EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS
Vol. 12, No. 3, 2019, 857-869
ISSN 1307-5543 – www.ejpam.com
Published by New York Business Global

Soft semi local functions in soft ideal topological spaces

F. A. Gharib1,∗, A. M. Abd El-latif1,2

1 Faculty of Arts and Science, Northern Border University, Rafha, P. O. Box, 840,
K. S. A.
2 Mathematics Department, Faculty of Education, Ain Shams University, Roxy, 11341,
Cairo, Egypt

Abstract. In this paper, we define a soft semi local function (F,E)∗s(Ĩ , τ) by using semi open soft
sets in a soft ideal topological space (X, τ,E, Ĩ). This concept is discussed with a view to find new
soft topologies from the original one, called ∗s-soft topology. Some properties and characterizations
of soft semi local function are explored. Finally, the notion of soft semi compatibility of soft
ideals with soft topologies is introduced and some equivalent conditions concerning this topic are
established here.

2010 Mathematics Subject Classifications: 54A05, 54A40, 06D72

Key Words and Phrases: Soft ideal, Soft semi local function, ∗s-Soft topology, Soft semi
compatibility

1. Introduction

The notion of ideal topological spaces can be found in some classical texts of Kura-
towski [16, 17] and Vaidyanathaswamy [29]. Some early applications of ideal topological
spaces can be found in various branches of mathematics, like measure theory by Schein-
berg [27]. In 1990 Jankovic and Hamlett [6] wrote a paper in which they, among their
results, included many other results in this area using modern notation, and logically and
systematically arranging them. This paper rekindled the interest in this topic, resulting
in many generalizations of the ideal topological space and many generalizations of the no-
tion of open sets, like in papers of Jafari and Rajesh [5] and Manoharan and Thangavelu
[22]. In 1966, Velicko [30] introduced the notions of θ-open and θ-closed sets, and also a
θ-closure, examining H-closed spaces in terms of an arbitrary filter base.

∗Corresponding author.
DOI: https://doi.org/10.29020/nybg.ejpam.v12i3.3442

Email addresses: fatouhalmg@yahoo.com (F. A. Gharib),
alaa 8560@yahoo.com, alaa.ali@nbu.edu.sa (A. M. Abd El-latif)

http://www.ejpam.com 857 c© 2019 EJPAM All rights reserved.



F. A. Gharib, A. M. Abd El-latif / Eur. J. Pure Appl. Math, 12 (3) (2019), 857-869 858

In [2], Al-Omari and Noiri introduced the local closure function as a generalization of
the θ-closure and the local function in an ideal topological space. They proved some basic
properties for the local closure function, and also introduced two new topologies obtained
from the original one using the local closure function. Abd El Monsef et al. [1] introduced
semi local function in 1992 and defined a topologyτ∗s , which investigated recently in 2010
in [15]. In 2012, S. Mistry and S. Modak [20] defined pre local function.

The concept of soft sets was first introduced by Molodtsov [24] in 1999 as a general
mathematical tool for dealing with uncertain objects. After presentation of the operations
of soft sets [21], the properties and applications of soft set theory have been studied
increasingly [3, 18, 23, 26]. Recently, in 2011, Shabir and Naz [28] initiated the study of
soft topological spaces. The notion of soft ideal was initiated for the first time by Kandil
et al. [10]. They also introduced the concept of soft local function. These concepts are
discussed with a view to find new soft topologies from the original one, called soft ideal
topological spaces (X, τ,E, Ĩ). Applications to various fields were further investigated by
Kandil et al.[8, 9, 11–14].

In this paper, we will introduce and study two different notions via soft ideals namely,
soft semi local function and soft semi compatibility of τ with Ĩ and investigate their
relationships with other types of similar operators.

2. Preliminaries

In this section, we will present the basic definitions and results of soft set theory which
will be needed in the sequel.

Definition 1. [24] Let X be an initial universe and E be a set of parameters. Let P (X)
denote the power set of X and A be a non-empty subset of E. A pair (F,A) denoted by
FA is called a soft set over X , where F is a mapping given by F : A→ P (X) i.e
FA = {(e, F (e)) : e ∈ A ⊆ E, F : A → P (X)}. The family of all these soft sets denoted
by SS(X)A.

Definition 2. [28] Let τ be a collection of soft sets over a universe X with a fixed set of
parameters E, then it is called a soft topology on X if:

(1) X̃, φ̃ ∈ τ , where φ̃(e) = φ and X̃(e) = X, ∀e ∈ E,

(2) the union of any number of soft sets in τ belongs to τ ,

(3) the intersection of any two soft sets in τ belongs to τ .

The triplet (X, τ,E) is called a soft topological space over X. A soft set (F,A) over X is
said to be closed soft set in X, if its relative complement (F,A)′ is an open soft set. We
denote the set of all open soft sets over X by OS(X, τ,E), or OS(X) and the set of all
closed soft sets by CS(X, τ,E), or CS(X).

Definition 3. [28] Let (X, τ,E) be a soft topological space and (F,E) ∈ SS(X)E. The
soft closure of (F,E), denoted by cl(F,E) is the intersection of all closed soft super sets
of (F,E) .



F. A. Gharib, A. M. Abd El-latif / Eur. J. Pure Appl. Math, 12 (3) (2019), 857-869 859

Definition 4. [31] Let (X, τ,E) be a soft topological space and (F,E) ∈ SS(X)E. The
soft interior of (G,E), denoted by int(G,E) is the union of all open soft subsets of (G,E).

Definition 5. [31] The soft set (F,E) ∈ SS(X)E is called a soft point in X̃ if there exist
x ∈ X and e ∈ E such that F (e) = {x} and F (e′) = φ for each e′ ∈ E − {e}, and the soft
point (F,E) is denoted by xe. We denote the set of all soft point of the universal set X
by ε.

Definition 6. [31] The soft point xe is said to be belonging to the soft set (G,A), denoted
by xe∈̃(G,A), if for the element e ∈ A, F (e) ⊆ G(e).

Definition 7. [25] Let (X, τ,E) be a soft topological space and (F,E) ∈ SS(X)E. Define
τ(F,E) = {(G,E)∩̃(F,E) : (G,E) ∈ τ}, which is a soft topology on (F,E). This soft
topology is called a soft relative topology of τ on (F,E), and [(F,E), τ(F,E)] is called a soft
subspace of (X, τ,E).

Definition 8. [4, 7] A soft set (F,E) of a soft topological space (X, τ,E) is called semi
open soft, if FE⊆̃cl(int(FE)) (resp., semi closed soft, if int(cl(FE))⊆̃FE). The set of all
semi open soft sets is denoted by SOS(X) and the set of all semi closed soft sets is denoted
by SCS(X). Also, The semi soft closure of (F,E), denoted by scl(F,E), is defined by the
intersection of all semi closed soft sets containing (F,E).

Definition 9. [6]. A non-empty collection I of subsets of a set X is called an ideal on
X, if it is closed under finite unions and subsets.

3. Soft semi local functions and generated a new soft topology

In this section, we will introduce a soft semi local function (F,E)∗s(Ĩ , τ) by using semi
open soft sets in a soft ideal topological space (X, τ,E, Ĩ). This concept is discussed with
a view to find new soft topologies from the original one, called ∗s-soft topology. Some
properties and characterizations of soft semi local function will be studied.

Definition 10. [10] Let Ĩ be a non-null collection of soft sets over a universe X with the
same set of parameters E. Then, Ĩ ⊆ SS(X)E is called a soft ideal on X if it is closed
under finite soft unions and soft subsets.

Definition 11. [10] Let (X, τ,E) be a soft topological space and Ĩ be a soft ideal over X
with the same set of parameters E. Then,

(F,E)∗(Ĩ , τ) ((F,E)∗(Ĩ) or (F,E)∗) = ∪̃{xe ∈ ε : Oxe∩̃(F,E)˜6∈Ĩ ∀ Oxe ∈ τ}

is called the soft local function of (F,E) with respect to Ĩ and τ , where Oxe is an open soft
set containing xe.

Definition 12. Let (X, τ,E) be a soft topological space and Ĩ be a soft ideal over X with
the same set of parameters E. Then,



F. A. Gharib, A. M. Abd El-latif / Eur. J. Pure Appl. Math, 12 (3) (2019), 857-869 860

(F,E)∗s(Ĩ , τ) ((F,E)∗s(Ĩ) or (F,E)∗s) = ∪̃{xe ∈ ε : Oxe∩̃(F,E)˜6∈Ĩ ∀ Oxe ∈ SOS(X)}

is called the soft semi local function of (F,E) with respect to Ĩ and τ , where Oxe is a semi
open soft set containing xe.

Theorem 1. Let Ĩ be a soft ideal with the same set of parameters E on a soft topological
space (X, τ,E) and (F,E) ∈ SS(X)E. Then, (F,E)∗s⊆̃(F,E)∗.

Proof. Let xe∈̃(F,E)∗s . Then, Oxe∩̃(F,E) 6∈ Ĩ ∀ Oxe ∈ SOS(X). Since τ ⊆ SOS(X)
and Oxe∩̃(F,E) 6∈ Ĩ, Oxe∩̃(F,E) 6∈ Ĩ ∀ Oxe ∈ τ from Definition 10. Hence, xe∈̃(F,E)∗.

Remarks 1. The converse of the above theorem is not true in general as will shown in
the following example.

Example 1. Suppose that there are three cars in the universe X given by X = {c1, c2, c3}.
Let E = {e1, e2} be the set of decision parameters which are stands for ”expensive” and
”color” respectively.
Let (F1, E), (F2, E), (F3, E) be soft sets over the common universe X, which describe the
composition of the cars, where
F1(e1) = {c1}, F1(e2) = {c1}.
F2(e1) = {c2}, F2(e2) = {c2}.
F3(e1) = {c1, c2}, F3(e2) = {c1, c2}.
Then, τ = {X̃, φ̃, (F1, E), (F2, E), (F3, E)} defines a soft topology on X. Let Ĩ = {φ̃}
and (G,E) be a soft set defined by G(e1) = {c2, c3}, G(e2) = {c2}. Hence, (G,E)∗s =
scl(G,E) = scl(G,E)∗s = (G,E) and (G,E)∗ = cl(G,E) = cl(G,E)∗ = (H,E), where
H(e1) = {c2, c3}, H(e2) = {c2, c3}.

Remarks 2. The collection of all semi open soft sets of a soft topological space (X, τ,E)
fails to form a soft ideal on X as will shown in the following example.

Example 2. Suppose that there are two houses in the universe X given by X = {h1, h2}.
Let E = {e1, e2} be the set of decision parameters which are stands for ”wooden” and
”position” respectively.
Let (F1, E), (F2, E) be soft sets over the common universe X, which describe the composi-
tion of the houses, where
F1(e1) = {h1}, F1(e2) = {h2}.
F2(e1) = {h2}, F2(e2) = {h1}.
Then, τ = {X̃, φ̃, (F1, E), (F2, E)} defines a soft topology on X. Also, SOS(X) = {φ̃, (F1, E),
(F2, E), X̃} is not soft ideal, because it is not closed under soft subset.

Remarks 3. (1) If (F,E) ∈ Ĩ, then (F,E)∗s = φ̃.

(2) If Ĩ = SS(X)E, then (F,E)∗s = φ̃ = (F,E)∗.

(3) If Ĩ = {φ̃}, then (F,E)∗s = scl(F,E) 6= cl(F,E).



F. A. Gharib, A. M. Abd El-latif / Eur. J. Pure Appl. Math, 12 (3) (2019), 857-869 861

(4) Neither (F,E)∗s⊆̃(F,E) nor (F,E)⊆̃(F,E)∗s in general.

(5) If τ = SOS(X), then (F,E)∗s = (F,E)∗.

Theorem 2. Let Ĩ and J̃ be any two soft ideals with the same set of parameters E on a
soft topological space (X, τ,E). Let (F,E), (G,E) ∈ SS(X)E. Then,

(1) (φ̃)∗s = φ̃,

(2) If (F,E)⊆̃(G,E), then (F,E)∗s⊆̃(G,E)∗s,

(3) If Ĩ ⊆ J̃ , then (F,E)∗s(J̃)⊆̃(F,E)∗s(Ĩ),

(4) (F,E)∗s = scl(F,E)∗s⊆̃scl(F,E), where scl is the semi soft closure w.r.t. τ , (F,E)∗s

is semi closed soft set.

(5) ((F,E)∗s)∗s⊆̃(F,E)∗s,

(6) ((F,E)∪̃(G,E))∗s = (F,E)∗s∪̃(G,E)∗s,

(7) ∪̃j(F,E)∗s = (∪̃j(F,E))∗s,

(8) ((F,E)∩̃(G,E))∗s ⊆ (F,E)∗s∩̃(G,E)∗s,

(9) (F,E)∗s − (G,E)∗s = ((F,E)− (G,E))∗s − (G,E)∗s⊆̃((F,E)− (G,E))∗s,

(10) If (I, E) ∈ Ĩ, then ((F,E)− (I, E))∗s = (F,E)∗s = ((F,E)∪̃(I, E))∗s.

Proof.

(1) Obvious from Definition 12.

(2) Assume that xe∈̃(F,E)∗s and xe˜6∈(G,E)∗s . Then, there exist Oxe ∈ SOS(X) such
thatOxe∩̃(G,E) ∈ Ĩ. Since (F,E)⊆̃(G,E), Oxe∩̃(F,E)⊆̃Oxe∩̃(G,E) andOxe∩̃(F,E) ∈
Ĩ. Hence, xe˜6∈(F,E)∗s , which is a contradiction. Thus, xe∈̃(G,E)∗s and so (F,E)∗s⊆̃(G,E)∗s .

(3) Let xe∈̃(F,E)∗s(J̃). Then, Oxe∩̃(F,E) 6∈ J̃ ∀ Oxe ∈ SOS(X). Since Ĩ ⊆ J̃ . Then,
Oxe∩̃(F,E) 6∈ Ĩ ∀ Oxe ∈ SOS(X). Hence, xe∈̃(F,E)∗s(Ĩ). Thus, (F,E)∗s(J̃)⊆̃(F,E)∗s(Ĩ).

(4) We first prove that (F,E)∗s = scl(F,E)∗s . Let xe∈̃scl(F,E)∗s . Then, Oxe∩̃(F,E)∗s 6=
φ̃ for every Oxe ∈ SOS(X). Therefore, there exists a soft point ye′ such that
ye′∈̃Oxe∩̃(F,E)∗s and so ye′∈̃(F,E)∗s . Hence, Oy′e∩̃(F,E) 6∈ Ĩ. Thus, xe∈̃(F,E)∗s .
This shows that, scl(F,E)∗s⊆̃(F,E)∗s but we have (F,E)∗s⊆̃scl(F,E)∗s .
Now, we prove that scl(F,E)∗s = (F,E)∗s⊆̃scl(F,E). Assume that, xe˜6∈scl(F,E).
Then, there existsOxe ∈ SOS(X) such thatOxe∩̃(F,E) = φ̃ ∈ Ĩ. Hence, xe˜6∈(F,E)∗s .
Thus, (F,E)∗s⊆̃scl(F,E).

(5) Since (F,E)∗s⊆̃scl(F,E) from (4). Replace (F,E) with (F,E)∗s , we get ((F,E)∗s)∗s⊆̃scl
(F,E)∗s = (F,E)∗s from (4).



F. A. Gharib, A. M. Abd El-latif / Eur. J. Pure Appl. Math, 12 (3) (2019), 857-869 862

(6) Let xe∈̃((F,E)∪̃(G,E))∗s . Then, Oxe∩̃((F,E)∪̃(G,E)) = (Oxe∩̃(F,E))
∪̃(Oxe∩̃(G,E)) 6∈ Ĩ ∀ Oxe ∈ SOS(X). Hence, eitherOxe∩̃(F,E) 6∈ Ĩ orOxe∩̃(G,E) 6∈
Ĩ from Definition 10 ∀ Oxe ∈ SOS(X). This means that, either xe∈̃(F,E)∗s or
xe∈̃(G,E)∗s . Thus, xe∈̃(F,E)∗s∪̃(G,E)∗s . It follows that, ((F,E)∪̃(G,E))∗s⊆̃(F,E)∗s∪̃
(G,E)∗s . For the reverse inclusion, since (F,E), (G,E)⊆̃((F,E)∪̃(G,E)). Then,
(F,E)∗s⊆̃((F,E)∪̃(G,E))∗s and (G,E)∗s⊆̃((F,E)∪̃(G,E))∗s from (2). Hence, (F,E)∗s∪̃(G,E)∗s

⊆̃((F,E)∪̃(G,E))∗s and it is implies that ((F,E)∪̃(G,E))∗s = (F,E)∗s∪̃(G,E)∗s .

(7) Obvious from (6).

(8) Since ((F,E)∩̃(G,E))⊆̃(F,E), (G,E). Then, ((F,E)∩̃(G,E))∗s⊆̃(F,E)∗s and ((F,E)∩̃(G,E))
⊆̃(G,E)∗s from (2). Hence, ((F,E)∩̃(G,E))∗s⊆̃(F,E)∗s∩̃(G,E)∗s .

(9) We first prove that (F,E)∗s−(G,E)∗s = ((F,E)−(G,E))∗s−(G,E)∗s . Since (F,E)−
(G,E)⊆̃(F,E). Then, ((F,E) − (G,E))∗s⊆̃(F,E)∗s . Hence, ((F,E) − (G,E))∗s −
(G,E)∗s⊆̃(F,E)∗s − (G,E)∗s . For the reverse inclusion, since (F,E) = [(F,E) −
(G,E)]∪̃[(F,E)∩̃(G,E)]. Then, (F,E)∗s = [[(F,E)− (G,E)]∪̃[(F,E)∩̃(G,E)]]∗s . It
follows, (F,E)∗s = [[(F,E)−(G,E)]∪̃[(F,E)∩̃(G,E)]]∗s = [(F,E)−(G,E)]∗s∪̃[(F,E)∩̃
(G,E)]∗s⊆̃[(F,E)−(G,E)]∗s∪̃(G,E)∗s from (6). Hence, (F,E)∗s−(G,E)∗s⊆̃[[(F,E)−
(G,E)]∗s∪̃(G,E)∗s ]− (G,E)∗s = [[(F,E)− (G,E)]∗s∪̃(G,E)∗s ]∩̃(G,E)∗

′
= [(F,E)−

(G,E)]∗s − (G,E)∗s . Thus, (F,E)∗s − (G,E)∗s = ((F,E)− (G,E))∗s − (G,E)∗s .
Now, if xe∈̃((F,E)− (G,E))∗s − (G,E)∗s . Then, xe∈̃((F,E)− (G,E))∗s and conse-
quently (F,E)∗s − (G,E)∗s = ((F,E)− (G,E))∗s − (G,E)∗s⊆̃((F,E)
− (G,E))∗s .

(10) We first prove that (F,E)∗s = ((F,E) − (I, E))∗s . Since ((F,E) − (I, E))⊆̃(F,E),
((F,E) − (I, E))∗s⊆̃(F,E)∗s from (2). For the reverse inclusion, let xe˜6∈((F,E) −
(I, E))∗s . Then, there exists Oxe ∈ SOS(X) such that Oxe∩̃((F,E) − (I, E)) ∈ Ĩ.
Since (I, E) ∈ Ĩ. Then, (I, E)∪̃(Oxe∩̃((F,E)−(I, E))) ∈ Ĩ. Hence, (I, E)∪̃(Oxe∩̃(F,E)) ∈
Ĩ. Thus, Oxe∩̃(F,E)) ∈ Ĩ for some Oxe ∈ SOS(X). It follows that, xe˜6∈(F,E)∗s .
This means that, (F,E)∗s⊆̃((F,E)− (I, E))∗s .
Now, we prove that ((F,E)∪̃(I, E))∗s = (F,E)∗s . From (2), ((F,E)∪̃(I, E))∗s =
(F,E)∗s∪̃(I, E)∗s = (F,E)∗s∪̃φ̃ = (F,E)∗s from Remark 3 (1). This completes the
proof.

Corollary 1. Let Ĩ be a soft ideal with the same set of parameters E on a soft topological
space (X, τ,E). Let (F,E), (G,E) ∈ SS(X)E. Then,

(1) (F,E)∗s is semi closed soft set.

(2) ((F,E)∗s)∗s⊆̃(F,E)∗,

(3) ((F,E)∗s)∗⊆̃(F,E)∗,

(4) ((F,E)∗)∗s⊆̃(F,E)∗,



F. A. Gharib, A. M. Abd El-latif / Eur. J. Pure Appl. Math, 12 (3) (2019), 857-869 863

(5) scl(F,E)∗s⊆̃cl(F,E),

(6) (F,E)∗s⊆̃(F,E)∗⊆̃cl(F,E),

(7) If (I, E) ∈ Ĩ, then [X̃ − (I, E)]∗s = X̃∗s.

Proof. It is follows from Theorem 1 and Theorem 2.

Theorem 3. Let Ĩ and J̃ be two soft ideals with the same set of parameters E on a soft
topological space (X, τ,E) and (F,E) ∈ SS(X)E. Then, (F,E)∗s(Ĩ∩J̃) = (F,E)∗(Ĩ)∪̃(F,E)∗(J̃).

Proof. Since Ĩ ∩ J̃ ⊆ Ĩ and Ĩ ∩ J̃ ⊆ J̃ , (F,E)∗(Ĩ), (F,E)∗(J̃)⊆̃(F,E)∗s(Ĩ ∩ (J̃))
from Theorem 2 (3). Therefore, (F,E)∗(Ĩ)∪̃(F,E)∗(J̃)⊆̃(F,E)∗s(Ĩ ∩ J̃). For the reverse
inclusion, let xe∈̃(F,E)∗s(Ĩ ∩ J̃). Then, Oxe∩̃(F,E) 6∈ (Ĩ ∩ J̃) ∀ Oxe ∈ SOS(X) and
so Oxe∩̃(F,E) 6∈ Ĩ or Oxe∩̃(F,E) 6∈ J̃ . It follows, xe∈̃(F,E)∗s(Ĩ) or xe∈̃(F,E)∗s(J̃) and
consequently xe∈̃(F,E)∗s(Ĩ)∪̃(F,E)∗s(J̃). Thus, (F,E)∗s(Ĩ ∩ J̃)⊆̃(F,E)∗(Ĩ)∪̃(F,E)∗(J̃).
This completes the proof.

Theorem 4. Let (X, τ,E) be a soft topological space and Ĩ be a soft ideal over X with
the same set of parameters E. Then the operator cl∗s : SS(X)E → SS(X)E defined by:

cl∗s(F,E) = (F,E)∪̃(F,E)∗s . (1)

is a soft closure operator.

Proof. cl∗s(φ̃) = φ̃∪̃(φ̃)∗s = φ̃∪̃φ̃ = φ̃ from Theorem 1 (1), and obviously (F,E)⊆̃cl∗s(F,E)
∀(F,E) ∈ SS(X)E . Now, cl∗s((F,E)∪̃(G,E)) = ((F,E)∪̃(G,E))∪̃((F,E)∪̃(G,E))∗s =
((F,E)∪̃(G,E))∪̃((F,E)∗s∪̃(G,E)∗s) = ((F,E)∪̃(F,E)∗s)∪̃((G,E)∪̃(G,E)∗s) = cl∗s(F,E)∪̃cl∗s
(G,E) from Theorem 2 (6). Also, for any (F,E) ∈ SS(X)E , cl∗s(cl∗s(F,E)) = cl∗s((F,E)∪̃(F,E)∗s)
= ((F,E)∪̃(F,E)∗s)∪̃((F,E)∪̃(F,E)∗s)∗s = ((F,E)∪̃(F,E)∗s)∪̃((F,E)∗s∪̃((F,E)∗s)∗s)⊆̃((F,E)∪̃
(F,E)∗s))∪̃((F,E)∗s∪̃((F,E)∗s)) = cl∗s(F,E) from Theorem 2 (5).

Definition 13. Let (X, τ,E) be a soft topological space, Ĩ be a soft ideal over X with
the same set of parameters E and cl∗s : SS(X)E → SS(X)E be the soft closure operator.
Then there exists a unique soft topology over X with the same set of parameters E, finer
than τ , called the ∗-soft topology, denoted by τ∗s(Ĩ) or τ∗s, given by

τ∗s(Ĩ) = {(F,E) ∈ SS(X)E : cl∗s(F,E)′ = (F,E)′}. (2)

Example 3. (1) If Ĩ = {φ̃}, then (F,E)∗s(Ĩ , τ) = scl(F,E) ∀(F,E) ∈ SS(X)E. Hence,
cl∗s(F,E) = scl(F,E).

(2) If Ĩ = SS(X)E, then (F,E)∗s(Ĩ , τ) = φ̃ ∀(F,E) ∈ SS(X)E. Hence, cl∗s(F,E) =
(F,E) and τ∗s = SS(X)E (the soft discrete topology).

Theorem 5. Let Ĩ and J̃ be any two soft ideals with the same set of parameters E on a
soft topological space (X, τ,E). If Ĩ ⊆ J̃ , then τ∗s(Ĩ) ⊆ τ∗s(J̃).



F. A. Gharib, A. M. Abd El-latif / Eur. J. Pure Appl. Math, 12 (3) (2019), 857-869 864

Proof. Let (F,E) be a τ∗s(Ĩ)-closed soft set. Since Ĩ ⊆ J̃ , (F,E)∗s(J̃ , τ)⊆̃(F,E)∗s(Ĩ , τ).
Therefore, cl∗s

J̃
(F,E)⊆̃cl∗s

Ĩ
(F,E). Hence, (F,E)⊆̃cl∗s

J̃
(F,E)⊆̃cl∗s

Ĩ
(F,E) = (F,E) and con-

sequently (F,E) is τ∗s(J̃)-closed soft.

Proposition 1. Let (X, τ,E) be a soft topological space and Ĩ be a soft ideal over X with
the same set of parameters E. Then,

β(Ĩ , τ) = {(F,E)− (G,E) : (F,E) ∈ SOS(X), (G,E) ∈ Ĩ}

is a soft basis for the soft topology τ∗s(Ĩ).

Proof. Since X̃ ∈ τ, φ̃ ∈ Ĩ. Then, X̃ − φ̃ ∈ β. Hence, X̃ ∈ β and
⋃̃

j∈J((Fj , E) −
(Gj , E)) = X̃. Also, let ((F1, E)−(G1, E)), ((F2, E)−(G2, E)) ∈ β such that xe∈̃((F1, E)−
(G1, E))∩̃((F2, E)−(G2, E)). Then xe∈̃((F1, E)−(G1, E))∩̃((F2, E)−(G2, E)) = ((F1, E)∩̃(F2, E))−
((G1, E)∪̃(G2, E)) ∈ β(Ĩ , τ). Thus, β is a soft basis of τ∗s .

Corollary 2. Let (X, τ,E) be a soft topological space and Ĩ be a soft ideal over X with
the same set of parameters E. Then, τ ⊆ β(Ĩ , τ) ⊆ τ∗(Ĩ) ⊆ τ∗s(Ĩ).

Proof. It is obvious from Theorem 2(3) and Proposition 1.

4. Soft semi-compatibility of τ with Ĩ

In this section, we will introduce the notion of soft semi compatibility of soft ideals with
soft topologies and some equivalent conditions concerning this topic will be investigated
here.

Definition 14. Let (X, τ,E) be a soft topological space and Ĩ be a soft ideal over X with
the same set of parameters E. We say that τ is semi compatible with Ĩ, denoted by τ ∼s Ĩ,
if the following holds for each (F,E) ∈ SS(X)E:
if for each soft point xe and xe∈̃(F,E) there exists Oxe ∈ SOS(X) such that Oxe∩̃(F,E) ∈
Ĩ, then (F,E) ∈ Ĩ.

Theorem 6. Let (X, τ,E) be a soft topological space, Ĩ be a soft ideal over X with the
same set of parameters E and τ ∼s Ĩ. Then, the following are equivalent for each (F,E) ∈
SS(X)E:

(1) If (F,E)∩̃(F,E)∗s = φ̃, then (F,E)∗s = φ̃.

(2) ((F,E)− (F,E)∗s)∗s = φ̃.

(3) ((F,E)∩̃(F,E)∗s)∗s = (F,E)∗s.

Proof.

(1) =⇒ (2) Let (F,E) ∈ SS(X)E . Since ((F,E) − (F,E)∗s)∩̃((F,E) − (F,E)∗s)∗s =
[((F,E)− (F,E)∗s)∩̃(F,E)∗s ]∩̃[((F,E)− (F,E)∗s)∩̃(X̃− (F,E)∗s)∗s ] = φ̃∩̃[((F,E)−
(F,E)∗s)∩̃(X̃ − (F,E)∗s)∗s ] = φ̃, ((F,E)− (F,E)∗s)∗s = φ̃ by (1).



F. A. Gharib, A. M. Abd El-latif / Eur. J. Pure Appl. Math, 12 (3) (2019), 857-869 865

(2) =⇒ (3) Let (F,E) ∈ SS(X)E . Since (F,E) = ((F,E)−((F,E)∩̃(F,E)∗s))∪̃((F,E)∩̃(F,E)∗s),
(F,E)∗s = [(F,E)−((F,E)∩̃(F,E)∗s)∪̃((F,E)∩̃(F,E)∗s)]∗s = [(F,E)−((F,E)∩̃(F,E)∗s)]∗s

∪̃[(F,E)∩̃(F,E)∗s ]∗s = [φ̃∪̃((F,E)−(F,E)∗s)]∗s∪̃[(F,E)∩̃(F,E)∗s ]∗s = [(F,E)∩̃(F,E)∗s ]∗s

from (2).

(3) =⇒ (1) Let (F,E) ∈ SS(X)E and (F,E)∩̃(F,E)∗s = φ̃. Then, (F,E)∗s = [(F,E)∩̃(F,E)∗s ]∗s =
(φ̃)∗s = φ̃.

Corollary 3. Let (X, τ,E) be a soft topological space, Ĩ be a soft ideal over X with the
same set of parameters E, (F,E) ∈ SS(X)E and τ ∼s Ĩ. Then, ((F,E)∗s)∗s = (F,E)∗s.

Proof. Let (F,E) ∈ SS(X)E . Since (F,E)∗s = ((F,E)∩̃(F,E)∗s)∗s⊆̃(F,E)∗s)∗s from
Theorem 6. But, we have ((F,E)∗s)∗s⊆̃(F,E)∗s from Theorem 2 (5). Thus, ((F,E)∗s)∗s =
(F,E)∗s .

Theorem 7. Let (X, τ,E) be a soft topological space and Ĩ be a soft ideal over X with
the same set of parameters E. Then, the following are equivalent:

(1) τ ∼s Ĩ.

(2) If (F,E) ∈ SS(X)E has a cover of semi open soft sets each of whose soft intersection
with (F,E) is in Ĩ, then (F,E) ∈ Ĩ.

(3) For every (F,E) ∈ SS(X)E such that (F,E)∩̃(F,E)∗s = φ̃ implies (F,E) ∈ Ĩ.

(4) For every (F,E) ∈ SS(X)E, (F,E)− (F,E)∗s ∈ Ĩ.

(5) For every τ∗s-closed soft subset (F,E), (F,E)− (F,E)∗s ∈ Ĩ.

(6) For every (F,E) ∈ SS(X)E, if (F,E) contains no non-null soft set (G,E) with
(G,E)⊆̃(G,E)∗s, then (F,E) ∈ Ĩ.

Proof.

(1) =⇒ (2) Obvious from Definition 14.

(2) =⇒ (3) Let (F,E) ∈ SS(X)E such that (F,E)∩̃(F,E)∗s = φ̃ and xe∈̃(F,E). Then,
xe˜6∈(F,E)∗s and thenOxe∩̃(F,E) ∈ Ĩ for someOxe ∈ SOS(X). Therefore, (F,E)⊆̃∪̃{Oxe :
xe∈̃(F,E) and Oxe ∈ SOS(X)}. By (2), (F,E) ∈ Ĩ.

(3) =⇒ (4) Let (F,E) ∈ SS(X)E . Since ((F,E) − (F,E)∗s)∩̃((F,E) − (F,E)∗s)∗s =
((F,E)∩̃(F,E)∗

′
)

∩̃((F,E)−(F,E)∗s)∗s⊆̃((F,E)∩̃(F,E)∗
′
)∩̃(F,E)∗s = φ̃. Then , (F,E)−(F,E)∗s ∈ Ĩ

by (3).

(4) =⇒ (5) Immediate.



F. A. Gharib, A. M. Abd El-latif / Eur. J. Pure Appl. Math, 12 (3) (2019), 857-869 866

(5) =⇒ (1) Let (F,E) ∈ SS(X)E and assume that for every xe∈̃(F,E) there exists Oxe ∈
SOS(X) such that Oxe∩̃(F,E) ∈ Ĩ. Then, xe˜6∈(F,E)∗s . Hence, ((F,E)∩̃(F,E)∗s) =
φ̃ and since (F,E)∪̃(F,E)∗s is τ∗s-closed soft, we have ((F,E)∪̃(F,E)∗s)−((F,E)∪̃(F,E)∗s)∗s

∈ Ĩ by (5). Hence, ((F,E)∪̃(F,E)∗s)−((F,E)∗s∪̃((F,E)∗s)∗s) = ((F,E)∪̃(F,E)∗s)−
((F,E)∗s) = (F,E) ∈ Ĩ by Theorem 2 (5,6). Therefore, τ ∼s Ĩ.

(4) =⇒ (6) Let (F,E) ∈ SS(X)E such that (F,E) contains no non-null soft set (G,E)
with (G,E)⊆̃(G,E)∗s . Since (F,E)∩̃(F,E)∗s⊆̃(F,E)∗s = ((F,E)∩̃(F,E)∗s)∗s from
Theorem 6 (3). It follows, (F,E)∩̃(F,E)∗s⊆̃((F,E)∩̃(F,E)∗s)∗s . By assumption,
(F,E)∩̃(F,E)∗s = φ̃. Thus, (F,E) = (F,E)− (F,E)∗s ∈ Ĩ by (4).

(6) =⇒ (4) Let (F,E) ∈ SS(X)E . Since ((F,E)−(F,E)∗s)∩̃((F,E)−(F,E)∗s)∗s = φ̃ and
(F,E)− (F,E)∗s) contains no non-null soft set (G,E) with (G,E)⊆̃(G,E)∗s . Hence,
(F,E)− (F,E)∗s ∈ Ĩ by (6).

Theorem 8. If (X, τ,E) is a soft topological space, Ĩ be a soft ideal over X with the same
set of parameters E and semi compatible with τ . Then, a soft set is τ∗s-closed if and only
if it is the union of a τ -semi closed soft set and a soft set in Ĩ.

Proof.
Let (F,E) be a τ∗s-closed soft set. Then, cl∗s(F,E) = (F,E) = (F,E)∪̃(F,E)∗s =

(F,E). It follows, (F,E)∗s⊆̃(F,E). Hence, (F,E) = ((F,E) − (F,E)∗s)∪̃(F,E)∗s , where
(F,E) − (F,E)∗s ∈ Ĩ from Theorem 7 and (F,E)∗s is τ -semi closed soft from Corollary
1. Conversely, let (F,E) = (G,E)∪̃(I, E), for some τ -semi closed soft set (G,E) and
(I, E) ∈ Ĩ. Then, (F,E)∗s = ((G,E) − (I, E))∗s = (G,E)∗s⊆̃scl(G,E) = (G,E)⊆̃(F,E)
from Theorem 2 (4,12). It follows, (F,E)∪̃(F,E)∗s = (F,E). Hence, cl∗s(F,E) = (F,E).
Therefore, (F,E) is τ∗s-closed soft.

Definition 15. [19] A soft topological space (X, τ,E) is called a soft semi compact space
if each semi open soft cover of X̃ has a finite soft subcover.

Theorem 9. [19] Each semi closed soft subspace of a soft semi compact topological space
is a soft semi compact.

Theorem 10. Let (X, τ,E) be a soft semi compact space in which every soft subset of
X̃ is a semi closed soft and Ĩ be a soft ideal over X with the same set of parameters E.
Then, τ ∼s Ĩ.

Proof. Let (F,E) be a soft subset of X̃. Assume that, for each xe∈̃(F,E), there
exists a semi open soft set Oxe such that Oxe∩̃(F,E) ∈ Ĩ. Then, (F,E)⊆̃∪̃{Oxe :
xe∈̃(F,E) and Oxe ∈ SOS(X)}. It follows, the family ∪̃{Oxe : xe∈̃(F,E)} is a semi
open soft cover of (F,E). By assumption, (F,E) is a semi closed soft set . By Theorem
9, (F,E) is a soft semi compact. Hence, there exist a finite number of soft points, say,
x1e1 , x2e2 , ..., xnen in (F,E) such that (F,E)⊆̃∪̃ni=1Oxiei

and hence (F,E) = (F,E)∩̃[∪̃ni=1Oxiei
] =

∪̃ni=1[(F,E)∩̃Oxiei
]. Since Oxiei

∩̃(F,E) ∈ Ĩ for each i, (F,E) ∈ Ĩ. Therefore, τ ∼s Ĩ.



REFERENCES 867

5. Conclusion

In this paper, we will introduce and study two different notions via soft ideals namely,
soft semi local function and soft semi compatibility of τ with Ĩ and investigate their
relationships with other types of similar operators. Some properties and characterizations
of soft semi local function are explored. In future, we will introduce the notions of soft
θ-open and soft θ-closed sets and soft θ-closure. Also, we introduce the notion of soft
local closure functions as a generalization of the soft θ-closure and the soft (semi) local
function in a soft ideal topological space and the future research will be undertaken in this
direction.

Conflict of Interest

We declare that, there is no conflict of interest regarding the publication of this
manuscript.

Acknowledgements

The authors gratefully acknowledge the approval and the support of this research study
by the grant no. SAR-2017-1-8-F-7213, K. S. A. from the Deanship of Scientific Research
at Northern Border University, Arar, K. S. A.

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