New approach of soft M-open sets in soft topological spaces

1 Introduction

Soft set was created by Molodtsov (1999) in order to deal with challenges arising from incomplete information. He had indicated a few applications of soft theory for finding solutions of problems in economics, medical science and so on. Recently, there has been a significant increase in the number of papers published on soft sets and their applications in various fields (Al-Shami and Abo-Tabl, 2021, Alzahrani et al., 2022). Soft sets and their applications have advanced significantly in recent years (Alzahrani et al., 2022; Fatimah and Alcantud, 2021). The concept of S $\tau s$ was introduced by Shabir and Naz (2011), who study them as existing in an initial universe with a fixed set of parameters. Furthermore, (Maji et al., 2003) presented various operations on soft sets, and so far, several of the fundamental features of these operations have been exposed. Soft open ( $\mathcal{S}$ o) sets, soft interior, soft closed sets, soft closure, and soft separation axioms were defined by the authors. M-open sets were introduced into general topology by (EL-Maghrabi and AL-Juhani, 2011). The purpose of this article is to conduct a theoretical investigation of the new set termed $\mathcal{S}$ Mo and $\mathcal{S}$ Mc sets over $\mathcal{S}$ τs and to analysis some of their properties. Also, in this research, we express soft operations by ‘∼’, soft closed ( $\mathcal{S}$ c) set, soft open cover by $\mathcal{S}$ o cover.

Throughout this entire article, we will refer to ${\mathrm{U}}_1$ as an initial universal set, ( $U_1$ , τ, H) is a $\mathcal{S}$ τs and $(F_1,H_1$ ) is a soft set over $U_1$ .

2 Preliminaries

Except where else stated, $U_1$ and W denoted a $\mathcal{S}\tau s$ with ( $U_1$ , τ, S) and (W,ν, T). Additionally, a soft mapping f: $U_1$  → W, since f: ( $U_1$ , τ, S) →(W,ν, T), u: $U_1$  → W and p:S → T denote assumed mappings.

$\mathit{And}$ ( $F_1,H_1$ ) $\stackrel{\sim }{\cap }$ ( $F_2$ , D) =(V,J) is a soft sets defined as J = $H_1$ $\stackrel{\sim }{\cap }$ D, and V(s) = $F_1$ (s) $\stackrel{\sim }{\cap }$ $F_2$ (s), $\forall$ s ∈ J.

(i) The soft closure of ( $F_2$ , $H_1$ ) is $\stackrel{\sim }{s}$ cl( $F_2$ , $H_1$ ) = $\stackrel{\sim }{\cap }$ {(D, $H_1$ ): (D, $H_1$ ) is $\mathcal{S}$ c set and ( $F_2$ , $H_1$ ) $\stackrel{\sim }{\subseteq }$ (D, $H_1$ }.

(ii) The soft interior of ( $F_2$ , $H_1$ ) is s̃int( $F_2$ , $H_1$ ) = $\stackrel{\sim }{\cup }$ {(D, $H_1$ ): (D, $H_1$ ) is $\mathcal{S}$ o set and (D, $H_1$ ) $\stackrel{\sim }{\subseteq }$ ( $F_2$ , $H_1$ )}.

(i) ( $F_1,H_1$ ) is $\mathcal{S}$ c set iff ( $F_1,H_1$ ) = s̃cl( $F_1,H_1$ ).

(ii) ( $F_2$ , $H_1$ ) is $\mathcal{S}$ o set iff ( $F_2$ , $H_1$ ) = s̃int( $F_2$ , $H_1$ ).

Akdağ and Özkan (2014a,b) defined soft (pre-open(closed), α-open(closed), Chen (2013)​ defined soft (semi-open(closed)) and Arockiarani and Arokialancy (2013) defined soft (regular open(closed), β-open(closed)) sets, for example.

3 M-open and M-closed soft sets

We define soft θ-semi-open and $\mathcal{S}$ Mo sets in $\mathcal{S}\tau s$ and study some of their characteristics.

Mukherjee and Debnath (2017) defined soft (δ-pre open (closed), δ-pre interior and δ-pre closure) sets, for example.

(i) soft θ-semi ( $\mathcal{S}$ θ-semi) open set iff ( $F_1$ , H) $\stackrel{\sim }{\subseteq }$ cl(int_θ ( $F_1$ , H)).

(ii) $\mathcal{S}$ θ-semi ( $\mathcal{S}$ θ-semi) closed set iff ( $F_1$ , H) $\stackrel{\sim }{\supseteq }$ int(cl_θ ( $F_1$ , H)).

(i) soft M-open (briefly, $\mathcal{S}$ Mo) set if ( $F_1,H$ ) $\stackrel{\sim }{\subseteq }$ cl(int_θ( $F_1,H$ )) $\stackrel{\sim }{\cup }$ int(cl_δ( $F_1,H$ )),

(ii) soft M-closed (briefly, $\mathcal{S}$ Mc) set if (( $F_1,H$ ) $\stackrel{\sim }{\supseteq }$ int(cl_θ( $F_1,H$ )) $\stackrel{\sim }{\cap }$ cl(int_δ( $F_1,H$ )).

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In the next example, we will prove that $\mathcal{S}$ Mo se is not $\mathcal{S}$ δ-preopen.

F₁(s₁) = {u₁, u₂} F₁(s₂) = {u₁, u₂}.

F₂(s₁) = {u₂} F₂(s₂) = {u₁, u₃}.

F₃(s₁) = {u₂, u₃} F₃(s₂) = {u₁}.

F₄(s₁) = {u₂} F₄(s₂) = {u₁}.

F₅(s₁) = {u₁, u₂} F₅(s₂) = U.

F₆(s₁) = U F₆(s₂) = {u₁, u₂}.

F₇(s₁) = {u₂, u₃} F₇(s₂) = {u₁, u₃}.

( $U_1$ , τ, S) is a $\mathcal{S}$ τs, and $\mathcal{S}$ c sets are $\stackrel{\sim }{U_1}$ , $\stackrel{\sim }{\varphi }$ , (F₁, S)^c, (F₂, S)^c, (F₃, S)^c, …., (F₇, S)^c.

Let ( $F_{1,}{H}_1$ )= {(h₁, {u₁, u₂}, (h₂, {u₂}), (h₃, {u₁, u₃})}. Then.

cl(int_θ ( $F_1$ , $H_1$ )) $\stackrel{\sim }{\cup }$ int(cl_δ ( $F_1$ , $H_1$ )) =  $\stackrel{\sim }{U_1}$ and ( $F_1,H_1$ ) $\stackrel{\sim }{\subseteq }$ cl(int_θ( $F_1,H_1$ )) $\stackrel{\sim }{\cup }$ int(cl_δ( $F_1,H_1$ )), therefore, ( $F_1$ , $H_1$ ) is $\mathcal{S}$ Mo set but not $\mathcal{S}$ δ-preopen.

(i) ( $F_1,K$ ) is a $\mathcal{S}$ Mo set iff ( $F_1,K$ )^c is a $\mathcal{S}$ Mc set.

(ii) ( $F_1,K$ ) is a $\mathcal{S}$ Mc set iff ( $F_1,K$ )^c is a $\mathcal{S}$ Mo set.

Proof. Obvious.

1. $\mathcal{S}$ M-closure of a soft set is defined as $\mathcal{S}$ Mcl( $F_1,H_1$ ) = $\stackrel{\sim }{\cap }$ {(V, $H_1$ )⊇^̃ ( $F_1,H$ ): (V,H) is a $\mathcal{S}$ Mc set of $U_1$ }.

2. $\mathcal{S}$ M-interior of a soft set is defined as $\mathcal{S}$ Mint( $F_1,H_1$ ) = $\stackrel{\sim }{\cup }$ {( $F_2$ , $H_1$ ) $\stackrel{\sim }{\subseteq }$ ( $F_1,H_1$ ):( $F_2$ , $H_1$ ) is a $\mathcal{S}$ Mo set of $U_1$ }.

1. $\mathcal{S}$ Mcl( $F_1,H_1$ )^c =  $\stackrel{\sim }{U_1}$ - $\mathcal{S}$ Mint( $F_1,H_1$ ),

2. $\mathcal{S}$ Mint( $F_1,H_1$ )^c =  $\stackrel{\sim }{U_1}$ - $\mathcal{S}$ Mcl( $F_1,H_1$ ).

(ii) Let ( $F_2$ , $H_1$ ) be a $\mathcal{S}$ Mo set. Then for a $\mathcal{S}$ Mc set ( $F_1,H_1$ ) $\stackrel{\sim }{\subseteq }$ ${(F}_2$ , $H_1$ )^c,

${(F}_2$ , $H_1$ ) $\stackrel{\sim }{\subseteq }$ ( $F_1,H_1$ )^c. Now, $\mathcal{S}$ Mcl( $F_1,H_1$ ) = $\stackrel{\sim }{\cap }$ {( $F_2$ , $H_1$ )^c: $(F_2$ , $H_1$ ) is a $\mathcal{S}$ Mo set ,( $F_2$ , $H_1$ ) $\stackrel{\sim }{\subseteq }$ ( $F_1$ , $H_1$ )^c = $\stackrel{\sim }{U_1}$ - $\stackrel{\sim }{\cup }$ {( $F_2$ , $H_1$ ): ( $F_2$ , $H_1$ ) is a $\mathcal{S}$ Mo set and ( $F_2$ , $H_1$ ) $\stackrel{\sim }{\subseteq }$ ( $F_1,H_1$ )^c }= $\stackrel{\sim }{U_1}$ - $\mathcal{S}$ Mint( $F_1,H_1$ )^c. So, $\mathcal{S}$ Mint( $F_1,H_1$ )^c =  $\stackrel{\sim }{U_1}$ - $\mathcal{S}$ Mcl( $F_1,H_1$ ).

Proof. Obvious.

1. $\mathcal{S}$ Mcl( $\stackrel{\sim }{\varphi }$ ) =  $\stackrel{\sim }{\varphi }$ ,

2. $\mathcal{S}$ Mint( $\stackrel{\sim }{\varphi }$ ) =  $\stackrel{\sim }{\varphi }$ ,

3. $\mathcal{S}$ Mcl( $F_1,H$ ) is a $\mathcal{S}$ Mc set,

4. $\mathcal{S}$ Mint( $F_1,H$ ) is a $\mathcal{S}$ Mo set,

5. $\mathcal{S}\mathrm{M}cl(F_1,H)\stackrel{\sim }{\subseteq }\mathcal{S}\mathrm{M}cl(F_2,H)if(F_1,H)\stackrel{\sim }{\subseteq }(F_2,H),$

6. $\mathcal{S}\mathrm{M}int(F_1,H)\stackrel{\sim }{\subseteq }\mathcal{S}\mathrm{M}int(F_2,H)if(F_1,H)\stackrel{\sim }{\subseteq }(F_2,H),$

7. $\mathcal{S}$ Mcl( $\mathcal{S}$ Mcl( $F_1,H$ )) =  $\mathcal{S}$ Mcl( $F_1,H$ ),

8. $\mathcal{S}$ Mint( $\mathcal{S}$ Mint( $F_1,H$ )) =  $\mathcal{S}$ Mint( $F_1,H$ ).

1. $\mathcal{S}$ Mcl (( $G_1,K_1$ ) $\stackrel{\sim }{\cup }$ ( $G_2$ , $K_1$ )) $\stackrel{\sim }{\supseteq }$ $\mathcal{S}$ Mcl( $G_1,K_1$ ) $\stackrel{\sim }{\cup }$ $\mathcal{S}$ Mcl( $G_2$ , $K_1$ ).

2. $\mathcal{S}$ Mcl(( $G_1,K_1$ ) $\stackrel{\sim }{\cap }$ $(G_2$ , $K_1$ )) $\stackrel{\sim }{\subseteq }$ $\mathcal{S}$ Mcl( $G_1,K_1$ ) $\stackrel{\sim }{\cap }$ $\mathcal{S}$ Mcl( $G_2$ , $K_1$ ).

Therefore, $\mathcal{S}$ Mcl(( $G_1,K_1$ ) $\stackrel{\sim }{\cup }$ ${(G}_2$ , $K_1$ )) $\stackrel{\sim }{\supseteq }$ $\mathcal{S}$ Mcl( $G_1,K_1$ ) $\stackrel{\sim }{\cup }$ $\mathcal{S}$ Mcl( $G_2$ , $K_1$ ).

(ii)Similar to (i).

1. $\mathcal{S}$ Mint( $L_1,H$ ) $\stackrel{\sim }{\cup }$ ${(L}_2$ , H)) $\stackrel{\sim }{\supseteq }$ $\mathcal{S}$ Mint( $L_1,H$ ) $\stackrel{\sim }{\cup }$ $\mathcal{S}$ Mint( $L_1,H$ )

2. $\mathcal{S}$ Mint(( $L_1,H$ ) $\stackrel{\sim }{\cap }$ ${(L}_2$ , H)) $\stackrel{\sim }{\subseteq }$ $\mathcal{S}$ Mint( $L_1,H$ ) $\stackrel{\sim }{\cap }$ $\mathcal{S}$ Mint( $L_2$ , H)

Proof. Obvious.

(i) If int_θ( $F_1,H$ ) =  $\stackrel{\sim }{\varphi }$ , then ( $F_1,H$ ) is a $\mathcal{S}$ δ-preopen set.

(ii) If cl_δ( $F_1,H$ ) =  $\stackrel{\sim }{\varphi }$ , then ( $F_1,H$ ) is a $\mathcal{S}$ θ-semiopen set.

Proof. Obvious.

(i) The union of any $\mathcal{S}$ Mo sets is a $\mathcal{S}$ Mo set,

(ii) The intersection of any $\mathcal{S}$ Mc sets is a $\mathcal{S}$ Mc set.

[cl(int_θ( $\stackrel{\sim }{\cup }$ ( $F_1,H$ ) _α)) $\stackrel{\sim }{\cup }$ int(cl_δ( $\stackrel{\sim }{\cup }$ ( $F_1,H$ ) _α))]. Thus $\stackrel{\sim }{\cup }$ ( $F_1,H$ ) _α is $\mathcal{S}$ Mo set.

(ii) As (i) by taking the complements.

(F₁, S) = {(s₁,{u₁}), (s₂, {u₂})}, (F₂, S) = {(s, U), (s₂, { u₂})}.

(F₃, S) = {(s₁,{u₁}), (s₂, $U_1$ )}.

( $U_1$ , τ, S) is a $\mathcal{S}$ τs. So, the soft sets (G₁, S), (G₂, S) which defines as (G₁, S) = {(s₁,{u₂}), (s₂, {u₁})} and (G₂, S) ={(s₁,{u₁, u₂}), (s₂, {u₁})} are $\mathcal{S}$ Mo sets, but (G₁, S) $\stackrel{\sim }{\cap }$ (G₂, S)=(K, S) is not a $\mathcal{S}$ Mo set.

And, the soft sets (G₃, S) and (G₄, S) which defines as(G₃, S) = {(s₁,{u₁}), (s₂, {u₁})} and (G₄, S) ={((s₂, {u₂})} are $\mathcal{S}$ Mc sets, but (G₃, S) $\stackrel{\sim }{\cup }$ (G₄, S)= {(s₁,{u₁}), (s₂, {u₁, u₂})} =(M, S) is not a $\mathcal{S}$ Mc set.

1. Each $\mathcal{S}$ δ-preopen set is $\mathcal{S}$ Mo.

2. Every $\mathcal{S}$ θ-semi-open set is $\mathcal{S}$ Mo.

( $F_1,H$ ) $\stackrel{\sim }{\subseteq }$ int(cl_δ ( $F_1,H$ )), therefore ( $F_1,H$ ) $\stackrel{\sim }{\subseteq }$ [int(cl_δ ( $F_1,H$ )) $\stackrel{\sim }{\cup }$ int_δ ( $F_1,H$ ))] $\stackrel{\sim }{\subseteq }$ [int(cl_δ ( $F_1,H$ )) $\stackrel{\sim }{\cup }$ cl(int_δ ( $F_1,H$ ))]. Hence, ( $F_1,H$ ) is a $\mathcal{S}$ Mo set.

(ii) Suppose that ( $F_1,H$ ) is a $\mathcal{S}$ θ-semio-penset in a $\mathcal{S}$ τs. Then, ( $F_1,H$ ) $\stackrel{\sim }{\subseteq }$ cl(int_θ( $F_1,H$ )) which implies that ( $F_1,H$ ) $\stackrel{\sim }{\subseteq }$ [cl(int_θ( $F_1,H$ )) $\stackrel{\sim }{\cup }$ int( $F_1,H$ )] $\stackrel{\sim }{\subseteq }$ [cl(int_θ( $F_1,H$ )) $\stackrel{\sim }{\cup }$ int(cl_δ ( $F_1,H$ ))]. Thus ( $F_1,H$ ) is a $\mathcal{S}$ Mo set.

4 Soft M-continuity and soft M-functions

In this section, we define $\mathcal{S}$ M-continuous functions, $\mathcal{S}$ M-irresolute functions, $\mathcal{S}$ M-open function ( $\mathcal{S}$ Mof) and $\mathcal{S}$ M- closed function ( $\mathcal{S}$ Mcf). Also, we study some of their characteristics and separation axioms by using $\mathcal{S}$ Mo sets.

u: $U_1$  → W, p: S → T are functions, for soft sets ( $F_1$ , H), ( $F_2$ , B) and a family of soft sets {( $F_{1\alpha }$ , H _α):α ∈ Λ, an index set} in the soft class ( $U_1$ , S), then:

(1) f ( $\stackrel{\sim }{\varphi }$ ) =  $\stackrel{\sim }{\varphi }$ ,

(2) f ( $\stackrel{\sim }{U_1}$ ) =  $\stackrel{\sim }{W}$ and $f^{-1}$ ( $\stackrel{\sim }{W}$ ) = $\stackrel{\sim }{U_1}$ ,

(3) f (U _α∈Λ ( $F_{1\alpha }$ ,H _α)) = (U _α∈Λ f ( $F_{1\alpha }$ , H _α)),

(4) f (∩_α∈Λ ( $F_{1\alpha }$ , H _α)) $\stackrel{\sim }{\subseteq }$ (∩_α∈Λ f ( $F_{1\alpha }$ _α, H_α)),

(5) If ( $F_1$ , H) $\stackrel{\sim }{\subseteq }$ ( $F_2$ , B), then f ( $F_1$ , H) $\stackrel{\sim }{\subseteq }$ f ( $F_2$ , B),

(i) a $\mathcal{S}$ δ-precontinuous (Anjan Mukherjee and Bishnupada Debnath, 2017).

if $f^{-1}$ ( $F_1$ , H) is $\mathcal{S}$ δ-preopen in $U_1,$ for each $\mathcal{S}$ o set ( $F_1,$ H) in W,

(ii) a $\mathcal{S}$ θ-semi continuous if $f^{-1}$ ( $F_1$ , H) is $\mathcal{S}$ θ-semi-open in $U_1,$ for each $\mathcal{S}$ o set ( $F_1$ , H) in W.

1. a $\mathcal{S}$ M-continuous if $f^{-1}$ ( $F_1$ , H) is a $\mathcal{S}$ Mo in $U_1,$ for each $\mathcal{S}$ o set ( $F_1$ , H) in W.

2. a $\mathcal{S}$ M-irresolute if $f^{-1}$ ( $F_1$ , H) is a $\mathcal{S}$ Mo set in $U_1,$ for each $\mathcal{S}$ Mo set ( $F_1$ , H) in W.

1. f is a $\mathcal{S}$ M-continuous.

2. For each soft singleton $P_{\lambda }^{F_1}$  ∈  $U_1$ and each $\mathcal{S}$ o set ( $F_1$ , H) in W, where f ( $P_{\lambda }^{F_1}$ ) $\stackrel{\sim }{\subseteq }$ ( $F_1$ , H) ∃ a $\mathcal{S}$ Mo set ( $F_2$ , H) in U, since $P_{\lambda }^{F_1}$ ∈( $F_2$ , H) and f (( $F_2$ , H)) $\stackrel{\sim }{\subseteq }$ ( $F_1$ , H)

3. $f^{-1}$ ( $F_1$ , H) = cl(int_θ( $f^{-1}$ ( $F_1$ , H))) $\stackrel{\sim }{\cup }$ int(cl_δ ( $f^{-1}$ ( $F_1$ , H))), for each $\mathcal{S}$ o set ( $F_1$ , H) in E.

4. The inverse image of every $\mathcal{S}$ c set in E is $\mathcal{S}$ Mc set.

5. cl(int_θ( $f^{-1}$ ( $F_1$ , H))) $\stackrel{\sim }{\cap }$ int(cl_δ ( $f^{-1}$ ( $F_1$ , H))) $\stackrel{\sim }{\subseteq }$ $f^{-1}$ ((cl ( $F_1$ , H))) for every soft set ( $F_1$ , H) $\stackrel{\sim }{\subseteq }$ W.

6. f [cl(int_θ( $F_2$ , H)) $\stackrel{\sim }{\cap }$ int(cl_δ ( $F_2$ , H))] $\stackrel{\sim }{\subseteq }$ cl(f( $F_2$ , H)), for each soft set ( $F_2$ , H) in $U_1$ .

$f^{-1}$ ( $F_1$ , H). Suppose that ( $F_2$ , H) = $f^{-1}$ ( $F_1$ , H) which is a $\mathcal{S}$ Mo set in $U_1$ . So, we have $P_{\lambda }^{F_1}$ ∈( $F_2$ , H). Now, f( $F_2$ , H) = f ( $f^{-1}$ ( $F_1$ , H)) ⊆̃ ( $F_1$ , H) .

(ii)⇒(iii). Consider ( $F_1$ , H) is an arbitrary $\mathcal{S}$ o set in W, let $P_{\lambda }^{F_1}$ be an arbitrary soft point in $U_1,$ since f (Ρ^F_λ) $\stackrel{\sim }{\subseteq }$ ( $F_1$ , H), then $P_{\lambda }^{F_1}$ ∈ $f^{-1}$ ( $F_1$ , H). By (ii), there is a $\mathcal{S}$ Mo set( $F_2$ , H) in $U_1$ , where $P_{\lambda }^{F_1}$ ∈( $F_2$ , H) and f (( $F_2$ , H)) $\stackrel{\sim }{\subseteq }$ ( $F_1$ , H). Therefore, $P_{\lambda }^{F_1}$ ∈( $F_2$ , H) $\stackrel{\sim }{\subseteq }$ $f^{-1}$ (f (( $F_2$ , H))) $\stackrel{\sim }{\subseteq }$ $f^{-1}$ ( $F_1$ , H) $\stackrel{\sim }{\subseteq }$ cl(int_θ( $f^{-1}$ ( $F_1$ , H))) $\stackrel{\sim }{\cup }$ int(cl_δ ( $f^{-1}$ ( $F_1$ , H))).

(iii)⇒(iv). Suppose that ( $F_1$ , H) is an arbitrary $\mathcal{S}$ c set in W. Then $\stackrel{\sim }{W}$ - ( $F_1$ , H) is a $\mathcal{S}$ o set in W. By (iii), ( $f^{-1}$ ( $\stackrel{\sim }{W}$ - ( $F_1$ , H))) $\stackrel{\sim }{\subseteq }$ cl(int_θ( $f^{-1}$ ( $\stackrel{\sim }{W}$ - ( $F_1$ , H)))) $\stackrel{\sim }{\cup }$ int(cl_δ ( $f^{-1}$ ( $\stackrel{\sim }{W}$ - ( $F_1$ , H)))). This implies $\stackrel{\sim }{U_1}$ -( $f^{-1}$ ( $F_1$ , H)) $\stackrel{\sim }{\subseteq }$ cl(int_θ( $\stackrel{\sim }{U}$ - $f^{-1}$ ( $F_1$ , H))) $\stackrel{\sim }{\cup }$ int(cl_δ ( $\stackrel{\sim }{U_1}$ - $f^{-1}$ ( $F_1$ , H))) $\stackrel{\sim }{\subseteq }$ cl( $\stackrel{\sim }{U_1}$ -int_θ( $f^{-1}$ ( $F_1$ , H))) $\stackrel{\sim }{\cup }$ int( $\stackrel{\sim }{U_1}$ - cl_δ ( $f^{-1}$ ( $F_1$ , H))) $\stackrel{\sim }{\subseteq }$ [ $\stackrel{\sim }{U_1}$ -cl(int_θ( $f^{-1}$ ( $F_1$ , H)))] $\stackrel{\sim }{\cup }$ .

[ $\stackrel{\sim }{U_1}$ -int(cl_δ ( $f^{-1}$ ( $F_1$ , H)))] and hence $\stackrel{\sim }{U_1}$ - $f^{-1}$ ( $F_1$ , H)) $\stackrel{\sim }{\subseteq }$ $\stackrel{\sim }{U_1}$ - [cl(int_θ( $f^{-1}$ ( $F_1$ , H))) $\stackrel{\sim }{\cap }$ .

int(cl_δ ( $f^{-1}$ ( $F_1$ , H)))]. Hence, $f^{-1}$ ( $F_1$ , H) $\stackrel{\sim }{\supseteq }$ [cl(int_θ( $f^{-1}$ ( $F_1,$ H))) $\stackrel{\sim }{\cap }$ int(cl_δ ( $f^{-1}$ ( $F_1$ , H)))] then $f^{-1}$ ( $F_1$ , H) is $\mathcal{S}$ Mc in $U_1$ .

(iv)⇒(v). Consider ( $F_1$ , H) $\stackrel{\sim }{\subseteq }$ W, therefore $f^{-1}$ (cl ( $F_1$ , H)) is $\mathcal{S}$ Mc in $U_1$ . Now,

[cl(int_θ( $f^{-1}$ ( $F_1$ , H))) $\stackrel{\sim }{\cap }$ int(cl_δ ( $f^{-1}$ ( $F_1$ , H)))] $\stackrel{\sim }{\subseteq }$ [cl(int_θ( $f^{-1}$ (cl ( $F_1$ , H)))) $\stackrel{\sim }{\cap }$ int(cl_δ ( $f^{-1}$ (cl ( $F_1$ , H))))] $\stackrel{\sim }{\subseteq }$ $f^{-1}$ (cl ( $F_1$ , H)).

(v)⇒(vi). Consider ( $F_2$ , H) $\stackrel{\sim }{\subseteq }$ $U_1$ . Put, ( $F_1$ , H) = f( $F_2$ , H) in (v) implies that [cl(int_θ ( $f^{-1}$ (f ( $F_2$ , H)))) $\stackrel{\sim }{\cap }$ int(cl_δ ( $f^{-1}$ (f ( $F_2$ , H))))] $\stackrel{\sim }{\subseteq }$ $f^{-1}$ (cl(f ( $F_2$ , H))). This is implies that [cl(int_θ) ( $F_2$ , H) $\stackrel{\sim }{\cap }$ int (cl_δ ( $F_2$ , H))] $\stackrel{\sim }{\subseteq }$ $f^{-1}$ (cl(f( $F_2$ , H))), hence, f [cl(int_θ( $F_2$ , H)) $\stackrel{\sim }{\cap }$ int(cl_δ ( $F_2$ , H))] $\stackrel{\sim }{\subseteq }$ cl(f( $F_2$ , H)).

(vi)⇒(i). Consider ( $F_2$ , H) $\stackrel{\sim }{\subseteq }$ W is a $\mathcal{S}$ o set, let ( $F_2$ , H) = $f^{-1}$ ( $F_1$ , H) and ( $F_1$ , H) = $\stackrel{\sim }{W}$ -( $F_1$ , H). Then f [cl(int_θ( $f^{-1}$ ( $F_1$ , H))) $\stackrel{\sim }{\cap }$ int(cl_δ ( $f^{-1}$ ( $F_1$ , H)))] $\stackrel{\sim }{\subseteq }$ cl(f ( $f^{-1}$ ( $F_1$ , H))) $\stackrel{\sim }{\subseteq }$ cl ( $F_1$ , H) = ( $F_1$ , H).Thus, $f^{-1}$ ( $F_1$ , H) is $\mathcal{S}$ Mc in $U_1$ , so f is $\mathcal{S}$ M-continuous.

Proof. Clearly by Definition 4.2.

g: (W, τ 2,T) →(E, τ 3,J) is soft continuous function, then gof: ( $U_1$ , τ 1,S) →(E, τ 3,J) is.

a $\mathcal{S}$ M-continuous function.

Proof. Suppose that ( $F_1$ , H) is a $\mathcal{S}$ o set in E. Now, (gof)^-1 ( $F_1$ , H) = (f^-1og⁻¹) ( $F_1$ , H) =.

(f^-1(g⁻¹ ( $F_1$ , H)). Where, g is soft continuous, g⁻¹( $F_1$ , H) is a $\mathcal{S}$ o set, then (gof)^-1 ( $F_1$ , H) is $\mathcal{S}$ M-open in W. But f being $\mathcal{S}$ M-continuous, (gof)^-1 ( $F_1$ , H) is a $\mathcal{S}$ Mo set in $U_1$ . Thus gof is a $\mathcal{S}$ M-continuous mapping.

g: (W, τ 2,T) → (W, τ 3,J) is a $\mathcal{S}$ M-continuous function., then gof: ( $U_1$ , τ 1,S) →(E, τ 3,J) is also $\mathcal{S}$ M-continuous function.

Proof. Suppose that ( $F_1$ , H) is a $\mathcal{S}$ o set in J. Now, (gof)^-1 ( $F_1$ , H) = (f^-1og⁻¹) ( $F_1$ , H) =.

(f^-1(g⁻¹ ( $F_1$ , H)), where g is $\mathcal{S}$ M-continuous, g⁻¹ ( $F_1$ , H) is a $\mathcal{S}$ Mo set and hence.

(gof)^-1 ( $F_1$ , H) is $\mathcal{S}$ Mo in W. But f being M-irresolute, (gof)^-1 ( $F_1$ , H) is a $\mathcal{S}$ Mo set in $U_1$ . Thus gof is a $\mathcal{S}$ M-continuous function.

Proof. Clear.

1. $\mathcal{S}$ M-open function (briefly, $\mathcal{S}$ Mof) if the image of each $\mathcal{S}$ o set in $U_1$ is a $\mathcal{S}$ Mo set in W,

2. $\mathcal{S}$ M-closed function (briefly, $\mathcal{S}$ Mcf) if the image of each $\mathcal{S}$ c set in $U_1$ is a $\mathcal{S}$ Mc set in W.

Proof. For a $\mathcal{S}$ c set ( $F_1$ , H) in $U_1$ , $f(F_1,H)$ is $\mathcal{S}$ c set in W, where g: W → E is $\mathcal{S}$ Mcf, $g\left(f(F_1,H)\right)$ is a $\mathcal{S}$ Mc set in E $,g\left(f(F_1,H)\right)=\left(gof\right)(F_1,H)$ is a $\mathcal{S}$ Mc set in E. Hence, $\mathit{gof}$ is $\mathcal{S}$ Mcf.

5 Soft M-separation axioms

In this section, soft M-separation axioms has been introduced and investigated with the help of $\mathcal{S}$ Mo sets. Also, some properties of $\mathcal{S}$ M- separation axioms are studied.

Proof. Consider $P_{\lambda }^{F_1}$ and $P_{\mu }^{F_2}are$ two soft points $\mathit{and}{P}_{\lambda }^{F_1}$ ${\ne P}_{\mu }^{F_2}$ with distinct $\mathcal{S}$ M-closure in a $\mathcal{S}$ τs ( $U_1$ , τ,S). If possible, let $P_{\lambda }^{F_1}$  ∈  $\mathcal{S}$ Mcl{ $P_{\mu }^{F_2}$ }, then $\mathcal{S}$ Mcl{ $P_{\lambda }^{F_1}$ } ⊆  $\mathcal{S}$ Mcl{ $P_{\mu }^{F_2}$ } which is a contradiction. So, $P_{\lambda }^{F_1}$ ∉ $\mathcal{S}$ Mcl{ $P_{\lambda }^{F_1}$ }which implies ( $\mathcal{S}$ Mcl{ $P_{\lambda }^{F_1}$ })^c is a $\mathcal{S}$ Mo set containing $P_{\lambda }^{F_1}$ but not $P_{\mu }^{F_2}$ , therefore ( $U_1$ , τ, E) is a $\mathcal{S}$ M-T₀ structure.

Proof. Suppose that W is a soft T₁. For arbitrary-two soft points $P_{\lambda }^{F_1}$ , $P_{\mu }^{F_2}o$ f $U_1$ and $P_{\lambda }^{F_1}$ ${\ne P}_{\mu }^{F_2}$ , there exist $\mathcal{S}$ o sets (L,H) and (Q,H) in W since, f ( $P_{\lambda }^{F_1}$ ) ∈ (L,H), $P_{\mu }^{F_2}$ $\notin \left(\mathrm{L},\mathrm{H}\right)\mathrm{a}\mathrm{n}\mathrm{d}$ f ( $P_{\mu }^{F_2}$ ) ∈ (Q,H),f ( $P_{\lambda }^{F_1}$ ) ∉ (Q,H). Where f is an injective $\mathcal{S}$ M-continuous function, we have $f^{-1}$ (L,H) and $f^{-1}$ (Q,H) are $\mathcal{S}$ Mo sets in $U_1$ . Hence $U_1$ is a $\mathcal{S}$ M-T₁.

Hausdorff) if For every-two soft points $P_{\lambda }^{F_1}$ , $P_{\mu }^{F_2}o$ f $U_1$ and $P_{\lambda }^{F_1}$ , ${\ne P}_{\mu }^{F_2}$ , there exist disjoint $\mathcal{S}$ Mo sets (L,H) and (A,D) where $P_{\lambda }^{F_1}$ ∈ (L,H) and $P_{\mu }^{F_2}$ ∈ (A,D).

Proof. Obvious

Proof. Consider ( $F_1$ , H) is a soft closed set in W with a soft point $P_{\mu }^{F_2}$ ∉ ( $F_1$ , H). Take.

$P_{\mu }^{F_2}$  = f ( $P_{\lambda }^{F_1}$ ). Since W is soft regular, there exists disjoint $\mathcal{S}$ o sets (L,H) and (J,B) since $,P_{\lambda }^{F_1}$ ∈(L,H), $P_{\mu }^{F_2}$ = $f$ ( $P_{\lambda }^{F_1}$ ) ∈ f(L,H) and ( $F_1$ , H) $\stackrel{\sim }{\subseteq }$ f (J,B) such that $f$ (L,H), $f$ (J,B) and $f$ (L,H) $\stackrel{\sim }{\cap }$ $f$ (J,B) =  $\stackrel{\sim }{\varphi }$ are $\mathcal{S}$ o sets. Thus, $f^{-1}(F_1$ , H) $\stackrel{\sim }{\subseteq }$ (J,B). Where $f$ is $\mathcal{S}$ M-continuous, $f^{-1}$ ( $F_1$ , H) is a $\mathcal{S}$ Mc set in $U_1$ and $P_{\lambda }^{F_1}$ ∉ $f^{-1}$ ( $F_1$ , H). Hence $U_1$ is $\mathcal{S}$ M-regular.

Proof. Let W be a soft normal space, ( $F_1,H$ ) and (V,D) be $\mathcal{S}$ c sets in $U_1,$ where ( $F_1,H$ ) $\stackrel{\sim }{\cap }$ (V,D) =  $\stackrel{\sim }{\varphi }$ . Since $f$ is soft closed injection, $f(F_1,H$ ) and $f($ V,D) are $\mathcal{S}$ c sets in W and f( $F_1,H$ ) $\stackrel{\sim }{\cap }$ $f(V,D)=$ $\stackrel{\sim }{\varphi }$ , but W is soft normal, there exist $\mathcal{S}$ Mo sets (L,H) and (Q,B)in W where $f(F_1,H)$ $\stackrel{\sim }{\subseteq }$ $l,f(V,D)$ $\stackrel{\sim }{\subseteq }$ Q and L $\stackrel{\sim }{\cap }$ Q =  $\stackrel{\sim }{\varphi }$ . Thus we obtain, ( $F_1,H$ ) $\stackrel{\sim }{\subseteq }$ $f^{-1}$ (L), (V,D) $\stackrel{\sim }{\subseteq }$ $f^{-1}$ (Q) and $f^{-1}$ (L $\stackrel{\sim }{\cap }$ Q) =  $\stackrel{\sim }{\varphi }$ , where $f$ is $\mathcal{S}$ M-continuous, $f^{-1}\left(L\right)$ and $f^{-1}$ (Q) are $\mathcal{S}$ Mc sets. Hence $U_1$ is $\mathcal{S}$ M-normal.

6 Soft M-connectedness and soft M-compactness

The investigation of compactness (which is based on open sets) for a $\mathcal{S}$ τs was started by Zorlutuna et al. (2012). Peyghan et al. (2012) defined and investigated the concept of soft connectedness in $\mathcal{S}$ τs. This section is objective to present M-connectedness in $\mathcal{S}$ τs and to characterize it. Finally, we discuss the properties of M-compactness in a $\mathcal{S}$ τs.

Proof. Let $g(V,H)$ be not soft connected. Therefore, there is non empty $\mathcal{S}$ o sets ( $F_1,H$ ) and ( $F_2$ , H) in Y, where $g($ V,H)= ( $F_1,H$ ) $\stackrel{\sim }{\cup }$ ( $F_2$ , H), where $g$ is $\mathcal{S}$ M-continuous, $g^{-1}$ ( $F_1,H$ ), $g^{-1}$ ( $F_2$ , H) are $\mathcal{S}$ Mo sets in $U_1$ and (V,H) =  $g^{-1}$ [( $F_1,H$ ) $\stackrel{\sim }{\cup }$ $(F_2$ , H)]= $g^{-1}$ ( $F_1,H$ ) $\stackrel{\sim }{\cup }$ $g^{-1}$ ( $F_2$ , H). Hence, $g^{-1}$ ( $F_1,H$ ) and $g^{-1}(F_2$ , H) are $\mathcal{S}$ Mo sets in $U_1$ . Thus, $(V,H)$ is not $\mathcal{S}$ M-connected which is in the opposition to the proposed hypothesis. Hence $,g(V,H)$ is soft connected.

Proof. Consider f🙁 $U_1$ , τ 1,S) → (W, τ 2, S) is a M-continuous function, where ( $U_1$ , τ 1,S) is M-compact $S\tau s$ and (W, τ 2, S) is another $S\tau s$ . Let {( $F_2$ , H)_α:α ∈ Λ} be $\mathcal{S}$ o cover of W, therefore,{f^-1( $F_2$ ,H)_α: α ∈ Λ} is $\mathcal{S}$ Mo cover of $U_1$ , therefore there exists a finite subset Δ of Λ where {f^-1( $F_2$ , H)_α: α ∈ Δ} is a Mo cover of $U_1.$ Hence, {( $F_2$ , H)_α: α ∈ Δ} is a finite $\mathcal{S}$ o cover of W. Therefore, W is soft compact.