# Some Local Properties of Soft Semi-Open Sets

Authors: Bin Chen
Journal: Discrete Dynamics in Nature and Society
 (2013)

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## Abstract

We introduce some local properties by soft semi-open sets. For example, soft semi-neighborhoods of the soft point, soft semi-first-countable spaces and soft semi-pu-continuous at the soft point are given. Furthermore, we define soft semi-connectedness and prove that a soft topological space is soft semiconnected if and only if both soft semi-open and soft semi-closed sets are only ∅ and X~.


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## Body

## 1. Introduction

Soft set theory [1] was firstly introduced by Molodtsov in 1999 as a general mathematical tool for dealing with uncertain, fuzzy, not clearly defined objects. He has shown several applications of this theory in solving many practical problems in economics, engineering, social science, medical science, and so forth, in [1].In the recent years, papers about soft sets theory and their applications in various fields have been writing increasingly [2–6]. Shabir and Naz [7] introduced the notion of soft topological spaces which are defined over an initial universe with a fixed set of parameters. The authors introduced the definitions of soft open sets, soft closed sets, soft interior, soft closure, and soft separation axioms. And the authors got some important results for soft separation axioms. The results are valuable for research in this field. Tanay and Kandemir [8] introduced the definition of fuzzy soft topology over a subset of the initial universe set while Jun et al. [9] studied the ideal theory in BCK/BCI-Algebras based on soft sets. Along the line of Shabir and Naz, Chen introduced the notations of soft semi-open sets in soft topological spaces in [10].In the present study, we introduce some local properties by soft semi-open sets. For example, soft semi-neighborhoods of the soft point, soft semi-first-countable spaces, and soft semi-pu-continuous at the soft point are given. And some of their properties are studied.


## 2. Preliminaries

LetU be an initial universe set and EU be a collection of all possible parameters with respect to U, where parameters are the characteristics or properties of objects in U. We will call EU the universe set of parameters with respect to U.Definition 1 (see [1]).
A pair(F,A) is called a soft set over U if A⊂EU and F:A→P(U), where P(U) is the set of all subsets of U.Definition 2 (see [1]).
LetU be an initial universe set and EU be a universe set of parameters. Let (F,A) and (G,B) be soft sets over a common universe set U and A,B⊂E. Then (F,A) is a subset of (G,B), denoted by (F,A)⊂~(G,B), if: (i) A⊂B; (ii) for all e∈A, F(e)⊂G(e).
(
F
,
A
) equals (G,B), denoted by (F,A)=(G,B), if (F,A)⊂~(G,B) and (G,B)⊂~(F,A).Definition 3 (see [1]).
A soft set(F,A) over U is called a null soft set, denoted by ∅, if e∈A, F(e)=∅.Definition 4 (see [1]).
A soft set(F,A) over U is called an absolute soft set, denoted by A~, if e∈A, F(e)=U.Definition 5 (see [1]).
The union of two soft sets(F,A) and (G,B) over a common universe U is the soft set (H,C), where C=A∪B, and for all e∈C,
(1)H(e)={F(e),ife∈A-B,G(e),e∈B-A,F(e)∪G(e),e∈B∩A.
We write(F,A)∪(G,B)=(H,C).Definition 6 (see [1]).
The intersection of two soft sets of(F,A) and (G,B) over a common universe U is the soft set (H,C), where C=A∩B, and for all e∈C, H(e)=F(e)∩G(e). We write (F,A)∩(G,B)=(H,C).
Now we recall some definitions and results defined and discussed in [6, 7, 10]. Henceforth, let X be an initial universe set and E be the fixed nonempty set of parameter with respect to X unless otherwise specified.Definition 7.
For a soft set(F,A) over U, the relative complement of (F,A) is denoted by (F,A)′ and is defined by (F,A)′=(F′,A), where F′:A→P(U) is a mapping given by F′(e)=U-F(e) for all e∈A.Definition 8.
Letτ be the collection of soft sets over X, then τ is called a soft topology on X if τ satisfies the following axioms.(1)
∅,X~ belong to τ.(2)
The union of any number of soft sets inτ belongs to τ.(3)
The intersection of any two soft sets inτ belongs to τ.
The triplet(X,τ,E) is called a soft topological space over X.Definition 9.
Let(X,τ,E) be a soft space over X, then the members of τ are said to be soft open sets in X.Definition 10.
Let(X,τ,E) be a soft space over X. A soft set (F,E) over X is said to be a soft closed set in X, if its relative complement (F,E)′ belongs to τ.Proposition 11.
Let(X,τ,E) be a soft space over X. Then one has the following.(1)
∅,X~are soft closed sets overX.(2)
The intersection of any number of soft closed sets is a soft closed set overX.(3)
The union of any two soft closed sets is a soft closed set overX.Definition 12.
Let(X,τ,E) be a soft topological space and (A,E) be a soft set over X.(1)
The soft interior of(A,E) is the soft set Int(A,E)=⋃{(O,E):(O,E) is soft open and (O,E)⊂~(A,E)}.(2)
The soft closure of(A,E) is the soft set Cl(A,E)=⋂{(F,E):(F,E) is soft closed and (A,E)⊂~(F,E)}.
By property 2 for soft open sets,Int(A,E) is soft open. It is the largest soft open set contained in (A,E).
By property 2 for soft closed sets,Cl(A,E) is soft closed. It is the smallest soft closed set containing (A,E).Proposition 13.
Let(X,τ,E) be a soft topological space and let (F,E) and (G,E) be a soft set over X. Then(1)
Int(Int(F,E))=Int(F,E),(2)
(F,E)⊂~(G,E)impliesInt(F,E)⊂~Int(G,E),(3)
Cl(Cl(F,E))=Cl(F,E),(4)
(F,E)⊂~(G,E)impliesCl(F,E)⊂~xCl(G,E).Definition 14.
A soft set(A,E) in a soft topological space (X,τ,E) will be termed soft semi-open (written S.S.O) if and only if there exists a soft open set (O,E) such that (O,E)⊂~(A,E)⊂~Cl(O,E).Definition 15.
A soft set(B,E) in a soft topological space (X,τ,E) will be termed soft semi-closed (written S.S.C) if its relative complement is soft semi-open, for example, there exists a soft closed set (F,E) such that Int(F,E)⊆~(B,E)⊂~(F,E).Definition 16.
Let(X,τ,E) be a soft topological space and (A,E) be a soft set over X.(1)
The soft semi-interior of(A,E) is the soft set IntS(A,E)=⋃{(O,E):(O,E) is soft semi-open and (O,E)⊂~(A,E)}.(2)
The soft semi-closure of(A,E) is the soft set ClS(A,E)=⋂{(F,E):(F,E) is soft semi-closed and (A,E)⊂~(F,E)}.


## 3. Soft Semi-Neighborhoods

Definition 17.
A soft set(F,E) in a soft topological space (X,τ,E) is said to be a soft semi-neighborhood of the soft point eF if there is a soft semi-open set (B,E) s.t. eF∈~(B,E)⊆~(F,E).
The semi-neighborhood system of a soft pointeF which is denoted by 𝒰eF is the set of all its semi-neighborhoods.Proposition 18.
The semi-neighborhood system𝒰eF at a soft point eF in the soft topological space (X,τ,E) has the following results.(a)
If(F,E)∈𝒰eF, then one haseF∈~(F,E).(b)
If(F,E)∈𝒰eFand(F,E)⊆~(B,E), then one has(B,E)∈𝒰eF.(c)
If(F,E)∈𝒰eF, then there exists a soft set(B,E)∈𝒰eFs.t.(F,E)∈𝒰e′for eache′∈~(B,E).Proof.
(a) If(F,E)∈𝒰eF, then we have a soft semi-open set (B,E) s.t. eF∈~(B,E)⊆~(F,E). So we have eF∈~(F,E).
(b) If(F,E)∈𝒰eF and (F,E)⊆~(B,E). Because (F,E)∈𝒰eF, there exists a soft semi-open set (H,E) s.t. eF∈~(H,E)⊆~(F,E). So we have eF∈~(H,E)⊆~(F,E)⊆~(B,E) and we have (B,E)∈𝒰eF.
(c) If(F,E)∈𝒰eF, then there is a soft semi-open set (H,E) s.t. eF∈~(H,E)⊆~(F,E). Let (B,E)=(H,E) and for each e′∈~(B,E),e′∈~(B,E)⊆~(H,E)⊆~(F,E). This shows (F,E)∈𝒰e′.Definition 19.
Let(X,τ,E) be a soft topological space and 𝒰eF be a soft semi-neighborhood of a soft point eF. If for every soft semi-neighborhood (F,E) of eF, there is a (H,E)∈𝒱eF⊆𝒰eF such that eF∈~(H,E)⊆~(F,E), then 𝒱eF is said to be a soft semi-neighborhoods base of 𝒰eF at eF.Proposition 20.
Let(X,τ,E) be a soft topological space and 𝒰eF be a soft semi-neighborhood of a soft point eF. 𝒱eF is the soft semi-neighborhoods base of 𝒰eF at eF. Then one has the following.(a)
If(F,E)∈𝒱eF, then we haveeF∈~(F,E).(b)
If(F,E)∈𝒱eFand(F,E)⊆~(B,E), then one has(B,E)∈𝒱eF.(c)
If(F,E)∈𝒱eF, then there exists a soft set(B,E)∈𝒱eFs.t.(F,E)∈𝒱e′for eache′∈~(B,E).Proof.
These properties are easily verified by referring to the corresponding properties of soft seminbds in Proposition18.Definition 21.
Let(X,τ,E) be a soft topological space and eF be a soft point in (X,τ,E). If eF has a countable soft semi-neighborhoods base, then we say that (X,τ,E) is soft semi-first-countable at eF. If each soft point in (X,τ,E) is soft semi-first-countable, then we say that (X,τ,E) is a soft semi-first-countable space.Proposition 22.
Let(X,τ,E) be a soft topological space and eF be a soft point in (X,τ,E). Then (X,τ,E) is soft semi-first-countable at eF if and only if there is a countable soft semi-neighborhoods base {(Fn,E),n∈N} at eF such that (Fn+1,E)⊆~(Fn,E) for each n∈N.Proof.
⇐ Obvious.
⇒ Let {(Un,E),n∈N} be a countable soft semi-neighborhoods base at eF. For each n∈N, put (Fn,E)=⋂i=1n‍(Ui,E). Then it is easy to see that {(Fn,E),n∈N} is a soft semi-neighborhoods base at eF and (Fn+1,E)⊆~(Fn,E) for each n∈N.Definition 23.
Let(X,τ,E) be a soft topological space, (B,E) be a soft set, and eF be a soft point in (X,τ,E). Then eF is said to be a soft semi-interior point of (B,E) if there is a soft semi-open set (H,E) satisfies eF∈~(H,E)⊆~(B,E).Proposition 24.
LeteF be a soft point in (X,τ,E) and (B,E) be a soft semi-open set. Then one has the following results.(a)
Each soft pointeF∈~(B,E)is a soft semi-interior point.(b)
For eache∈E, one can define(B,{e})as follows:(2)B(e′)={B(e),ife′=e,∅,ife′≠e.Then(B,{e})is a soft semi-interior point of(B,E)and(B,{e})=∪eFfor each soft semi-interior pointeFof(B,E).(c)
∪e∈E(B,{e})=(B,E).Proof.
(a), (c) are obvious by definitions.
(b) For(B,E) is a soft semi-open set, the soft semi-interior point (B,{e}) is the largest soft semi-interior point of (B,E) by e∈E and we have (B,{e})=∪eF for every soft semi-interior point eF of (B,E).Proposition 25.
Let(X,τ,E) be a soft topological space, (B,E) be a soft set. Then IntS(B,E)=⋃e∈E‍{eF:eF is any soft semi-interior point of (B,E)}.Proof.
By the definition of soft semi-interior and Proposition24.Definition 26.
A soft set(F,E) in a soft topological space (X,τ,E) is called a soft semi-neighborhood (briefly: snbd) of the soft set (G,E) if there is a soft semi-open set (B,E) s.t. (G,E)⊆~(B,E)⊆~(F,E).Proposition 27.
Let(X,τ,E) be a soft topological space, (B,E) be a soft set. Then (B,E) is soft semi-open if and only if for each soft set (F,E) contained in (B,E), (B,E) is a soft semi-nbd of (F,E).Proof.
⇒ obvious
⇐ Because (B,E)⊆~(B,E), then we have a soft semi-open set (H,E) s.t. (B,E)⊆~(H,E)⊆~(B,E). So we have (B,E)=(H,E) and (B,E) is soft semi-open.Definition 28 (see [6]).
Let the sequence{(Fn,E),n∈N} be a soft sequence in a soft topological space (X,τ,E). Then {(Fn,E),n∈N} is eventually contained in a soft set (B,E) if and only if there is an integer m such that, if n≥m, then (Fn,E)⊆~(B,E). The sequence is frequently contained in (B,E) if and only if for each integer m, there is an integer n such that n≥m and (Fn,E)⊆~(B,E).Definition 29.
Let the sequence{(Fn,E),n∈N} be a soft sequence in a soft topological space (X,τ,E), then one says {(Fn,E),n∈N} semiconverges to a soft point eF if it is eventually contained in each semi-nbd of eF. And the soft point eF is said to be a semicluster soft point of {(Fn,E),n∈N} if the sequence is frequently contained in every semi-nbd of eF.Proposition 30.
Let(X,τ,E) be soft semi-first-countable, then one has the following.(a)
(B,E)is soft semi-open⇔for every soft sequence{(Fn,E),n∈N}which semiconverges toeFin(B,E)is eventually contained in(B,E).(b)
IfeFis a semicluster soft point of the soft sequence{(Fn,E),n∈N}, then one has a subsequence of{(Fn,E),n∈N}which semi-converges toeF.Proof.
(a)⇒ Because (B,E) is soft semi-open, (B,E) is a semi-nbd of eF, and {(Fn,E),n∈N} semi-converges to eF. Then we have {(Fn,E),n∈N} is eventually contained in (B,E).
⇐ For each eF contained in (B,E), let {(Fn,E),n∈N} be the semi-nbd systems such that (Fn+1,E)⊆~(Fn,E) for each n∈N by Proposition 22. Then {(Fn,E),n∈N} is eventally contained in each semi-nbd of eF that is, {(Fn,E),n∈N} semi-converges to eF. So we have an integer m such that, if n≥m, (Fn,E)⊆~(B,E). Then (B,E) is a semi-nbd of eF, and by Proposition 27, (B,E) is soft semi-open.
(b) Let{(Fn,E)},n∈N be the seminbds such that (Fn+1,E)⊆~(Fn,E) for each n∈N by Proposition 22. For every nonnegative integer i, find f(i) satisfies f(i)≥i and (Ff(i),E)⊆~(Fi,E). Then {(Ff(i),E),i∈N} is a subsequence of the sequence {(Fn,E),n∈N}. Obviously this subsequence semiconverges to eF.


## 4. Soft Semi-pu-Continuous Functions

In this part, we define the soft semi-pu-continuous functions induced by two mapping u:X→Y and p:E1→E2 on soft topological spaces (X,τ1,E1), (Y,τ2,E2) and study some properties on them. We use SS(X)E1 and SS(Y)E2 denote all soft sets on (X,E1) and (Y,E2). SSS(X)E1 and SSS(Y)E2 denote all semi-open soft sets on soft topological spaces (X,τ1,E1) and (Y,τ2,E2).Definition 31.
Let(X,τ1,E1) and (Y,τ2,E2) be two soft topological spaces. Let u:X→Y and p:E1→E2 be mappings. Let fpu:SS(X)E1→SS(Y)E2 be a function and eF be a soft point in (X,τ1,E1).(a)
fpu is soft semi-pu-continuous at eF if for any (B,E2)∈𝒰fpu(eF)τ2, there is a (F,E1)∈𝒰eFτ1 s.t. fpu((F,E1))⊆~(B,E2).(b)
fpu is soft semi-pu-continuous from (X,τ1,E1) to (Y,τ2,E2) if fpu is soft semi-pu-continuous for every soft point of (X,τ1,E1).Proposition 32.
Let(X,τ1,E1) and (Y,τ2,E2) be two soft topological spaces. Let u:X→Y and p:E1→E2 be mappings. Let fpu:SS(X)E1→SS(Y)E2 be a function and eF be a soft point in (X,τ1,E1). Then the following statements are equal.(a)
fpuis soft semi-pu-continuous ateF.(b)
For any(B,E2)∈𝒰fpu(eF)τ2, there is a(F,E1)∈𝒰eFτ1satisfies(F,E1)⊆~fpu-1(B,E2).(c)
For any(B,E2)∈𝒰fpu(eF)τ2,fpu-1(B,E2)∈𝒰eFτ1.Proof.
From Definition31, this is obvious.Proposition 33.
Let(X,τ1,E1) and (Y,τ2,E2) be two soft topological spaces. Let u:X→Y and p:E1→E2 be mappings. fpu:SS(X)E1→SS(Y)E2 be a function. Then the following statements are equal.(a)
fpuis soft semi-pu-continuous.(b)
If(B,E2)∈SSS(Y)E2, one hasfpu-1((B,E2))∈SSS(X)E1.(c)
For each soft semi-closed set(F,E2)∈(Y,τ2,E2), one hasfpu-1((F,E2))is soft semi-closed in(X,τ1,E1).Proof.
(a)⇒ (b) Let (B,E2)∈SSS(Y)E2 and eF∈~fpu-1((B,E2)). Next, we will prove that fpu-1((B,E2))∈𝒰eFτ1. Because fpu(eF)∈~(B,E2) and (B,E2)∈SSS(Y)E2, we have (B,E2)∈𝒰fpu(eF)τ2. Because fpu is soft semi-pu-continuous at eF, there is a (F,E1)∈𝒰eFτ1 s.t. fpu((F,E1))⊆~(B,E2). So we have eF∈~(F,E1)⊆~fpu-1((B,E2)) and fpu-1((B,E2))∈𝒰eFτ1. By Proposition 27, fpu-1((B,E2))∈SSS(X)E1.
(b)⇒ (a) Let eF be a soft point, (B,E2)∈𝒰fpu(eF)τ2. Then we can get a soft semi-open set (F,E2)∈SSS(Y)E2 s.t. fpu(eF)∈~(F,E2)⊆~(B,E2). By (b) fpu-1((F,E2))∈SSS(X)E1 and eF∈~fpu-1((F,E2))⊆~fpu-1((B,E2)). And we have fpu-1((B,E2))∈𝒰eFτ1. So, we prove that fpu is soft semi-pu-continuous at each soft point.
(b)⇒ (c) Let (F,E2) be soft semi-closed in (Y,τ2,E2). Then (F,E2)′∈SSS(Y)E2 and by (b), fpu-1((F,E2)′)∈SSS(X)E1. Because fpu-1((F,E2)′)=(fpu-1((F,E2)))′, we prove fpu-1((F,E2)) is soft semi-closed in (X,τ1,E1).
(c)⇒ (b) Using the same method in (b) ⇒ (c), we can get the result.Proposition 34.
Let(X,τ1,E1) and (Y,τ2,E2) be two soft topological spaces. Let u:X→Y and p:E1→E2 be mappings. fpu:SS(X)E1→SS(Y)E2 be a function. If (X,τ,E) is soft semi-first-countable, then the following statements are equal.(a)
fpuis soft semi-pu-continuous.(b)
If(B,E1)∈(X,τ1,E1), the inverse image of any semi-nbd offpu((B,E1))is a semi-nbd of(B,E1).(c)
If(B,E1)∈(X,τ1,E1)and for every semi-nbd(F,E2)offpu((B,E1)), there exists a semi-nbd(G,E1)of(B,E1)satisfiesfpu((G,E1))⊆~(F,E2).(d)
Let the soft sequence{(Fn,E1),n∈N}semi-converges toeF, then one has{fpu(Fn,E1),n∈N}semi-converges tofpu(eF).Proof.
(a)⇒ (b) Let fpu be soft semi-pu-continuous. If (F,E2) is a semi-nbd of fpu((B,E1)), then (F,E2) contains a soft semi-open nbd (H,E2) of fpu((B,E1)). Because fpu((B,E1))⊆~(H,E2)⊆~(F,E2), we have fpu-1(fpu((B,E1)))⊆~fpu-1((H,E2))⊆~fpu-1((F,E2)). Since (B,E1)⊆~fpu-1(fpu((B,E1))) and fpu-1((H,E2)) is soft semi-open. So, we have fpu-1((F,E2)) is a semi-nbd of (B,E1).
(b)⇒ (c) Let (B,E1)∈(X,τ1,E1) be a soft set and (F,E2) be any semi-nbd of fpu((B,E1)). Then by (b),fpu-1((F,E2)) is a semi-nbd of (B,E1). So there is a soft semi-open set (G,E1)∈(X,τ1,E1) s.t. (B,E1)⊆~(G,E1)⊆~fpu-1((F,E2)). So we have fpu((G,E1))⊆~(F,E2).
(c)⇒ (d) Let (B,E2) be a semi-nbd of {fpu(eF)}, then there exists a semi-nbd (H,E1) of eF s.t. fpu((H,E1))⊆~(B,E2). Since the sequence of soft sets {(Fn,E1),n∈N} semi-converges to eF, there is an m such that n≥m, for the semi-nbd (H,E1) of eF, (Fn,E1)⊆~(H,E1). So we have fpu((Fn,E1))⊆~fpu((H,E1))⊆~(B,E2). Thus, {fpu(Fn,E1),n∈N} semi-converges to fpu(eF).
(d)⇒ (a) Let (H,E2) be any semi-open soft set over (Y,τ2,E2). We show fpu-1((H,E2)) is semi-open soft set over (X,τ1,E1). Let eF be any soft point of fpu-1((H,E2)). Because (X,τ,E) is semi-soft first-countable, then there exists a countable soft semi-neighborhoods base {(Fn,E),n∈N} at eF such that (Fn+1,E)⊆~(Fn,E) for each n∈N. Because {(Fn,E),n∈N} is decreasing, there exists an m such that for n≥m, (Fn,E)⊆~fpu-1((H,E2)). We know each (Fn,E) is soft semi-neighborhood of eF, so fpu-1((H,E2)) is a soft semi-neighborhood of eF by Proposition 18 (b). This complete the proof.Proposition 35.
Let(X,τ1,E1) and (Y,τ2,E2) be two soft topological spaces. Let u:X→Y and p:E1→E2 be mappings. fpu:SS(X)E1→SS(Y)E2 be a function. Then the following statements are equal.(a)
fpuis soft semi-pu-continuous.(b)
If(A,E1)∈(X,τ1,E1), one hasfpu(ClS(A,E1))⊂~ClSfpu((A,E1)).(c)
If(B,E2)∈(Y,τ2,E2), one hasClSfpu-1((B,E2))⊂~fpu-1(ClS(B,E2)).(d)
If(B,E2)∈(Y,τ2,E2), one hasfpu-1(IntS(B,E2))⊂~IntSfpu-1((B,E2)).Proof.
(a)⇒ (b) For fpu is soft semi-pu-continuous, then by Proposition 33 (c), fpu-1(ClSfpu((A,E1))) is soft semi-closed containing (A,E1), and thus ClS(A,E1)⊂~fpu-1(ClSfpu((A,E1))), which gives fpu(ClS(A,E1))⊂~fpufpu-1(ClSfpu((A,E1)))⊂~ClSfpu ((A,E1)).
To prove that (b)⇒ (c), we apply (b) to (A,E1)=fpu-1((B,E2)) and we obtain the inclusion fpu(ClSfpu-1((B,E2)))⊂~ClSfpu(fpu-1((B,E2)))⊂~ClS(B,E2), which gives ClSfpu-1((B,E2))⊂~fpu-1(ClS(B,E2)).
To prove that (c)⇒ (d), we apply (c) to (B,E2)′ and we obtain the inclusion ClSfpu-1((B,E2)′)⊂~fpu-1(ClS(B,E2)′), which gives fpu-1(IntS(B,E2))=fpu-1((ClS(B,E2)′)′)=(fpu-1(ClS(B,E2)′))′⊂~(ClSfpu-1((B,E2)′))′=(ClS((fpu-1((B,E2)))′))′=IntSfpu-1((B,E2)).
To complete (d)⇒ (a), for every (B,E2)∈SSS(Y)E2 we have (B,E2)=IntS(B,E2), and it follows from (d) that fpu-1((B,E2))⊂~IntSfpu-1((B,E2)). Thus, we have fpu-1((B,E2))=IntSfpu-1((B,E2)), that is, fpu-1((B,E2)) is soft semi-open in (X,τ1,E1). Then fpu is soft semi-pu-continuous.


## 5. Soft Semi-Connectedness

Definition 36.
Let(X,τ,E) be a soft topological space. A soft semiseparation on X~ is a pair (A,E) and (B,E) of nonnull soft semi-open sets s.t. X~=(A,E)⋃(B,E), (A,E)⋂(B,E)=∅.Definition 37.
A soft topological space(X,τ,E) is called soft semi-connected space if there is no soft semiseparations on X~. And if (X,τ,E) has such soft semiseparations, then (X,τ,E) is called soft semi-disconnected space.Theorem 38.
Let(X,τ,E) be a soft topological space. (A,E) and (B,E) are semiseparations on X~. If (F,E) is a soft semi-connected subspace of (X,τ,E), then one has (F,E)⊆~(A,E) or (F,E)⊆~(B,E).Proof.
Because(A,E) and (B,E) are soft semi-open sets, then we have (F,E)∩(A,E) and (F,E)∩(B,E) are also soft semi-open sets. Hence (F,E)∩(A,E) and (F,E)∩(B,E) are semiseparations of (F,E). And this is a contradiction. So, one of (F,E)∩(A,E) and (F,E)∩(B,E) is ∅ and thus (F,E)⊆~(A,E) or (F,E)⊆~(B,E).Theorem 39.
Let(X,τ,E) be a soft topological space and (F,E) is a soft semi-connected subspace of (X,τ,E). If (F,E)⊆~(G,E)⊆~ClS(F,E), then (G,E) is soft semi-connected.Proof.
Suppose(G,E) is not soft semi-connected, then there exist nonnull soft semi-open sets (A,E) and (B,E) which form a soft semiseparation of (G,E). Then by Theorem 38, we have (F,E)⊆~(A,E) or (F,E)⊆~(B,E). Suppose (F,E)⊆~(A,E), then we have (G,E)⊆~ClS(F,E)⊆~ClS(A,E). So (G,E)∩(B,E)⊆~ClS(A,E)∩(B,E)=(A,E)∩(B,E)=∅. Hence (B,E)=(B,E)∩(G,E)=∅, a contradiction. Thus, we have (G,E) is soft semi-connected.Theorem 40.
A soft topological space(X,τ,E) is soft semi-connected if and only if the both soft semi-open and soft semi-closed soft sets are only ∅ and X~.Proof.
⇒ Let the soft topological space (X,τ,E) be soft semi-connected. If (A,E) is both soft semi-open and soft semi-closed in (X,τ,E) which is different from ∅ and X~. So (A,E)′ is a soft semi-open set in (X,τ,E) which is different from ∅ and X~. Then (A,E) and (A,E)′ is a soft semiseparation of (X,τ,E). This is a contradiction. So the both soft semi-open and soft semi-closed soft sets are only ∅ and X~.
⇐ Let (A,E) and (B,E) be a soft semiseparation of (X,τ,E). Let (A,E)≠X~ and by Definition 36(A,E)=(B,E)′. This proves that (A,E) is both soft semi-open and soft semi-closed in (X,τ,E) which is different from ∅ and X~. This is a contradiction. So (X,τ,E) is soft semi-connected.Corollary 41.
A soft topological space(X,τ1,E1) is soft semi-disconnected if and only if there exists a nonnull proper soft subset which is both soft semi-open and semi-closed.Proof.
By Theorem40.Theorem 42.
Let(X,τ1,E1) and (Y,τ2,E2) be two soft topological spaces. fpu:SS(X)E1→SS(Y)E2 be a soft semi-pu-continuous function. If (X,τ1,E1) is soft semi-connected, then (Y,τ2,E2) is soft semi-connected.Proof.
Suppose that(Y,τ2,E2) is not soft semi-connected. Then there is a soft semiseparation (A,E2) and (B,E2) of (Y,τ2,E2). So we have X~=fpu-1((A,E2)⋃(B,E2))=fpu-1(A,E2)⋃fpu-1(B,E2) and fpu-1(A,E2)∩fpu-1(B,E2)=fpu-1(Y~)=∅X. Obviously, fpu-1(A,E2) and fpu-1(B,E2) are different from ∅X. So fpu-1(A,E2) and fpu-1(B,E2) are a soft semiseparation of (X,τ1,E1). This forms a contradiction. Thus, (Y,τ2,E2) is soft semi-connected.


## 6. Concluding Remarks

In this paper, we introduce the concept of soft semi-neighborhoods of the soft point, soft semi-pu-continuous at the soft point and soft semi-connectedness. And some of their properties are studied. In the study, soft semi-first-countable space is also given. And in the following papers, soft Frechét space, soft sequential space, soft set-Frechét space and the tightness of a soft point and their connection with soft semi-first-countable space need further study. We hope that the results in this paper will be useful in practical life and nature society.



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*Source: 298032-2013-04-23.xml*