International Journal of Analysis and Applications
ISSN 2291-8639
Volume 12, Number 2 (2016), 198-206
http://www.etamaths.com

AN APPROACH TO THE CONCEPT OF SOFT VIETORIS TOPOLOGY

İZZETTİN DEMİR∗

Abstract. In the present paper, we study the Vietoris topology in the context of soft set. Firstly,
we investigate some aspects of first countability in the soft Vietoris topology. Then, we obtain some

properties about its second countability.

1. Introduction

In 1999, Molodtsov [22] initiated the concept of a soft set theory as a new approach for coping with
uncertainties and also presented the basic results of the new theory. In [22], Molodtsov successfully ap-
plied the soft set theory in several directions, such as smoothness of functions, game theory, operations
research, Riemann integration, Perron integration and theory of measurement. After presentation of
the operations of soft sets [21], the properties and applications of this theory have been studied in-
creasingly ([4], [23], [25]).

Aktaş and Çağman [3] introduced the soft group and also compared soft sets to fuzzy set and rough
set. Shabir and Naz [29] initiated the study of soft topological spaces. Rong [28] presented the no-
tions of soft first countable and soft second countable spaces and investigated some their fundamental
properties. Recently, many papers concerning the soft set theory have been published ([7], [11], [14],
[16], [18], [24], [26]).

Hyperspace theory had its early beginnings in the 1900’s, with the work of Hausdorff and Vietoris.
This theory plays a fundamental role in mathematics and applied sciences, such as Convex Analysis,
Optimization, Economics and Image Processing. Over the years, a lot of research has been performed
on this subject ([5], [6], [9], [17], [20]).

As hyperspace of a topological space (X,τ), it means C(X), the set of closed subsets of X, equipped
with a topology τh such that the function i : (X,τ) → (C(X),τh) defined by i(x) = {x} is a homeo-
morphism onto its image. One of the most important and well-studied hyperspace topologies on C(X)
is the Vietoris topology. The Vietoris topology is a basic construct due to its usefulness in different
areas of mathematics and applications. Therefore, this topology has attracted the attention of many
mathematicians in the last few decades ([8], [12], [13], [15], [19], [27], [32]).

Extensions of hypertopologies to the soft sets have been studied by some authors. Akdağ and Erol
[1] and Shakir [30] defined independently a hyperspace of soft sets, called soft Vietoris topological
space. Later, Akdağ and Erol [2] studied on some hyperspaces of soft sets such as co-quasi H-closed
soft topological spaces and D-soft topological spaces.

In this paper, firstly we present a brief synopsis of all necessary definitions and results that will be
required. Next, we continue studying the soft Vietoris topology and obtain some results about its first
and second countability.

2. Preliminaries

In this section, we recollect some basic notions regarding soft sets. Throughout this work, let X be
an initial universe, P(X) be the power set of X and E be a set of parameters for X,

2010 Mathematics Subject Classification. 06D72, 54A40, 54B20.
Key words and phrases. soft set; soft Vietoris topological space; soft first countable space; soft second countable

space; soft separable space.

c©2016 Authors retain the copyrights of
their papers, and all open access articles are distributed under the terms of the Creative Commons Attribution License.

198



AN APPROACH TO THE CONCEPT OF SOFT VIETORIS TOPOLOGY 199

Definition 2.1 ([22]). A soft set F on the universe X with the set E of parameters is defined by the
set of ordered pairs

F = {(e,F(e)) : e ∈ E,F(e) ∈ P(X)}

where F is a mapping given by F : E → P(X).

Throughout this paper, the family of all soft sets over X is denoted by S(X,E) [7].

Definition 2.2 ([4], [21], [25]). Let F,G ∈ S(X,E). Then,
(i) The soft set F is called a null soft set, denoted by ∅̃, if F(e) = ∅ for every e ∈ E.
(ii) The soft set F is called an absolute soft set, denoted by X̃, if F(e) = X for every e ∈ E.
(iii) F is a soft subset of G if F(e) ⊆ G(e) for every e ∈ E. It is denoted by F v G.
(iv) The complement of F is denoted by Fc, where Fc : E → P(X) is a mapping defined by

Fc(e) = X −F(e) for every e ∈ E. Clearly, (Fc)c = F.
(v) The union of F and G is a soft set H defined by H(e) = F(e) ∪ G(e) for every e ∈ E. H is

denoted by F tG.
(vi) The intersection of F and G is a soft set H defined by H(e) = F(e)∩G(e) for every e ∈ E. H

is denoted by F uG.
(vii) The difference of F and G is a soft set H defined by H(e) = F(e) −G(e) for every e ∈ E. H

is denoted by F ^ G.

Definition 2.3 ([10], [19], [24]). A soft set P over X is said to be a soft point if there exists an e ∈ E
such that P(e) = {x} for some x ∈ X and P(e′) = ∅ for every e′ ∈ E\{e}. This soft point is denoted
as xe.

A soft point xe is said to belongs to a soft set F , denoted by xe ∈̃F , if x ∈ F(e).

From now on, let SP(X) be the family of all soft points over X.

Definition 2.4 ([29]). Let τ be a collection of soft sets over X, then τ is said to be a soft topology on
X if

(st1) ∅̃,X̃ belong to τ.
(st2) the union of any number of soft sets in τ belongs to τ.
(st3) the intersection of any two soft sets in τ belongs to τ
(X,τ,E) is called a soft topological space. The members of τ are called soft open sets in X. A soft

set F over X is called a soft closed in X if Fc ∈ τ.

Definition 2.5 ([7], [24]). Let (X,τ,E) be a soft topological space. A subcollection B of τ is called a
soft base for τ if every member of τ can be expressed as the union of some members of B.

Definition 2.6. Let (X,τ,E) be a soft topological space and F ∈ S(X,E).
(i) The soft interior of F is the soft set Fo = t{G : G is soft open set and G v F} [31].
(ii) The soft closure of F is the soft set F = u{G : G is soft closed set and F v G} [29].

Definition 2.7 ([24]). A soft set F in a soft topological space (X,τ,E) is called a soft neighborhood
of the soft point xe if there exists a soft open set G such that xe ∈̃ G v F .

The soft neighborhood system of a soft point xe, denoted by N(xe), is the family of all its soft
neighborhoods.

Definition 2.8 ([29]). Let (X,τ,E) be a soft topological space and Y be a non-empty subset of X.

Then, τY = {Ỹ u F : F ∈ τ} is called the soft relative topology on Y and (Y,τY ,E) is called a soft
subspace of (X,τ,E).



200 DEMİR

Here, Ỹ is the soft set over X defined by Ỹ (e) = Y for all e ∈ E.

Definition 2.9 ([28]). Let (X,τ,E) be a soft topological space and xe ∈̃X̃. A subcollection B of τ is
called a soft local base at a soft point xe if for every soft open set F containing xe, there exists a G ∈B
such that xe ∈̃G v F .

Definition 2.10 ([11]). Let (X,τ,E) be a soft topological space, {xenn : n ∈ N} be a sequence of soft
points in (X,τ,E) and xe ∈ SP(X). The sequence {xenn : n ∈ N} is said to converge to xe, and we
write xenn → xe, if for every F ∈N(xe), there exists an n0 ∈ N such that xenn ∈̃ F for all n ≥ n0.

Definition 2.11 ([24]). Let (X,τ,E) be a soft topological space. A soft point xe ∈ SP(X) is called a
limiting soft point of a soft set F over X if every soft open set containing xe contains at least one soft

point of F other than xe, i.e., if F u (G ^ xe) 6= ∅̃ for every G ∈ τ containing xe.
The union of all limiting soft points of F is called the derived soft set of F and is denoted by F ′.

Definition 2.12 ([28]). Let (X,τ,E) be a soft topological space.
(i) If each soft point in X has a countable soft local base, then it is called a soft first countable space.
(ii) If there exists a countable soft base for τ, then it is a soft second countable space.

Definition 2.13 ([28]). Let (X,τ,E) be a soft topological space. If there exists a family {xenn : n ∈ N}
of countable many soft points in X such that

⊔
n∈N x

en
n = X̃, then (X,τ,E) is called a soft separable

space.

Definition 2.14. Let (X,τ,E) be a soft topological space. Then,

(i) It is called a soft T1-space if every soft point in X is a soft closed set [18].

(ii) It is called a soft Hausdorff space or a soft T2-space if for any two distinct soft points x
e1
1 ,

xe22 ∈ SP(X) there exist soft open sets F and G such that x
e1
1 ∈̃F,x

e2
2 ∈̃G and F uG = ∅̃ [11].

(iii) It is called a soft regular space if for every xe ∈̃X̃ and every soft closed set F such that xe /̃∈F ,
there exist soft open sets F1 and F2 such that x

e ∈̃F1, F v F2 and F1 uF2 = ∅̃ [16].

Theorem 2.15 ([16]). Let (X,τ,E) be a soft topological space. Then, the following statements are
equivalent:

(1) (X,τ,E) is a soft regular space.
(2) For any soft open set F in (X,τ,E) and xe ∈̃F , there exists a soft open set G containing xe such
that xe ∈̃G v F .
(3) For any soft closed set H in (X,τ,E) and xe /̃∈H, there exists a soft open set G containing xe such
that GuH = ∅̃.

Definition 2.16 ([7], [31]). Let (X,τ,E) be a soft topological space and F ∈ S(X,E).
(i) A family C = {Fi : i ∈ I} of soft sets over X is called a cover of F if it satisfies F v

⊔
i∈I Fi. It

is called a soft open cover if each member of C is a soft open set. A subfamily of C is called a subcover
of C if it is also a cover of F .

(ii) (X,τ,E) is called a soft compact space if every soft open cover of X̃ has a finite subcover.

Theorem 2.17 ([26]). Let (Y,τY ,E) be a soft subspace of a soft topological space (X,τ,E). Then,

(Y,τY ,E) is a soft compact space if and only if every cover of Ỹ by soft open sets over X contains a
finite subcover.

Theorem 2.18 ([26]). Every soft compact subspace of a soft Hausdorff space is soft closed.



AN APPROACH TO THE CONCEPT OF SOFT VIETORIS TOPOLOGY 201

Definition 2.19 ([1]). Let (X,τ,E) be a soft topological space and F ∈ τ. Then, the families of soft
sets F+ and F− are defined as follows:

F+ = {G ∈ SC(X) : G v F} and F− = {G ∈ SC(X) : F uG 6= ∅̃},
where SC(X) is the family of non-null soft closed sets over X.

Proposition 2.20 ([1]). Let (X,τ,E) be a soft topological space. For non-null soft sets F and G, the
following statements are true:

(i) F+ ∩G+ = (F uG)+.
(ii) F+ ∪G+ ⊆ (F tG)+.
(iii) (F uG)− ⊆ F− ∩G−.
(iv) F− ∪G− = (F tG)−.
(v) F v G if and only if F+ ⊆ G+.
(vi) F v G if and only if F− ⊆ G−.

Proposition 2.21 ([1]). Let (X,τ,E) be a soft topological space. Then, the families

δ+SV = {F
+ : F ∈ τ} and δ−SV = {F

− : F ∈ τ}
are subbases for the topological spaces τ+SV and τ

−
SV on SC(X), respectively.

Definition 2.22 ([1]). The topological spaces τ+SV and τ
−
SV on SC(X) which mentioned in above

proposition are called a soft upper Vietoris topological space and a soft lower Vietoris topological space,
respectively.

Definition 2.23 ([1]). A topological space on SC(X) with δSV = δ
+
SV ∪δ

−
SV as subbase is called a soft

Vietoris topological space, denoted by τSV .

3. First and Second Countability of the Soft Vietoris Topology

The aim of this section is to present some properties related to countability of soft Vietoris topology.
Firstly, we focus on its first countability.

Theorem 3.1. Let (X,τ,E) be a soft T1-space. Then, the following statements are equivalent:

(1) (SC(X),τSV ) is a first countable space.
(2) (SC(X),τ+SV ) and (SC(X),τ

−
SV ) are first countable spaces.

Proof. (2) ⇒ (1) is obvious from Definition 2.22 and 2.23.
To prove (1) ⇒ (2), let (SC(X),τSV ) be a first countable space and take H ∈ SC(X). Let

L =
{
` =

⋂
i∈I

F+i ∩
⋂
j∈J

G−j : I,J is finite

}
be a countable local base at H for τSV . Then, the family

L+ =
{⋂

i∈I
F+i :

⋂
i∈I

F+i occurs in some ` ∈L
}
∪{SC(X)}

forms a countable local base at H for τ+SV . Indeed, if there is no
⋂

i∈I F
+
i ∈ ` with H ∈

⋂
i∈I F

+
i , then

SC(X) is the only open set in τ+SV containing H. If H ∈ F
+ and F+ ∈ τ+SV , where F 6= X̃, then there

exists an ` ∈L such that H ∈ ` ⊆ F+. Therefore, ` must be of the form
⋂

i∈I F
+
i ∩

⋂
j∈J G

−
j where I

is nonempty. Suppose that I = ∅. Then, since H ∈
⋂

j∈J G
−
j , we obtain (H tF

c) ∈
⋂

j∈J G
−
j . Also,

we know that (H tFc) /∈ F+. But this contradicts the fact that
⋂

j∈J G
−
j ⊆ F

+. Next, observe that

we have Fc v
⊔

i∈I F
c
i , since otherwise, there would be an x

e ∈̃X̃ such that xe ∈̃Fc, xe /̃∈
⊔

i∈I F
c
i and

(H txe) ∈ `−F+. Thus, by Proposition 2.20 (i) and (v), we obtain
⋂

i∈I F
+
i ⊆ F

+.

Now, we shall show that there exists a countable local base at H for τ−SV . Let H ∈ G
− and



202 DEMİR

G− ∈ τ−SV , where G 6= X̃. Since G
− ∈ τSV , there exists an ` ∈L such that H ∈ ` ⊆ G−. Without loss

of generality we can suppose that in the expression of every element from L the family J is nonempty(
for example,

⋂
i∈I F

+
i =

⋂
i∈I F

+
i ∩ X̃

−
)
. So, we have ` =

⋂
i∈I F

+
i ∩

⋂
j∈J G

−
j . For each j ∈ J, let

us take a soft set Kj = Gj ^
⊔

i∈I F
c
i and define K` =

⋂
j∈J K

−
j . It easy to see that H ∈ K`. Then,

L− =
{
K` : ` ∈L

}
forms a countable local base at H for τ−SV . Indeed, since there exists a j ∈ J with

Kj v G, we obtain K` ⊆ G−. Suppose that for each j ∈ J there exists an x
ej
j ∈̃Kj ^ G. Therefore,

we get
⊔

j∈J x
ej
j ∈ `−G

−, which yields a contradiction.

Theorem 3.2. Let (X,τ,E) be a soft T1-space. Then,

(SC(X),τ−SV ) is a first countable space if and only if (X,τ,E) is a soft first countable space and
each soft closed set over X is a soft separable.

Proof. Let (SC(X),τ−SV ) be a first countable space. From the fact that each soft point in X is a soft
closed set it follows that (X,τ,E) is a soft first countable space. Let H ∈ SC(X) and {Fn : n ∈ N} be
a countable family of nonempty soft open sets which determines a countable local base at H for τ−SV .

Because H u Fn 6= ∅̃ for each n ∈ N, we may choose a soft point xenn ∈̃H u Fn. Now, we shall show
that

⊔
n∈N x

en
n = H. It is easy to see that

⊔
n∈N x

en
n v H. Let xe ∈̃H. For each G ∈ τ containing xe,

we have GuH 6= ∅̃. Then, there exist n1, ...,nk ∈ N such that F−n1 ∩ ...∩F
−
nk
⊆ G−. Therefore, we

obtain Fnj v G for some j ∈{1, ...,k}. Thus, since x
ej
j ∈̃Fnj v G, we get Gu

(⊔
n∈N x

en
n

)
6= ∅̃ and so

that xe ∈̃
⊔

n∈N x
en
n .

On the other hand, let H ∈ SC(X). Then, by hypothesis, there exists a family {xenn : n ∈ N}
of countable many soft points in H such that

⊔
n∈N x

en
n = H. Now, let B(xenn ) be a countable soft

local base at xenn for each n ∈ N. Thus, one can readily verify that B(H) =
{⋂

j∈J G
−
j : Gj ∈

B(xejj ),J isfinite
}

is a countable local base at H for τ−SV .

For each H ∈ SC(X), we define H∗ = {F ∈ SC(X) : F u H = ∅̃}. Then, we get the following
theorem.

Theorem 3.3. Let (X,τ,E) be a soft T1-space. Then,

(SC(X),τ+SV ) is a first countable space if and only if for each H ∈ SC(X), there exists a countable
family LH ⊆ H∗ such that for each F ∈ H∗ there exist F1,F2, ...,Fn ∈LH with F v F1 tF2 t ...tFn.

Proof. The sufficiency is clear.
To prove necessity, let (SC(X),τ+SV ) be a first countable space and let H ∈ SC(X). Then, the

family

H =
{⋂

i∈I
F+i : F

c
i ∈ H

∗,I isfinite
}
∪{SC(X)}

is a countable local base at H for τ+SV . Now, put

LH =
{
G ∈ H∗ : Goccursinsomeelementof H

}
.

It is easy to see that LH is a countable family of H∗. Let F ∈ H∗. Then, we obtain H ∈ (Fc)+
and (Fc)+ ∈ τ+SV . Therefore, there exists a member

⋂n
i=1 F

+
i of H such that H ∈

⋂n
i=1 F

+
i ⊆ (F

c)+.
By Proposition 2.20 (i) and (v), we have F v Fc1 t ...tFcn. Thus, it follows from Fci ∈ LH for each
i ∈{1, ...,n} that the family LH has the required properties.

Definition 3.4. Let (X,τ,E) be a soft topological space, F ∈ S(X,E) and let F =
{
H : H v F

}
be a

family of non-null soft closed sets. Then, F is called a hemi-SC(X) if there exists a countable cofinal
subfamily of F with respect to inclusion relation.



AN APPROACH TO THE CONCEPT OF SOFT VIETORIS TOPOLOGY 203

Example 3.5. Let X = {x,y,z}, E = {e1,e2} and let F be a soft set over X, where F =
{(e1,{x,z}), (e2,{y})}. Consider a discrete soft topology τ on X and a family

F =
{
xe1,ze1,ye2,{(e1,{x}), (e2,{y})},{(e1,{z}), (e2,{y})},{(e1,{x,z})},F

}
of non-null soft closed sets contained in F . Then, a subfamily G = {F}⊂F is cofinal in F since there
exists an F ∈G such that H v F for every H ∈F. Thus, F is a hemi-SC(X).

Definition 3.6. The character of a soft set F ∈ S(X,E) in a soft topological space (X,τ,E) is defined
as the smallest cardinal number of the form |B(F)|, where B(F) is a soft local base at F for τ. This
cardinal number is denoted by χ(F).

Example 3.7. Let X = {x,y,z}, E = {e1,e2} and let F be a soft set over X, where F =
{(e1,{z}), (e2,{x,y})}. Let us define a soft topology on X as the following:

τ = {G : xe1 ∈̃G}∪{∅̃}.

Then, B(F) = {H} is a soft local base at F for τ, where H = {(e1,{x,z}), (e2,{x,y})}. Thus, the
character of F is χ(F) = 1.

Using the above definitions and theorems, we can easily prove the following corollaries.

Corollary 3.8. Let (X,τ,E) be a soft T1-space. Then, the following statements are satisfied:

(1) (SC(X),τ+SV ) is a first countable space if and only if each soft open set F , where F 6= X̃, is a
hemi-SC(X).

(2) (SC(X),τ+SV ) is a first countable space if and only if χ(F) ≤ |N| for each F ∈ SC(X), where |N|
denotes the cardinal number of N.

Corollary 3.9. Let (X,τ,E) be a soft T1-space. Then, (SC(X),τSV ) is a first countable space if and
only if the following three conditions hold:

(i) (X,τ,E) is a soft first countable space.
(ii) Each soft closed set over X is a separable.
(iii) Each soft open set over X is a hemi-SC(X).

Lemma 3.10. Let (X,τ,E) be a soft topological space and let {xenn : n ∈ N} be a sequence of soft
points in X. If xenn → xe ∈ SP(X), then F =

⊔
n∈N x

en
n txe is a soft compact set.

Proof. Let C = {Fi : i ∈ I} be a cover of F by soft open sets over X. Then, there exists an i0 ∈ I
such that xe ∈̃Fi0 . Since xenn → xe, there exists an n0 ∈ N such that xenn ∈̃Fi0 for all n ≥ n0. Now, let
us take an Fn ∈C satisfying xenn ∈̃Fn for all n < n0. Therefore, we have xenn ∈̃

⊔n0−1
i=1 Fi for all n < n0

and so that F v Fi0 t
⊔n0−1

i=1 Fi. Hence, the family {Fi0}∪{Fi : i = 1, ...,n0 − 1} is a finite subcover
of C. Thus, from Theorem 2.17 it follows that F is a compact set.

Theorem 3.11. Let (X,τ,E) be a soft Hausdorff space. If (SC(X),τ+SV ) is a first countable space,
then (X,τ,E) is a soft regular space.

Proof. Let F be a soft open set and xe ∈̃F. By Theorem 2.15, we shall show that there exists a
soft open set G such that xe ∈̃G v G v F. Without loss of generality we can suppose that F 6= X̃.
The first countability of (SC(X),τ+SV ) implies that F is a hemi-SC(X). Let {F1,F2, ...,Fn, ...} be a
countable cofinal subfamily in the family {H ∈ SC(X) : H v F}. Also, one can easily verify that
(X,τ,E) is a soft first countable space. Let us denote by {Un : n ∈ N} a countable soft local base at
xe with Un v F for each n ∈ N. Now, we claim that there exists an n ∈ N such that xe ∈̃Fon. Indeed,
suppose that xe /̃∈Fon for each n ∈ N. Then, there exists a soft point xenn ∈̃Un ^ Fn for each n ∈ N.
Therefore, we see that the sequence {xenn : n ∈ N} converges to xe. From Lemma 3.10 it follows that
L =

⊔
n∈N x

en
n t xe is a soft compact set. Also, we have L v F and by Theorem 2.18, we obtain



204 DEMİR

L ∈ SC(X). Hence, using the cofinality condition, we get L v Fn for some n ∈ N, which yields a
contradiction. Thus, there exists an n ∈ N such that xe ∈̃Fon v Fon v F.

Definition 3.12. Let (X,τ,E) be a soft topological space and {xeii : i ∈ I} a family of infinitely soft
points in X. (X,τ,E) is called a countably soft compact space if the soft set

⊔
i∈I x

ei
i has a limiting

soft point.

Theorem 3.13. Let (X,τ,E) be a soft Hausdorff space. If (SC(X),τ+SV ) is a first countable space,

then the derived soft set X̃′ of X̃ is a countably soft compact.

Proof. Suppose that X̃′ is not countably soft compact. Then, there exists a family {xenn : n ∈ N} of
countable many soft points in X̃′ such that F =

⊔
n∈N x

en
n does not have a limiting soft point. By

Theorem 3.11, (X,τ,E) is a soft regular space and therefore there exists a pairwise disjoint countable
family of soft open sets {Un : n ∈ N} such that xenn ∈̃Un for every n ∈ N. Also, from Corollary 3.8(2)
it follows that there exists a countable soft local base {Gn : n ∈ N} of soft open sets at F.

Now, for every n ∈ N, we can choose a yknn ∈ SP(X) with yknn ∈̃(Un uGn) ^ F because xenn is a
limiting soft point of X̃. Let G =

⊔
n∈N y

kn
n . Then, we have F uG = ∅̃. Indeed, suppose that there

exists a soft point x
en0
n0 such that x

en0
n0 ∈̃F and x

en0
n0 ∈̃G. Take a soft set H = Un0 ^ y

kn0
n0 . Since the

family {Un : n ∈ N} is pairwise disjoint, H is a soft open neighborhood of x
en0
n0 such that H uG = ∅̃.

This is a contradiction since x
en0
n0 ∈̃G.

Hence, since F uG = ∅̃, there exists a Gn such that F v Gn v G
c
. But, this is a contradiction to

the fact that yknn ∈̃G. Thus, X̃′ is a countably soft compact.

We now consider the second countability of the soft Vietoris topology.

Theorem 3.14. Let (X,τ,E) be a soft topological space. Then, the following statements are equiva-
lent:

(1) (SC(X),τ−SV ) is a second countable space.
(2) (X,τ,E) is a soft second countable space.

Proof. It is clear from Proposition 2.20 (iii)-(iv) and (vi).

Theorem 3.15. Let (X,τ,E) be a soft T1-space. Then, the following statements are equivalent:

(1) (SC(X),τSV ) is a second countable space.
(2) (SC(X),τ+SV ) and (SC(X),τ

−
SV ) are second countable spaces.

Proof. (2) ⇒ (1) follows immediately from Definition 2.22 and 2.23.
To prove (1) ⇒ (2), let (SC(X),τSV ) be a second countable space. From the fact that every soft

point in X is a soft closed set it follows that (X,τ,E) is a soft second countable space. Hence, by
Theorem 3.14, (SC(X),τ−SV ) is a second countable space.

Let

L =
{
` =

⋂
i∈I

F+i ∩
⋂
j∈J

G−j : I,J is finite

}
be a countable base for τSV . Then, the family

L+ =
{⋂

i∈I
F+i :

⋂
i∈I

F+i occurs in some ` ∈L
}
∪{SC(X)}

forms a countable base for τ+SV . Indeed, if there is no
⋂

i∈I F
+
i ∈ ` with H ∈

⋂
i∈I F

+
i , then SC(X) is

the only open set in τ+SV containing H. Let H ∈ F
+ and F+ ∈ τ+SV , where F 6= X̃. Since F

+ ∈ τSV ,
there exists an ` ∈ L such that H ∈ ` ⊆ F+. Therefore, ` must be of the form

⋂
i∈I F

+
i ∩

⋂
j∈J G

−
j

where I is nonempty (see the proof of Theorem 3.1). Thus, as is shown in the proof of Theorem 3.1,



AN APPROACH TO THE CONCEPT OF SOFT VIETORIS TOPOLOGY 205

we get
⋂

i∈I F
+
i ⊆ F

+, which completes the proof.

Theorem 3.16. Let (X,τ,E) be a soft T1-space. Then,

(SC(X),τ+SV ) is a second countable space if and only if there exists a countable family ∆ ⊆ SC(X)
such that for each F ∈ SC(X) and for each G ∈ τ satisfying F v G there exist F1,F2, ...,Fn ∈ ∆ with
F v F1 tF2 t ...tFn v G.

Proof. Let L be a countable base for τ+SV . We know that every element in L can be written as⋂{
H+j : j ∈ J

}
, where J is a finite set. Take

∆ =
{
F ∈ SC(X) : F occurs in the presentation of some element from L

}
.

One can readily verify that ∆ is a countable family of SC(X). Let F ∈ SC(X) and G ∈ τ such that
F v G. If F = X̃, then we are done. So, suppose that F 6= X̃ and also G 6= X̃. Now, let us take a soft
set K = Gc. Then, we have K ∈ SC(X) and K ∈ (Fc)+. Therefore, there exists a member

⋂
j∈J H

+
j

of L such that K ∈
⋂

j∈J H
+
j ⊆ (F

c)+. Hence, from Proposition 2.20 (i) and (v) it follows that

F v
⊔
j∈J

Hcj v K
c = G.

Thus, since Hcj ∈ ∆ for each j ∈ J, the family ∆ ⊆ SC(X) has the required properties.
For the converse, the family of sets of the form

⋂{
(Fcj )

+ : j ∈ J
}

, where Fj ∈ ∆ and J is a finite
set, together with SC(X) is a countable base for τ+SV .

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Department of Mathematics, Duzce University, 81620, Duzce-Turkey

∗Corresponding author: izzettindemir@duzce.edu.tr