EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS
Vol. 7, No. 1, 2014, 97-108
ISSN 1307-5543 – www.ejpam.com

Compact Soft Multi Spaces

İsmail Osmanoğlu and Deniz Tokat∗

Department of Mathematics, Faculty of Arts and Sciences, Nevşehir Hacı Bektaş Veli University,
50300 Nevşehir, Turkey

Abstract. In this work, first we recall the concepts of soft multiset and soft multi topology. Then we
will examine the concept of soft multi function which is defined between two soft multi class. Finally
we will introduce soft multi compactness on soft multi topological space and give basic definitions and
theorems about it.

2010 Mathematics Subject Classifications: 03E70, 54A99, 54D30.

Key Words and Phrases: Soft multiset, Soft multi function, Soft multi topology, Soft multi compact-
ness.

1. Introduction

Classical mathematical methods are not enough to solve the problems of daily life and
also are not enough to meet the new requirements. Therefore, some theories such as Fuzzy
set theory [17], Rough set theory [9], Soft set theory [8] and Multiset (or Bag) theory [16]
have been developed to solve these problems.

Applications of these theories have been many areas of mathematics. Shabir and Naz
[11] defined the soft topological space and studied the concepts of soft open set, soft multi
interior point, soft neighborhood of a point, soft separation axioms, and subspace of a soft
topological space. Aygunoglu and Aygun [2] introduced the soft continuity of soft mapping,
soft product topology and studied soft compactness and generalized Tychono theorem to the
soft topological space. Min [7] gave some results on soft topological spaces. Zorlutuna et
al. [18] also investigated soft interior point and soft neighborhood. There are some other
studies on the structure of soft topological spaces [3, 15]. Maji et al. [6] also initiated the
more generalized concept of fuzzy soft sets which is a combination of fuzzy set and soft set.
Tanay and Kandemir introduced topological structure of fuzzy soft set in [12] and gave a
introductory theoretical base to carry further study on this concept. Following this study,
some others [1, 5, 10, 14] studied on the concept of fuzzy soft topological spaces.

∗Corresponding author.

Email addresses: ismailosmanoglu@yahoo.com (̇I. Osmanoğlu), dtokat@nevsehir.edu.tr (D. Tokat)

http://www.ejpam.com 97 c© 2014 EJPAM All rights reserved.



İ. Osmanoğlu, D. Tokat / Eur. J. Pure Appl. Math, 7 (2014), 97-108 98

The concept of soft multisets which is combining soft sets and multisets can be used to
solve some real life problems. Also this concept can be used in many areas, such as data
storage, computer science, information science, medicine, engineering, etc. The concept of
soft multisets was introduced in [13]. Moreover, in [13] soft multi topology and its some
properties was given. Also [13] soft multi connectedness was given.

In this work we will defined soft multi function between two soft multiset. After we will
introduce soft multi compactness on soft multi topological space and will give basic definitions
and theorems of soft multi compactness.

2. Preliminaries

2.1. Soft Set, Multiset and Soft Multiset

In this section, we present the basic definitions of soft set, multiset and soft multiset which
may be found in earlier studies [4, 8, 13].

Definition 1 (Soft set). Let U be an initial universe set and E be set of parameters. Let P (U)
denotes the power set of U and A ⊆ U. A pair (F, A) is called a soft set over U, where F is a
mapping given by F : A→ P(U).

Definition 2 (Multiset). An mset M drawn from the set X is represented by a function Count
M or CM defined as CM : X → N where N represented the set of non negative integers. The word
“multiset” is often shortened to “mset”.

Let M be an mset from X with x appearing n times in M . It is denoted by x ∈n M .
M = {k1/x1, k2/x2, . . . , kn/xn} where M is an mset with x1 appearing k1 times, x2 appearing
k2 times and so on.

Definition 3. Let M be an mset drawn from a set X . The support set of M denoted by M∗ is a
subset of X and M∗ = {x ∈ X : CM (x) > 0}. i.e., M∗ is an ordinary set and it is also called root
set.

The power set of an mset is the support set of the power mset and is denoted by P∗ (M).

Example 1. Let M =
�

2/x , 3/y
	

be an mset. Then M∗ =
�

x , y
	

is the support set of M and

P∗ (M) =
��

2/x , 1/y
	

,
�

2/x , 2/y
	

,
�

1/x , 1/y
	

, {1/x , 2/y},
�

1/x , 3/y
	

, {2/x} , {1/x} ,
�

3/y
	

,
�

2/y
	

,
�

1/y
		

is the support set of P (M).

Definition 4 (Soft multiset). Let U be an universal multiset , E be set of parameters and A⊆ E.
Then a pair (F, A) is called a soft multiset where F is a mapping given by F : A→ P∗ (U). For
∀e ∈ A, multiset F (e) represent by count function CF(e) : U∗ → N where N represents the set of
non negative integers.



İ. Osmanoğlu, D. Tokat / Eur. J. Pure Appl. Math, 7 (2014), 97-108 99

Example 2. Let multiset and the parameter set be U = {1/x , 5/y, 3/z, 4/w} and E =
�

p, q, r
	

.
Define a mapping F : E→ P∗ (U) as follows:

F
�

p
�

=
�

1/x , 2/y, 3/z
	

,

F
�

q
�

= {4/w} and

F (r) =
�

3/y, 1/z, 2/w
	

.

Then (F, A) is a soft multiset where for ∀e ∈ A, F (e) multiset represent by count function
CF(e) : U∗→ N, which are defined as follows:

CF(p) (x) = 1, CF(p)
�

y
�

= 2, CF(p) (z) = 3, CF(p) (w) = 0,
CF(q) (x) = 0, CF(q)

�

y
�

= 0, CF(q) (z) = 0, CF(q) (w) = 4,
CF(r) (x) = 0, CF(r)

�

y
�

= 3, CF(r) (z) = 1, CF(r) (w) = 2.

Then (F, A) =
�

F
�

p
�

, F
�

q
�

, F (r)
	

= {
�

1/x , 2/y, 3/z
	

, {4/w} , {3/y, 1/z,
2/w}}.

Definition 5. For two soft multisets (F, A) and (G, B) over U, we say that (F, A) is a soft submul-
tiset of (G, B) if

i. A⊆ B

ii. CF(e) (x)≤ CG(e) (x), ∀x ∈ U∗, ∀e ∈ A

We write (F, A)⊂̃(G, B). In addition to (F, A) is a whole soft submultiset of (G, B) if
CF(e) (x) = CG(e) (x), ∀x ∈ U∗, ∀e ∈ A.

Definition 6. Let (F, A) and (G, B) be two soft multisets over U.

Equal (F, A) = (G, B)⇔ (F, A) ⊆̃ (G, B) and (F, A) ⊇̃ (G, B).

Union (H, C) = (F, A)∪̃(G, B) where C = A ∪ B and CH(e) (x) = max{CF(e) (x), CG(e) (x)},
∀e ∈ A∪ B, ∀x ∈ U∗.

Intersection (H, C) = (F, A)∩̃(G, B) where C = A∩B and CH(e) (x) =min{CF(e) (x) , CG(e) (x)},
∀e ∈ A∩ B, ∀x ∈ U∗. We write (F, A)∩̃(G, B).

Difference (H, E) = (F, E)\(G, E) where CH(e) (x) =max
¦

CF(e) (x)− CG(e) (x) , 0
©

, ∀x ∈ U∗.

Null A soft multiset (F, A) is said to be a NULL soft multiset denoted by Φ if for all e ∈ A, F(e) = ;.

Complement The complement of a soft multiset (F, A) is denoted by (F, A)c and is defined by
(F, A)c = (F

c
, A) where F

c
: A→ P∗(U) is a mapping given by F c(e) = U\F(e) for all e ∈ A

where CF c(e) (x) = CU (x)− CF(e) (x), ∀x ∈ U∗.

Definition 7. Let (F, E) be a soft multiset over U and a ∈ U∗. We say that a ∈ (F, E) read as a
belongs to the soft multiset (F, E) whenever a ∈ F(e) for all e ∈ E.

Note that for any a ∈ U, a /∈ (F, E), if a /∈ F(e) for some e ∈ E.



İ. Osmanoğlu, D. Tokat / Eur. J. Pure Appl. Math, 7 (2014), 97-108 100

Let (F, E) be soft multiset over U .If for all e ∈ E and a ∈ U∗, CF(e) (a) = n (n≥ 1) then we
will write a ∈ F (e) instead of a ∈n F (e).

Definition 8. Let V be a non-empty submultiset of U, then Ṽ denotes the soft multiset (V, E) over
U for which V (e) = V , for all e ∈ E.

In particular, (U , E) will be denoted by Ũ .

Definition 9. Let a ∈ U∗, then (a, E) denotes the soft multiset over U for which a(e) = {a}, for
all e ∈ E.

Definition 10. Let (F, E) be a soft multiset over U and V be a non-empty submultiset of U. Then
the sub soft multiset of (F, E) over V denoted by (V F, E), is defined as follows

V F(e) = V ∩ F(e), for all e ∈ E where CV F(e) (x) =min{CV (x) , CF(e) (x)},∀x ∈ U∗

In other words (V F, E) = Ṽ ∩̃(F, E).

2.2. Soft Multi Topology

In this section, we recall soft multi topology which given in [13].

Definition 11. Let X be universal multiset and E be set of parameters. Then the collection of all
soft multisets over X with parameters from E is called a soft multi class and is denoted as XE .

Definition 12. Let τ ⊆ XE , then τ is said to be a soft multi topology on X if the following
conditions hold.

i. Φ, X̃ belong to τ.

ii. The union of any number of soft multisets in τ belongs to τ.

iii. The intersection of any two soft multisets in τ belongs to τ.

τ is called a soft multi topology over X and the binary (XE ,τ) is called a soft multi topological
space over X .

The members of τ are said to be soft multi open sets in X .
A soft multiset (F, E) over X is said to be a soft multi closed set in X , if its complement (F, E)c

belongs to τ.

Example 3. et X = {2/x , 3/y, 4/z, 5/w}, E = {p, q} and τ = {Φ, X̃ , (F1, E), (F2, E), (F3, E)}
where (F1, E), (F2, E), (F3, E) are soft multisets over X , defined as follows

F1
�

p
�

=
�

1/x , 2/y, 3/z
	

, F1
�

q
�

= {4/w}
F2
�

p
�

= X , F2
�

q
�

=
�

1/x , 3/y, 4/z, 5/w
	

F3
�

p
�

=
�

2/x , 3/y, 3/z, 1/w
	

, F3
�

q
�

= {1/x , 4/w} .

Then τ defines a soft multi topology on X and hence (XE ,τ) is a soft multi topological space
over X .



İ. Osmanoğlu, D. Tokat / Eur. J. Pure Appl. Math, 7 (2014), 97-108 101

Definition 13. Let (XE ,τ1) and (XE ,τ2) be soft multi topological spaces. Then, the following
hold.

• If τ2 ⊃ τ1, then T2 is soft multi finer than τ1.

• If τ2 ⊃ τ1, then τ2 is soft multi strictly finer than τ1.

• If either τ2 ⊇ τ1 or τ2 ⊆ τ1, then τ1 is comparable with τ2.

Definition 14. Let X be universal multiset, E be the set of parameters.

• let τ be the collection of all soft multisets which can be defined over X . Then τ is called the
soft multi discrete topology on X and (XE ,τ) is said to be a soft multi discrete space over X .

• τ = {Φ, X̃ } is called the soft multi indiscrete topology on X and (XE ,τ) is said to be a soft
indiscrete space over X .

Definition 15. Let (XE ,τ) be a soft multi topological space over X and Y be a non-empty subset
of X . Then

τY = {(Y F, E) : (F, E) ∈ τ}

is said to be the soft multi topology on Y and (YE ,τY ) is called a soft multi subspace of (XE ,τ).
We can easily verify that τY is, in fact, a soft multi topology on Y .

3. Soft Multi Function

In this section, we defined soft multi function and examined its basic theorems.

Definition 16. Let XE and YK be two soft multi class. Let ϕ : X ∗ → Y ∗ and ψ : E → K be two
functions. Then the pair (ϕ,ψ) is called a soft multi function and denoted by
f = (ϕ,ψ) : XE → YK is defined as follows:

Let (F, E) be a soft multiset in XE . Then the image of (F, E) under soft multi function f is soft
multiset in YK defined by f (F, E), where for k ∈ψ(E)⊆ K and y ∈ Y ∗,

C f (F,E)(k)(y) =







sup
e∈ψ−1(k)∩E,x∈ϕ−1(y)

CF(e) (x) ,if ψ−1(k) 6= ;,ϕ−1(y) 6= ;;

0 , otherwise.

Let (G, K) be a soft multiset in YK . Then the inverse image of (G, K) under soft multi function f
is soft multiset in XE defined by f −1 (G, K), where for e ∈ψ−1(K)⊆ E and x ∈ X ∗,

C f −1(G,K)(e)(x) = CG(ψ(e))
�

ϕ(x)
�

.



İ. Osmanoğlu, D. Tokat / Eur. J. Pure Appl. Math, 7 (2014), 97-108 102

Example 4. Let X = {2/a, 3/b, 4/c, 5/d}, Y =
�

5/x , 4/y, 3/z, 2/w
	

, E =
�

e1, e2, e3, e4
	

,
K =

�

k1, k2, k3
	

and XE , YK , classes of soft multisets. Let ϕ : X ∗ → Y ∗ and ψ : E → K be two
function defined as

ϕ(a) = z, ϕ(b) = y, ϕ(c) = y, ϕ(d) = x ,
ψ(e1) = k1, ψ(e2) = k3, ψ(e3) = k2, ψ(e4) = k1.

Choose two soft multisets in XE and YK , respectively, as

(F, A) ={e1 = {1/a, 2/b, 1/d}, e3 = {3/b, 2/c, 1/d}, e4 = {2/a, 5/d}},
(G, B) ={k1 = {4/x , 2/w}, k2 = {1/x , 1/y, 2/z, 2/w}}

Then soft multiset image of (F, A) under f : XE → YK is obtained as

C f (F,A)(k1)(x) =







sup
e∈ψ−1(k1)∩A,a∈ϕ−1(x)

CF(e) (a) , if ψ−1(k1) 6= ;,ϕ−1(x) 6= ;;

0 , otherwise.

=







sup
e∈{e1,e4},a∈{d}

CF(e) (a) , if ψ−1(k1) 6= ;,ϕ−1(x) 6= ;;

0 , otherwise.

= sup
¦

CF(e1) (d) , CF(e4) (d)
©

=5

C f (F,A)(k1)(y) = sup
¦

CF(e1) (b) , CF(e4) (b) , CF(e1) (c) , CF(e4) (c)
©

= 2,

C f (F,A)(k1)(z) = sup
¦

CF(e1) (a) , CF(e4) (a)
©

= 2,

C f (F,A)(k1)(w) =0 (since ϕ−1(w) = ;),

C f (F,A)(k2)(x) =







sup
e∈ψ−1(k2)∩A,a∈ϕ−1(x)

CF(e) (a) , if ψ−1(k2) 6= ;,ϕ−1(x) 6= ;;

0 , otherwise.

=







sup
e∈{e3},a∈{d}

CF(e) (a) , if ψ−1(k2) 6= ;,ϕ−1(x) 6= ;;

0 , otherwise.

= sup
¦

CF(e3) (d)
©

=1,

C f (F,A)(k2)(y) = sup
¦

CF(e3) (b) , CF(e3) (c)
©

= 3,

C f (F,A)(k2)(z) = sup
¦

CF(e3) (a)
©

= 0,

C f (F,A)(k2)(w) =0 (since ϕ−1(w) = ;).



İ. Osmanoğlu, D. Tokat / Eur. J. Pure Appl. Math, 7 (2014), 97-108 103

Consequently, we have

( f (F, A), B) = {k1 = {5/x , 2/y, 2/z}, k2 = {1/x , 3/y}}.

Soft multiset inverse image of (G, B) under f : XE → YK is obtained as

C f −1(G,B)(e3)(a) =CG(ψ(e3))
�

ϕ(a)
�

= CG(k2) (z) = 2,

C f −1(G,B)(e3)(b) =CG(ψ(e3))
�

ϕ(b)
�

= CG(k2)
�

y
�

= 1,

C f −1(G,B)(e3)(c) =CG(ψ(e3))
�

ϕ(c)
�

= CG(k2)
�

y
�

= 1,

C f −1(G,B)(e3)(d) =CG(ψ(e3))
�

ϕ(d)
�

= CG(k2) (x) = 1,

C f −1(G,B)(e4)(a) =CG(ψ(e4))
�

ϕ(a)
�

= CG(k1) (z) = 0,

C f −1(G,B)(e4)(b) =CG(ψ(e4))
�

ϕ(b)
�

= CG(k1)
�

y
�

= 0,

C f −1(G,B)(e4)(c) =CG(ψ(e4))
�

ϕ(c)
�

= CG(k1)
�

y
�

= 0,

C f −1(G,B)(e4)(d) =CG(ψ(e4))
�

ϕ(d)
�

= CG(k1) (x) = 4.

Consequently, we have

( f −1(G, B), D) = {e3 = {2/a, 1/b, 1/c, 1/d}, e4 = {4/d}}.

Theorem 1. Let f : XE → YK be a soft multi function, (F, A), (Fi , A) soft multisets in XE and
(G, B), (Gi , B) soft multisets in YK .

(1) f (Φ) = Φ, f (X̃ )⊆̃Ỹ ,

(2) f −1(Φ) = Φ, f −1(Ỹ ) = X̃ ,

(3) f ((F1, A1)∪̃(F2, A2)) = f (F1, A1)∪̃ f (F2, A2).
In general, f (∪̃i∈I(Fi , Ai)) = ∪̃i∈I f (Fi , Ai),

(4) f −1((G1, B)∪̃(G2, B)) = f −1(G1, B)∪̃ f −1(G2, B).
In general, f −1(∪̃i∈I(Gi , B)) = ∪̃i∈I f −1(Gi , B),

(5) f ((F1, A)∩̃(F2, A))⊆̃ f (F1, A)∩̃ f (F2, A).
In general, f (∩̃i∈I(Fi , A))⊆̃∩̃i∈I f (Fi , A),

(6) f −1((G1, B)∩̃(G2, B)) = f −1(G1, B)∩̃ f −1(G2, B).
In general, f −1(∩̃i∈I(Gi , B)) = ∩̃i∈I f −1(Gi , B),

(7) If (F1, A)⊆̃(F2, A), then f (F1, A)⊆̃ f (F2, A),

(8) If (G1, B)⊆̃(G2, B), then f −1(G1, B)⊆̃ f −1(G2, B).

Proof. By using Definition 16 , we only prove (3)− (8)



İ. Osmanoğlu, D. Tokat / Eur. J. Pure Appl. Math, 7 (2014), 97-108 104

(3) Suppose that (F, A) = (F1, A1)∪̃(F2, A1). We we should show that f (F, A) = f (F1, A1)∪̃ f (F2, A2).
Then for k ∈ K and y ∈ Y ∗,

C f (F,A)(k)(y) = sup
e∈ψ−1(k)∩A,x∈ϕ−1(y)

CF(e) (x)

= sup
e∈ψ−1(k)∩A,x∈ϕ−1(y)

max{CF1(e) (x) , CF2(e) (x)}

=max{ sup
e∈ψ−1(k)∩A,x∈ϕ−1(y)

CF1(e) (x) , sup
e∈ψ−1(k)∩A,x∈ϕ−1(y)

CF2(e) (x)}

=max{C f (F1,A1)(k)(y), C f (F2,A2)(k)(y)}.

Therefore f (F, A) = f (F1, A1)∪̃ f (F2, A2). Hence f ((F1, A1)∪̃(F2, A2)) = f (F1, A1)∪̃ f (F2, A2).
Similarly, f (∪̃i∈I(Fi , Ai)) = ∪̃i∈I f (Fi , Ai).

(4) Suppose that (G, B) = (G1, B)∪̃(G2, B). We we should show that f −1(G, B) = f −1(G1, B)∪̃ f −1(G2, B).
Then for e ∈ E and x ∈ X ∗,

C f −1(G,B)(e)(x) =CG(ψ(e))(ϕ(x))

=max{CG1(ψ(e))(ϕ(x)), CG2(ψ(e))(ϕ(x))}

=max{C f −1(G1,B1)(e)(x), C f −1(G2,B2)(e)(x)}.

Therefore f −1((G1, B)∪̃(G2, B)) = f −1(G1, B)∪̃ f −1(G2, B). Similarly,
f −1(∪̃i∈I(Gi , B)) = ∪̃i∈I f −1(Gi , B).

(5) Suppose that (F, A) = (F1, A1)∩̃(F2, A2). Then for k ∈ K and y ∈ Y ∗,

C f (F,A)(k)(y) = sup
e∈ψ−1(k)∩(A1∩A2),x∈ϕ−1(y)

CF(e) (x)

= sup
e∈ψ−1(k)∩(A1∩A2),x∈ϕ−1(y)

min{CF1(e) (x) , CF2(e) (x)}

=min{ sup
e∈ψ−1(k)∩(A1∩A2),x∈ϕ−1(y)

CF1(e) (x) , sup
e∈ψ−1(k)∩(A1∩A2),x∈ϕ−1(y)

CF2(e) (x)}

≤min{ sup
e∈ψ−1(k)∩A1,x∈ϕ−1(y)

CF1(e) (x) , sup
e∈ψ−1(k)∩A2,x∈ϕ−1(y)

CF2(e) (x)}

=min{C f (F1,A1)(k)(y), C f (F2,A2)(k)(y)}.

Therefore f ((F1, A1)∩̃(F2, A2))⊆̃ f (F1, A1)∩̃ f (F2, A2). Similarly, f (∪̃i∈I(Fi , Ai))⊆̃∪̃i∈I f (Fi , Ai).

(6) Suppose that (G, B) = (G1, B)∩̃(G2, B). We we should show that f −1(G, B) = f −1(G1, B)∩̃ f −1(G2, B).
Then for e ∈ E and x ∈ X ∗,

C f −1(G,B)(e)(x) =CG(ψ(e))(ϕ(x))

=min{CG1(ψ(e))(ϕ(x)), CG2(ψ(e))(ϕ(x))}

=min{C f −1(G1,B1)(e)(x), C f −1(G2,B2)(e)(x)}.

Therefore f −1((G1, B)∩̃(G2, B)) = f −1(G1, B)∩̃ f −1(G2, B). Similarly,
f −1(∩̃i∈I(Gi , B)) = ∩̃i∈I f −1(Gi , B).



İ. Osmanoğlu, D. Tokat / Eur. J. Pure Appl. Math, 7 (2014), 97-108 105

(7) Suppose that (F1, A1)⊆̃(F2, A2). Then for all e ∈ E and x ∈ X ∗, CF1(e) (x) ≤ CF2(e) (x).
Hence for k ∈ K and y ∈ Y ∗,

C f (F1,A1)(k)(y) = sup
e∈ψ−1(k)∩A1,x∈ϕ−1(y)

CF1(e) (x)

≤ sup
e∈ψ−1(k)∩A2,x∈ϕ−1(y)

CF2(e) (x)

=C f (F2,A2)(k)(y).

This show that f (F1, A1)⊆̃ f (F2, A2).

(8) Suppose that (G1, B1)⊆̃(G2, B2). Then for all k ∈ K and y ∈ Y ∗, CG1(k)
�

y
�

≤ CG2(k)
�

y
�

.
Hence for e ∈ E and x ∈ X ∗,

C f −1(G1,B1)(e)(x) =CG1(ψ(e))(ϕ(x))

≤CG2(ψ(e))(ϕ(x))

=C f −1(G2,B2)(e)(x)

where ψ(e) ∈ K and ϕ(x) ∈ Y ∗. This show that f −1(G1, B1)⊆̃ f −1(G2, B2).

4. Compact Soft Multi Spaces

In this section, we introduced soft multi compactness on soft multi topological space and
give basic definitions and theorems about it.

Definition 17. Let
�

XE ,τ
�

and
�

YK ,σ
�

be two soft multi topological spaces.

(1) A soft multi function f :
�

XE ,τ
�

→
�

YK ,σ
�

is called soft multi continuous if for all
(G, B) ∈ σ, f −1(G, B) ∈ τ.

(2) A soft multi function f :
�

XE ,τ
�

→
�

YK ,σ
�

is called soft multi open if for all (F, A) ∈ τ,
f (F, A) ∈ σ.

Definition 18. A family Ψ of soft multisets is a cover of a soft multiset (F, A) if

(F, A)⊆ ∪
��

Fi , A
�

:
�

Fi , A
�

∈Ψ, i ∈ I
	

.

It is a soft multi open cover if each member of Ψ is a soft multi open set. A subcover of Ψ is a
subfamily of Ψ which is also a cover.

Definition 19. Let (XE ,τ) be soft multi topological space and (F, A) ∈ FS(X , E). Soft multiset
(F, A) is called compact if each soft multi open cover of (F, A) has a finite subcover. Also soft multi
topological space (XE ,τ) is called compact if each soft multi open cover of X̃ has a finite subcover.



İ. Osmanoğlu, D. Tokat / Eur. J. Pure Appl. Math, 7 (2014), 97-108 106

Example 5. A soft multi topological space (XE ,τ) is compact if X is finite.

Example 6. Let (XE ,τ) and (YK ,σ) be two soft multi topological spaces and τ ⊂ σ. Then, soft
multi topological space (XE ,τ) is compact if (YK ,σ) is compact.

Proposition 1. Let (G, B) be a whole soft multi closed set in soft multi compact space (XE ,τ).
Then (G, B) is also compact.

Proof. Let
�

Fi , A
�

be any open covering of (G, B). Then

X̃ ⊆
�

∪i∈I
�

Fi , A
��

∪ (G, B)c ;

that is,
�

Fi , A
�

together with soft multi open set (G, B)c is a open covering of X̃ . Therefore
there exists a finite subcovering

�

F1, A
�

,
�

F2, A
�

, . . . ,
�

Fn, A
�

, (G, B)c . So

X̃ ⊆
�

F1, A
�

∪
�

F2, A
�

∪ . . .∪
�

Fn, A
�

∪ (G, B)c .

Therefore
(G, B)⊆

�

F1, A
�

∪
�

F2, A
�

∪ . . .∪
�

Fn, A
�

∪ (G, B)c

which clearly implies
(G, B)⊆

�

F1, A
�

∪
�

F2, A
�

∪ . . .∪
�

Fn, A
�

since (G, B)∩ (G, B)c = Φ. Hence (G, B) has a finite subcovering and so is compact.

Definition 20 ([13]). Let (XE ,τ) be a soft multi topological space over X and x , y ∈ X such
that x 6= y. If there exist soft multi open sets (F, A) and (G, A) such that x ∈ (F, A), y ∈ (G, A)
and (F, A)∩̃(G, A) = Φ, then (XE ,τ) is called a soft multi Hausdorff space.

Proposition 2. Let (G, B) be a whole soft multi compact set in soft multi Hausdorff space (XE ,τ).
Then (G, B) is closed.

Proof. Let x ∈ (G, B)c . For each y ∈ (G, B), we have x 6= y , so there are disjoint soft multi
open sets

�

Fy , A
�

and
�

Fy , A
�

so that x ∈
�

Fy , A
�

and y ∈
�

H y , A
�

. Then

{
�

H y , A
�

: y ∈ (G, B)} is an soft multi open cover of (G, B) Let {
�

H y1
, A
�

,
�

H y2
, A
�

, . . . ,
�

H yn
, A
�

}
be a finite subcover. Then ∩n

i=1

�

Fyi
, A
�

is an open set containing x and contained in (G, B)c .
Thus (G, B)c is soft multi open and (G, B) is closed.

Theorem 2. Let (XE ,τ) and (YK ,σ) be soft multi topological spaces and f : (XE ,τ)→ (YK ,σ)
continuous and onto soft multi function. If (XE ,τ) is soft multi compact, then (YK ,σ) is soft
multi compact.

Proof. We will use Theorem 1. Let
�

Fi , A
�

be any open covering of Ỹ ; i.e., Ỹ ⊆ ∪i∈I
�

Fi , A
�

.
Then f −1

�

Ỹ
�

⊆ f −1 �∪i∈I
�

Fi , A
��

; and X̃ ⊆ ∪−1
i∈I f

�

Fi , A
�

. So f −1 ��Fi , A
��

is an open cover-
ing of X̃ . As (XE ,τ) is compact, there are 1,2, . . . , n in I such that

X̃ ⊆ f −1 �F1, A
�

∪ f −1 ��F2, A
��

∪ . . .∪ f −1 ��Fn, A
��

.



REFERENCES 107

Since
�

ϕ,ψ
�

is surjective, we have

Ỹ = f
�

X̃
�

⊆ f
�

f −1 ��F1, A
��

∪ . . .∪ f −1 ��Fn, A
��

�

= f
�

f −1 ��F1, A
��

�

∪ . . .∪ f
�

f −1 ��Fn, A
��

�

=
�

F1, A
�

∪
�

F2, A
�

∪ . . .∪
�

Fn, A
�

.

So we have Ỹ ⊆
�

F1, A
�

∪
�

F2, A
�

∪ . . . ∪
�

Fn, A
�

; i.e., Ỹ is covered by a finite number of
�

Fi , A
�

.
Hence (YK ,σ) is compact.

Definition 21. Let (XE ,τ) and (YK ,σ) be two soft multi topological spaces. A soft multi function
f : (XE ,τ)→ (YK ,σ) is called soft multi closed if f ((F, A)) is soft multi closed set in (YK ,σ), for
all soft multi closed set (F, A) in (XE ,τ).

Theorem 3. Let (XE ,τ) be a soft multi topological space and (YK ,σ) be a soft multi Hausdorff
space. Soft multi function f is closed if soft multi function f : (XE ,τ)→ (YK ,σ) is continuous.

Proof. Let (G, B) be any soft multi closed set in (XE ,τ). By Proposition 1 we have (G, B) is
compact. Since soft multi function

�

ϕ,ψ
�

is continuous, soft multiset f ((G, B)) is compact in
(YK ,σ). As (YK ,σ) is soft multi Hausdorff space, soft multiset f ((G, B)) is closed. Then soft
multi function f is closed.

Definition 22. A family Ψ of whole soft multisets has the finite intersection property if the
intersection of the members of each finite subfamily of Ψ is not the null soft multiset.

Theorem 4. A soft multi topological space is compact if and only if each family of whole soft
multi closed sets with the finite intersection property has a nonnull intersection.

Proof. ⇒: Let Ψ be any family of whole soft multi closed subset such that
∩{
�

Fi , A
�

:
�

Fi , A
�

∈ Ψ, i ∈ I} = Φ. Consider Ω = {
�

Fi , A
�c :

�

Fi , A
�

∈ Ψ, i ∈ I}. So Ω is
a soft multi open cover of X̃ . As soft multi topological space is compact, there exists a finite
subcovering

�

F1, A
�c ,
�

F2, A
�c , . . . ,

�

F3, A
�c . Then ∩n

i=1

�

Fi , A
�

= X̃−∪n
i=1

�

Fi , A
�c = X̃−X̃ = Φ.

Thence Ψ cannot have finite intersection property.
⇐: Assume that a soft multi topological space is not compact. Then any soft multi open

cover of X̃ has not a finite subcover. Let {
�

Fi , A
�

: i ∈ I} be soft multi open cover of X̃ .
So ∪n

i=1

�

Fi , A
�

6= X̃ . Therefore ∩n
i=1

�

Fi , A
�c 6= Φ. Thus,

¦

�

Fi , A
�c : i = 1, . . . , n

©

have finite
intersection property. By using hypothesis, ∩

�

Fi , A
�c 6= Φ and so ∪

�

Fi , A
�

6= X̃ . This is a
contradiction. Thus soft multi topological space is compact.

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