure on X where $(F_1,E)$, $(F_2,E)$ and $(F_3,E)$ are soft sets defined as follows:

  $$\begin{aligned} F_1(e_1)= & {} \{h_1\},\quad F_1(e_2)=\{h_1\}, \\ F_2(e_1)= & {} \{h_1,h_2\},\quad F_2(e_2)=\{h_1,h_2\},\\ F_3(e_1)= & {} \{h_2,h_3\},\quad F_3(e_2)=\{h_3\}. \end{aligned}$$

  Then ${\mathcal {SW}}$ is ${\mathcal {W}}$-$D_2$ and ${\mathcal {W}}$-$D_1$ but is not ${\mathcal {W}}$-$T_1$.

• 4. ${\mathcal {SW}}=\{\tilde{\emptyset }, (F_1,E), (F_2,E), (F_3,E)\}$ is a soft weak structure on X where $(F_1,E)$, $(F_2,E)$ and $(F_3,E)$ are soft sets defined as follows:

  $$\begin{aligned} F_1(e_1)= & {} \{h_1, h_2\},\quad F_1(e_2)=\{h_2,h_3\}, \\ F_2(e_1)= & {} \{h_1,h_3\},\quad F_2(e_2)=\{h_1\},\\ F_3(e_1)= & {} \{h_2,h_3\},\quad F_3(e_2)=\{h_3\}. \end{aligned}$$

  Then ${\mathcal {SW}}$ is ${\mathcal {W}}$-$D_1$ but it is not ${\mathcal {W}}$-$D_2$.

Theorem 9

If a soft weak structure ${\mathcal {SW}}$ is ${\mathcal {W}}$-$T_0,$ then for each $x,y\in X$ such that $x\ne y,$ we have $\widehat{(x,E)}\,\tilde{\ne }\,\widehat{(y,E)}$.

Proof

Let ${\mathcal {SW}}$ be ${\mathcal {W}}$-$T_0$ and $x,y\in X$ such that $x\ne y$. Then there exists ${\mathcal {W}}$-open soft set (F, E) such that $x\in (F,E)$ and $y\not \in (F,E)$. Therefore $(F,E)'$ is ${\mathcal {W}}$-closed soft set such that $x\not \in (F,E)'$ and $y\in (F,E)'$. Since $\widehat{(y,E)}$ is the intersection of all ${\mathcal {W}}$-closed soft subsets that contain y, then $\widehat{(y,E)}\,\tilde{\subset }\, (F,E)'$ and hence $x\not \in \widehat{(y,E)}$. Thus $\widehat{(x,E)}\,\tilde{\ne }\,\widehat{(y,E)}$.

Corollary 2

If $\widehat{(x,E)}$ is ${\mathcal {W}}$-closed soft set for each $x\in X,$ then the converse of Theorem 9 is true.

Proof

Suppose that $z\in X$ such that $z\in \widehat{(x,E)}$ and $z\not \in \widehat{(y,E)}$. If $x\in \widehat{(y,E)}$, then $\widehat{(x,E)}\, \tilde{\subseteq }\,\widehat{(y,E)}$ which is a contradiction since $z\not \in \widehat{(y,E)}$. Thus $\widehat{(y,E)}'$ is ${\mathcal {W}}$-open soft set such that $x\in \widehat{(y,E)}'$ and $y\not \in \widehat{(y,E)}'$. Hence ${\mathcal {SW}}$ is ${\mathcal {W}}$-$T_0$.

Theorem 10

A soft weak structure ${\mathcal {SW}}$ is ${\mathcal {W}}$-$T_1$ if (x, E) is ${\mathcal {W}}$-closed soft set for all $x\in X$.

Proof

Let (z, E) be ${\mathcal {W}}$-closed soft set for each $z\in X$ and let $x,y\in X$ such that $x\ne y$. Then $(x,E)'$ and $(y,E)'$ are ${\mathcal {W}}$-open soft sets such $y\in (x,E)'$, $x\not \in (x,E)'$ and $y\not \in (y,E)'$, $x\in (y,E)'$. Hence ${\mathcal {SW}}$ is ${\mathcal {W}}$-$T_1$.

Corollary 3

The converse of Theorem 10 is true if the union of ${\mathcal {W}}$-open soft sets is ${\mathcal {W}}$-open soft set.

Theorem 11

The relative soft weak structure of ${\mathcal {W}}$-$T_2$ (resp. ${\mathcal {W}}$-$T_1,$ ${\mathcal {W}}$-$T_0)$ soft weak structure is ${\mathcal {W}}$-$T_2$ (resp. ${\mathcal {W}}$-$T_1,$ ${\mathcal {W}}$-$T_0).$

Proof

Let ${\mathcal {SW}}_Y$ be a relative soft weak structure of ${\mathcal {SW}}$ where $Y\subseteq X$ and let x, $y\in Y$ such that $x\ne y$. Since ${\mathcal {SW}}$ is a ${\mathcal {W}}$-$T_2$, then there exist two soft sets (F, E) and (G, E) on X such that $x\in (F,E)$, $y\in (G,E)$ and $(F,E)\,\tilde{\cap }\, (G,E)\,\tilde{=}\,\tilde{\emptyset }$. Thus $x\in (F,E)\,\tilde{\cap }\,{\tilde{Y}}=(^YF,E)$, $y\in (G,E)\,\tilde{\cap }\,{\tilde{Y}}(^YG,E)$ and

Since $(^YF,E)$ and $(^YG,E)$ are ${\mathcal {W}}_Y$-open soft sets, then ${\mathcal {SW}}_Y$ is a ${\mathcal {W}}_Y$-$T_2$.

Similarly, we can prove the theorem in the cases ${\mathcal {W}}$-$T_1$, ${\mathcal {W}}$-$T_0$ and ${\mathcal {W}}$-$D_i$; $i=0,1,2$ (note that for the cases ${\mathcal {W}}$-$D_i$; $i=0,1,2$, we will use the fact that if (F, E) is $D_{{\mathcal {W}}}$-soft set, then $(^YF,E)$ is $D_{{\mathcal {W}}_Y}$-soft set).

5 Soft compactness

A family $\mathcal {C}=\{(F_i,E)\}_{i\in \Gamma }$ of ${\mathcal {W}}$-open soft sets is called a ${\mathcal {W}}$-open cover of the soft set (F, E) if $(F,E)\, \tilde{\subseteq }\,\bigcup _{i\in \Gamma }(F_i,E)$. A finite subcover is a finite subfamily of $\mathcal {C}$ which also ${\mathcal {W}}$-open cover of (F, E). Throughout this section we consider that ${\mathcal {SW}}$ is a strong soft weak structure.

Definition 8

A soft weak structure ${\mathcal {SW}}$ is called soft compact if each ${\mathcal {W}}$-open cover of ${\tilde{X}}$ has a finite subcover. A soft set (F, E) is said to be soft compact if each ${\mathcal {W}}$-open cover of it has a finite subcover.

Theorem 12

Let ${\mathcal {SW}}$ be a soft compact,  then ${\mathcal {SW}}_e$ is compact weak structure for each $e\in E$.

Proof

Let ${\mathcal {SW}}$ be a soft compact and $\{F_i(e)\}_{i\in \Gamma }$ be a family of ${\mathcal {SW}}_e$-open soft sets that covers X, then $X\subseteq \bigcup _{i\in \Gamma }F_i(e)$ for each $e\in E$. Therefore ${\tilde{X}}\, \tilde{\subseteq }\,(F_i,E)$. Since ${\mathcal {SW}}$ is soft compact, then there exists a finite subfamily of ${\mathcal {W}}$-open soft sets $\{(F_1,E), (F_2,E),\dots ,(F_n,E)\}$ such that ${\tilde{X}}\, \tilde{\subseteq }\, (F_1,E)\tilde{\cup }(F_2,E)\tilde{\cup }\cdots \tilde{\cup }(F_n,E)$. Hence $\{F_1(e),F_2(e),\ldots ,F_n(e)\}$ is a finite subcover of X. Thus ${\mathcal {SW}}_e$ is compact.

Theorem 13

Let ${\mathcal {SW}}_1$ and ${\mathcal {SW}}_2$ are two soft weak structures on X with respect to E. If ${\mathcal {SW}}_1\subseteq {\mathcal {SW}}_2$ and ${\mathcal {SW}}_2$ is a soft compact,  then ${\mathcal {SW}}_1$ is soft compact.

Proof

Let $\{(F_i,E)\}_{i\in \Gamma }$ be a ${\mathcal {W}}_1$-open cover of ${\tilde{X}}$. Since ${\mathcal {SW}}_1\subseteq {\mathcal {SW}}_2$, then $\{(F_i,E)\}_{i\in \Gamma }$ be ${\mathcal {W}}_2$-open cover of ${\tilde{X}}$. But ${\mathcal {SW}}_2$ is soft compact, then ${\tilde{X}}\, \tilde{\subseteq }\, (F_{1}, E)\tilde{\cup }(F_{2}, E)\tilde{\cup }\cdots \tilde{\cup }(F_{n},E)$. Therefore, ${\mathcal {SW}}_1$ is soft compact.

Theorem 14

Let ${\mathcal {SW}}_1$ and ${\mathcal {SW}}_2$ are two soft weak structures on X with respect to E. If ${\mathcal {SW}}_1$ and ${\mathcal {SW}}_2$ are soft compact,  then ${\mathcal {SW}}_1\cup {\mathcal {SW}}_2$ is also soft compact.

Proof

Let $\mathcal {C}$ be a ${\mathcal {W}}$-open cover of ${\mathcal {SW}}_1\cup {\mathcal {SW}}_2$, then $\mathcal {C}$ is ${\mathcal {W}}$-open cover of ${\mathcal {SW}}_1$ and ${\mathcal {SW}}_2$. Since ${\mathcal {SW}}_1$ and ${\mathcal {SW}}_2$ are soft compact, then there exist finite subcovers $\{(F_1,E), (F_2,E),\ldots ,(F_n,E)\}$ and $\{(G_1,E),(G_2,E),\ldots ,(G_n,E)\}$ of ${\mathcal {SW}}_1$ and ${\mathcal {SW}}_2$, respectively. Then

$$\begin{aligned}&\{(F_1,E),(F_2,E),\ldots ,(F_n,E),(G_1,E),(G_2,E),\\&\quad \ldots ,(G_n,E)\} \end{aligned}$$

is a finite subcover of $\mathcal {C}$ for ${\mathcal {SW}}_1\cup {\mathcal {SW}}_2$. Thus ${\mathcal {SW}}_1\cup {\mathcal {SW}}_2$ is soft compact.

Theorem 15

Let ${\mathcal {SW}}_Y$ be the relative soft weak structure of ${\mathcal {SW}}$ where $Y\subseteq X$. Then ${\mathcal {SW}}_Y$ is soft compact if and only if ${\tilde{Y}}$ is soft compact.

Proof

Let ${\mathcal {SW}}_Y$ be a soft compact and $\{(F_i,E)\}_{i\in \Gamma }$ be a ${\mathcal {W}}$-open cover of ${\tilde{Y}}$. Then $\{(^YF_i,E)\}_{i\in \Gamma }$ is ${\mathcal {W}}_Y$-open cover of ${\tilde{Y}}$. Then ${\tilde{Y}}\,\tilde{\subset }\, (^YF_{1},E)\tilde{\cup }\cdots \tilde{\cup }(^YF_{n},E)$. Therefore $\{(F_{i},E)\}_{i=1}^{n}$ is a finite subcover of ${\tilde{Y}}$. Thus ${\tilde{Y}}$ is soft compact.

Conversely, let ${\tilde{Y}}$ be a soft compact and $\{(^YF_i,E)\}_{i\in \Gamma }$ be a ${\mathcal {W}}_Y$-open cover of ${\tilde{Y}}$. Then $\{(F_i,E)\}_{i\in \Gamma }$ is ${\mathcal {W}}$-open cover of ${\tilde{Y}}$. Therefore, ${\tilde{Y}}\,\tilde{\subset }\,(F_{1},E)\tilde{\cup }\cdots \tilde{\cup }(F_{n},E)$. Thus $\{(^YF_{i},E)\}_{i=1}^n$ is a finite subcover of ${\tilde{Y}}$. Hence ${\mathcal {SW}}_Y$ is soft compact.

Theorem 16

Every ${\mathcal {W}}$-closed relative soft weak structure of compact soft weak structure is soft compact.

Proof

Let ${\mathcal {SW}}_Y$ be a relative soft weak structure of ${\mathcal {SW}}$ where $Y\subseteq X$ and ${\tilde{Y}}$ is a ${\mathcal {W}}$-closed soft set and let $\{(F_i,E)\}_{i\in \Gamma }$ be a ${\mathcal {W}}$-open cover of ${\tilde{Y}}$. Since ${\tilde{Y}}'$ is a ${\mathcal {W}}$-open soft set, then $\{(F_i,E)\}_{i\in \Gamma }\cup {\tilde{Y}}'$ is a ${\mathcal {W}}$-open cover of ${\tilde{X}}$. Therefore ${\tilde{X}}\,\tilde{\subset }\, (F_{1},E)\tilde{\cup }\cdots \tilde{\cup }(F_{n},E)\tilde{\cup }{\tilde{Y}}'$. Thus $\{(F_{i},E)\}_{i=1}^n$ is a finite subcover of ${\tilde{Y}}$.

6 Conclusion

Soft set theory is an important and powerful mathematical tool to study the uncertainty. It can be applied in many different fields of mathematics which include the probability theory, game theory, smoothness of functions, operations research, algebra, Riemann integration, Perron integration, topology, measurement theory and also other scientific fields like computer science.

In this paper, we introduced the notion of soft weak structures as a generalization of soft topology, generalized soft topology and soft minimal structures and discussed some of its properties. Also, we introduced soft separation axioms and soft compactness and established several interesting theorems and examples. We hope that our findings help the researchers to enhance and promote their studies on soft topology to carry out a general framework for their applications in life.