EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS
Vol. 17, No. 2, 2024, 1168-1182
ISSN 1307-5543 – ejpam.com
Published by New York Business Global

Methods of Generating Soft Topologies and Soft
Separation Axioms

Zanyar A. Ameen1,∗, Ohud F. Alghamdi2, Baravan A. Asaad3,4,
Ramadhan A. Mohammed5

1 Department of Mathematics, College of Science, University of Duhok, Duhok 42001, Iraq
2 Department of Mathematics, Faculty of Science, Al-Baha University, Al-Baha, Saudi
Arabia
3 Department of Mathematics, Faculty of Science, University of Zakho, Zakho 42002, Iraq
4 Department of Computer Science, College of Science, Cihan University-Duhok, Duhok
42001, Iraq
5 Department of Mathematics, College of Basic Education, University of Duhok, Duhok
42001, Iraq

Abstract. The paper develops a novel analysis of mutual interactions between topology and soft
topology. It is known that each soft topology produces a system of crisp (parameterized) topologies.
The other way round is also possible. Namely, one can generate a soft topology from a system
of crisp topologies. Different methods of producing soft topologies are discussed by implementing
two formulas. Then, the relationships between the resulting soft topologies are obtained. With
the help of an example, it is demonstrated that one formula is more constructible than the other.
Now, it is reasonable to ask which (topological) properties of a soft topology can be transferred
to the set of crisp topologies, or the opposite. To address this question, we consider the standard
separation axioms and show how well these axioms can be preserved when moving from a system
of crisp topologies to the soft topology generated by it and contrariwise. Additionally, our findings
extend and disprove some results from the literature.

2020 Mathematics Subject Classifications: 54A05, 54H99

Key Words and Phrases: soft topology, soft T0, soft T1, soft T2, soft regular, soft normal, soft
T3, soft T4

1. Introduction

In its modern version, the Weierstrass Extreme Value Theorem demonstrates that
topological considerations can be useful in decision-making theory and economics, (see,
[4]). Indeed, the development of topological structures helps to enhance other disciplines.

∗Corresponding author.
DOI: https://doi.org/10.29020/nybg.ejpam.v17i2.5161

Email addresses: zanyar@uod.ac (Z. A. Ameen), ofalghamdi@bu.edu.sa (O. F. Alghamdi),
baravan.asaad@uoz.edu.kr (B. A. Asaad), ramadhan.hajani@uod.ac (R. A. Mohammed)

https://www.ejpam.com 1168 © 2024 EJPAM All rights reserved.



Z. A. Ameen et al. / Eur. J. Pure Appl. Math, 17 (2) (2024), 1168-1182 1169

General topology is the mathematical branch of topology that concerns itself with the
foundational set-theoretic notions and constructions. Motivated by the standard axioms
of classical topological space, Shabir and Naz [28], and Çağman et al. [15], separately,
introduced another branch of topology named “soft topology.”

Soft topology is a combination of soft set theory and topology. It is focused on the
construction of the system of all soft sets.

Soft sets were presented as a collection of relevant parameters to characterize a universe
of possibilities. Soft set theory has been a fruitful area of study and connection with various
disciplines since its establishment. Molodtsov [24], in 1999, originated the soft set theory
as a mathematical tool for dealing with uncertainty which is free of the challenges related
with other theories such as fuzzy set theory [32], rough set theory [26], and so on. In
particular, the nature of parameter sets associated with soft sets provides a standardized
foundation for modeling uncertain data. This leads to the rapid growth of soft set theory
and soft topology in a short amount of time and provides various applications of soft sets
in real life see [16, 17, 27, 30]).

There are various studies that have made significant contributions to the development
of soft topology since its foundation in [15, 28]. A soft topological approach was then
used to interpret the behavior of the most fundamental concepts in (general) topology.
To be specific, soft compactness [14], soft connectedness [20], soft submaximality [2], soft
extremal disconnectedness [13], soft clustering [9], soft simple extendedness [8], and soft
nodecness [6], and congruence representations [11] of soft spaces.

Different methods of generating soft topologies on a common universal set were dis-
cussed in [3, 4, 19, 25, 29, 33].

Soft continuity of mappings has been widely generalized to diverse classes, including
soft semi-continuity [21], soft β-continuity [31], soft U-continuity [7], soft SD-continuity
[10], and mappings of the Baire property [12].

Soft separation axioms are another significant aspect in the late development of soft
topology; see for example [1, 23, 28].

Two remarkable formulas for generating soft topologies from a system of crisp topolo-
gies have been given by Terepeta [29]. One of the formulas (Formula 2) is said to generate
a single set soft topology, while the other one generates a more general soft topology
(Formula 1). Terepeta mainly applied Formula 2 to study the inheritance of soft separa-
tion axioms after the system of crisp topologies. Recently, Alcantud [3] proposed a slight
extension of Formula 1 (we also call it Formula 1). He then employed such a formula to in-
vestigate the behavior of separability and second countability axioms between a system of
crisp topologies and the soft topology generated by it. Recently, Alcantud [4] established
crucial relationships between soft and fuzzy soft topologies. The work of Terepeta and
Alcantud inspired us to attempt this research. Following their direction, we first apply the
formulas to the system of crisp topologies taken from a soft topology in order to determine
the connections between the obtained soft topologies and the original one. In addition, we
use Formula 1 to verify how well the separation axioms are transferred between a system
of crisp topologies and the soft topology generated by it. The latter statement extends the
work of Terepeta (see, Section 2.1 in [29]), which is the main objective of this research.



Z. A. Ameen et al. / Eur. J. Pure Appl. Math, 17 (2) (2024), 1168-1182 1170

2. Preliminaries

Let X be an initial universe, P(X) be all subsets of X and Ω be a set of parameters.
An ordered pair (F,Ω) = {(ω, F (ω)) : ω ∈ Ω} is said to be a soft set over X, where
F : Ω → P(X) is a set value mapping. The family of all soft sets on X is represented
by SΩ(X). The soft set (X,Ω)\(F,Ω) (or simply (F,Ω)c = (F c,Ω)) is the complement
of (F,Ω), where F c : Ω → P(X) is given by F c(ω) = X\F (ω) for each ω ∈ Ω. A soft
subset (F,Ω) over X is called null, denoted by Φ̃, if F (ω) = ∅ for each ω ∈ Ω and is
called absolute, denoted by X̃, if F (ω) = X for each ω ∈ Ω. Notice that X̃c = Φ̃ and
Φ̃c = X̃. It is said that (A,Ω) is a soft subset of (B,Ω) (written by (A,Ω)⊆̃(B,Ω), [22])
if A(ω) ⊆ B(ω) for each ω ∈ Ω, and (A,Ω) = (B,Ω) if (A,Ω)⊆̃(B,Ω) and (B,Ω)⊆̃(A,Ω).
The union of soft sets (A,Ω), (B,Ω) is represented by (F,Ω) = (A,Ω)∪̃(B,Ω), where
F (ω) = A(ω)∪B(ω) for each ω ∈ Ω, and intersection of soft sets (A,Ω), (B,Ω) is given by
(F,Ω) = (A,Ω)∩̃(B,Ω), where F (ω) = A(ω) ∩ B(ω) for each ω ∈ Ω, see [5]. A soft point
[28] is a soft set (F,Ω) over X in which F (ω) = {x} for each ω ∈ Ω, where x ∈ X, and is
denoted by ({x},Ω). It is said that a soft point ({x},Ω) is in (F,Ω) (briefly, x ∈ (F,Ω))
if x ∈ F (ω) for each ω ∈ Ω. On the other hand, x /∈ (F,Ω) if x /∈ F (ω) for some ω ∈ Ω.

This implies that if ({x},Ω)
⋂̃

(F,Ω) = Φ̃, then x /∈ (F,Ω).

Definition 1. [28] A collection Σ of SΩ(X) is said to be a soft topology on X if the
following conditions are satisfied:

(i) Φ̃, X̃ ∈ Σ;

(ii) If (F1,Ω), (F2,Ω) ∈ Σ, then (F1,Ω)∩̃(F2,Ω) ∈ Σ; and

(iii) If each {(Fi,Ω) : i ∈ I}⊆̃Σ, then
⋃̃

i∈I(Fi,Ω) ∈ Σ.

Terminologically, we call (X,Σ,Ω) a soft topological space on X. The elements of Σ are
called soft open sets in Σ (or simply, soft open sets when no confusion arises), and their
complements are called soft closed sets in Σ (or shortly, soft closed sets).

We use (X,Σ,Ω) to mention a soft topological space. In addition, (F,Ω) and (G,Ω)
are disjoint soft sets over X if (F,Ω)∩̃(G,Ω) = Φ̃.

Definition 2. [15] A subcollection B ⊆ Σ is called a soft base for the soft topology Σ if
each element of Σ is a union of elements of B.

Definition 3. [15] Let Σ1,Σ2 be two soft topologies on X. It is said that Σ2 is finer than
Σ1 (or Σ1 is coarser than Σ2) if Σ1⊆̃Σ2.

Lemma 1. [28] Let (X,Σ,Ω) be a soft topology on X. For each ω ∈ Ω, Σω = {F (ω) :
(F,Ω) ∈ Σ} is a crisp topology on X.

Definition 4. [2] Let F⊆̃SΩ(X). The intersection of all soft topologies on X including
F is called a soft topology generated by F and is referred to T (F).



Z. A. Ameen et al. / Eur. J. Pure Appl. Math, 17 (2) (2024), 1168-1182 1171

Lemma 2. [2, Lemma 3.5] Let Σ1,Σ2 be two soft topologies on X. The resulting soft topol-
ogy T (Σ1∪̃Σ2) is identical to the soft topology T (F) generated by F = {(F1,Ω)∩̃(F2,Ω) :
(F1,Ω) ∈ Σ1, (F2,Ω) ∈ Σ2}.

Definition 5. [28] A soft space (X,Σ,Ω) is called

(i) soft T0 if for each x, y ∈ X with x ̸= y, there exist soft open sets (U,Ω), (V,Ω) such
that x ∈ (U,Ω), y /∈ (U,Ω) or x /∈ (V,Ω), y ∈ (V,Ω),

(ii) soft T1 if for each x, y ∈ X with x ̸= y, there exist soft open sets (U,Ω), (V,Ω) such
that x ∈ (U,Ω), y /∈ (U,Ω) and x /∈ (V,Ω), y ∈ (V,Ω),

(iii) soft T2 (soft Hausdorff) if for each x, y ∈ X with x ̸= y, there exist soft open sets

(U,Ω), (V,Ω) containing x, y respectively such that (U,Ω)
⋂̃

(V,Ω) = Φ̃.

(iv) soft regular if for each soft closed set (F,Ω) and each soft point x with x /∈ (F,Ω),
there exist soft open sets (U,Ω), (V,Ω) such that x ∈ (U,Ω), (F,Ω)⊆̃(V,Ω) and

(U,Ω)
⋂̃

(V,Ω) = Φ̃.

(v) soft normal if for each soft closed sets (F,Ω), (D,Ω) with (F,Ω)
⋂̃

(D,Ω) = Φ̃,
there exist soft open sets (U,Ω), (V,Ω) such that (F,Ω)⊆̃(U,Ω), (D,Ω)⊆̃(V,Ω) and

(U,Ω)
⋂̃

(V,Ω) = Φ̃.

(vi) soft T3 if it is soft T1 and soft regular.

(vii) soft T4 if it is soft T1 and soft normal.

Lemma 3. [23, Theorem 3.18] If (X,Σ,Ω) is a soft regular space, then Σω = Σω′ for each
ω, ω′ ∈ Ω.

3. Methods of generating soft topologies and their relationships

This section provides different methods of producing soft topologies via Formulas 1 &
2. An example is given which discusses the implementation of these formulas in detail. The
relationships between the original soft topology and the soft topologies that are produced
by Formulas 1 & 2.

Definition 6. [3, 29] Let Σ = {Σω : ω ∈ Ω} be a family of (crisp) topologies on a set X
for some index set Ω. Then following procedures produce different soft topologies on X:

(Formula 1) T (Σ) =
{
{(ω, F (ω)) : ω ∈ Ω} ∈ SΩ(X) : F (ω) ∈ Σω, ∀ω ∈ Ω

}
,

T (Σ) is called a soft topology generated by Σ. If for each ω, ω′ ∈ Ω, Σω = Σω′ = Σ, then
T (Σ) = T (Σ).

(Formula 2) T̂ (Σω) =
{
{(ω, F (ω)) : ω ∈ Ω} ∈ SΩ(X) : F (ω) = F (ω′) ∈ Σω,∀ω, ω′ ∈ Ω

}
,

T̂ (Σω) is called a single set soft topology generated by Σω.



Z. A. Ameen et al. / Eur. J. Pure Appl. Math, 17 (2) (2024), 1168-1182 1172

Definition 7. Let (X,Σ,Ω) be a soft topological space. If Σ = {Σω : ω ∈ Ω} is the family
of all crisp topologies from Σ, then T (Σ) is called the soft topology associated with Σ. Note
that T (Σ) is called an extended soft topology in [25].

Lemma 4. Let Σ = {Σω : ω ∈ Ω} be the family of all crisp topologies from (X,Σ,Ω).
Then

Σ⊆̃T (Σ).

Proof. It can be concluded from the definition of soft sets and the soft topology
generated by Σ.

Lemma 5. Let β̄ = {βω : ω ∈ Ω} be a family of bases for the topologies Σω on X. Then

B(β̄) =
{
{(ω, F (ω)) : ω ∈ Ω} ∈ SΩ(X) : F (ω) ∈ βω ∪ {∅},∀ω ∈ Ω

}
is a base for a soft

topology on X and T (Σ) = T (B(β̄)).

Proof. By using Corollary 3 in [3] and simple modifications to the proof of Theorem 3
in [3], we can conclude the proof.

The following result is a straightforward generalization of Lemma 2, so the proof is
omitted.

Lemma 6. Let {Σω : ω ∈ Ω} be a family of soft topologies on X. The resulting soft topol-

ogy T (
⋃̃

ω∈ΩΣω) is identical to the soft topology T (F) generated by F = {
⋂̃n

ωi=1(Fωi ,Ω) :

(Fωi ,Ω) ∈
⋃̃

ωi∈ΩΣωi}.

Lemma 7. Let Σ = {Σω : ω ∈ Ω} be a family of crisp topologies on X. Then

T (
⋃̃

ω∈Ω
T̂ (Σω)) = T̂ (T (

⋃
ω∈Ω

Σω)).

Proof. The Lemma 5 reduces the task of working with basic soft open sets rather than

soft open sets. Let (B0,Ω) ∈ T (
⋃̃

ω∈ΩT̂ (Σω)). Then (B0,Ω) =
⋂̃n

i=1(Bi,Ω) for (Bi,Ω) ∈⋃̃
T̂ (Σω), and so (B0,Ω) =

⋂̃n

i=1(Bi,Ω) such that (Bi,Ω) ∈ T̂ (Σω) for some ω ∈ Ω. By
Formula 2, one can detach Ω from (Bi,Ω) for i = 0, 1, · · · , n, and get B0 =

⋂n
i=1Bi, where

Bi ∈ Σω for some ω ∈ Ω. This implies that B0 =
⋂n

i=1Bi for Bi ∈
⋃

Σω. Therefore, by

Formula 2, (B0,Ω) ∈ T̂ (T (
⋃̃

Σω)). The reverse of the inclusion can be proved by a similar
technique.

The following example shows how the techniques in Definition 1 and the relations in
Lemmas 4−7 can be used in practice:

Example 1. Let X = {x1, x2, x3}, Ω = {ω1, ω2}. Consider the soft topology on X,

Σ = {Φ̃, (F1,Ω), (F2,Ω), (F3,Ω), (F4,Ω), X̃},

where

(F1,Ω) = {(ω1, {x1}), (ω2, ∅)},



Z. A. Ameen et al. / Eur. J. Pure Appl. Math, 17 (2) (2024), 1168-1182 1173

(F2,Ω) = {(ω1, {x1, x2}), (ω2, X)},
(F3,Ω) = {(ω1, ∅), (ω2, {x3})}, and

(F4,Ω) = {(ω1, {x1}), (ω2, {x3})}.

The crisp topologies from Σ are

Σω1 = {∅, {x1}, {x1, x2}, X} and Σω2 = {∅, {x3}, X}.

Applying the Formula 2, we obtain the following two soft topologies on X:

T̂ (Σω1) =
{

Φ̃, {(ω1, {x1}), (ω2, {x1})}, {(ω1, {x1, x2}), (ω2, {x1, x2})}, X̃
}

=
{

Φ̃, ({x1},Ω), ({x1, x2},Ω), X̃
}

(more compactly) and

T̂ (Σω2) =
{

Φ̃, {(ω1, {x3}), (ω2, {x3})}, X̃
}

= {Φ̃, (Ω, {x3}), X̃)}.

From Lemma 6, we can naturally generate a soft topology T on X by the union of
T̂ (Σω1) and T̂ (Σω2). That is,

T
(⋃̃2

i=1
T̂ (Σωi)

)
=

{
Φ̃, (G1,Ω), (G2,Ω), (G3,Ω), (G4,Ω), X̃

}
,

where

(G1,Ω) = {(ω1, {x1}), (ω2, {x1})},
(G2,Ω) = {(ω1, {x3}), (ω2, {x3})},
(G3,Ω) = {(ω1, {x1, x2}), (ω2, {x1, x2})}, and

(G4,Ω) = {(ω1, {x1, x3}), (ω2, {x1, x3})}.

The compact form of the above conclusion is

T
(⋃̃2

i=1
T̂ (Σωi)

)
=

{
Φ̃, ({x1},Ω), ({x3},Ω), ({x1, x2},Ω), ({x1, x3},Ω), X̃

}
.

By applying the Formula 1, the next soft topology on X will be obtained.

T (Σ) = T ({Σω1 ,Σω2}) =
{

Φ̃, (H1,Ω), (H2,Ω), · · · , (H10,Ω), X̃
}
,

where

(H1,Ω) = {(ω1, ∅), (ω2, X)},
(H2,Ω) = {(ω1, ∅), (ω2, {x3})},
(H3,Ω) = {(ω1, X), (ω2, ∅)},
(H4,Ω) = {(ω1, X), (ω2, {x3})},
(H5,Ω) = {(ω1, {x1}), (ω2, ∅)},
(H6,Ω) = {(ω1, {x1}), (ω2, X)},



Z. A. Ameen et al. / Eur. J. Pure Appl. Math, 17 (2) (2024), 1168-1182 1174

(H7,Ω) = {(ω1, {x1}), (ω2, {x3})},
(H8,Ω) = {(ω1, {x1, x2}), (ω2, ∅)},
(H9,Ω) = {(ω1, {x1, x2}), (ω2, X)}, and

(H10,Ω) = {(ω1, {x1, x2}), (ω2, {x3})}.

One can easily check that the above computations lead to the following observations:

• Σ is a subcollection of T (Σ).

• Σ is independent of T
(⋃̃2

i=1T̂ (Σωi)
)
.

• T
(⋃̃2

i=1T̂ (Σωi)
)
is independent of T (Σ).

The relationships between the soft topologies on a common universe obtained by the
methods described in this section is summarized in Diagram 1.

Σ

T (Σ) Σ = {Σω : ω ∈ Ω} T̂ (T (
⋃̃
Σ))

p
ro
d
u
ces

⊇̃

independent

generates generates

Figure 1: Relationships between different soft topologies

4. Non-uniqueness of soft topology T (Σ) associated with Σ

In this short section, we provide an example to witness that two different (even incom-
parable) soft topologies may have a common associated soft topology.

Example 2. Consider the soft topology on X, Σ = {Φ̃, (F1,Ω), (F2,Ω), (F3,Ω), (F4,Ω), X̃}
given in Example 1, there X = {x1, x2, x3}, Ω = {ω1, ω2}. Let Σ′ = {Φ̃, (R1,Ω), (R2,Ω),
(R3,Ω), (R4,Ω), X̃} be another soft topology on X, where

(R1,Ω) = {(ω1, {x1}), (ω2, ∅)},
(R2,Ω) = {(ω1, {x1, x2}), (ω2, X)},
(R3,Ω) = {(ω1, X), (ω2, {x3})}, and

(R4,Ω) = {(ω1, {x1, x2}), (ω2, {x3})}.

Then Σ and Σ′ are incomparable. Set Σ̂ = Σ
⋃̃

Σ′. Therefore, Σ̂ is finer than both Σ and Σ′.
On the other hand, Σ,Σ′ and Σ̂ have the same family of crisp topologies Σ = {Σω1 ,Σω2},
and thus they generate only one T (Σ).



Z. A. Ameen et al. / Eur. J. Pure Appl. Math, 17 (2) (2024), 1168-1182 1175

5. Separation axioms preservation between Σ and T (Σ)

With the exception of T (Σ), a single set soft topology T̂ (Σ) generated by Σ inherits
soft separation axioms after Σ, according to Terepeta [29]. In this section, we use T (Σ)
to see how well separation axioms are preserved when moving from Σ to T (Σ) and vice
versa.

Here, we start defining special types of soft sets that exist when constructing a soft
topology by using T (Σ) for future usage.

Definition 8. Let G,H ⊂ X and let ω ∈ Ω, we define

(i) (FG
ω ,Ω) to be a soft set over X such that FG

ω (ω) = G and FG
ω (ω′) = X for each

ω′ ̸= ω.

(ii) (Fω
H ,Ω) to be a soft set over X such that Fω

H(ω) = H and Fω
H(ω′) = ∅ for each

ω′ ̸= ω, (see, [3, Definition 5]).

Note that (FG
ω ,Ω)c = (Fω

H ,Ω) if and only if Gc = H.

Theorem 1. Let Σ = {Σω : ω ∈ Ω} be a family of crisp topologies on X. If Σω is a
T0-space for some ω ∈ Ω then T (Σ) is a soft T0-space.

Proof. Suppose that Σω is a T0-space for some ω ∈ Ω. Let x, y ∈ X with x ̸= y. Then
there exist open sets U, V ∈ Σω such that x ∈ U , y /∈ U or x /∈ V , y ∈ V . By Definition
8, there exist two corresponding soft sets (FU

ω ,Ω), (F V
ω ,Ω) ∈ T (Σ) for which x ∈ (FU

ω ,Ω),
y /∈ (FU

ω ,Ω) or x /∈ (F V
ω ,Ω), y ∈ (F V

ω ,Ω). Thus, T (Σ) is soft T0.

The following example shows that the converse of Theorem 1 is not true in general.

Example 3. Let X = {x1, x2, x3} and let Σω1 = {∅, {x1}, {x2, x3}, X} and Σω2 =
{∅, {x3}, {x1, x2}, X} be crisp topologies on X indexed by Ω = {ω1, ω2}. By using the
Formula 1, the following soft topology on X will be obtained:

T (Σ) = T ({Σω1 ,Σω2}) =
{

Φ̃, (H1,Ω), (H2,Ω), (H3,Ω), · · · , (H14,Ω), X̃
}
,

where

(H1,Ω) = {(ω1, X), (ω2, ∅)},
(H2,Ω) = {(ω1, {x1}), (ω2, ∅)},
(H3,Ω) = {(ω1, {x2, x3}), (ω2, ∅)},
(H4,Ω) = {(ω1, ∅), (ω2, X)},
(H5,Ω) = {(ω1, {x1}), (ω2, X)},
(H6,Ω) = {(ω1, {x2, x3}), (ω2, X)},
(H7,Ω) = {(ω1, ∅), (ω2, {x1, x2})},
(H8,Ω) = {(ω1, X), (ω2, {x1, x2})},
(H9,Ω) = {(ω1, {x1}), (ω2, {x1, x2})},



Z. A. Ameen et al. / Eur. J. Pure Appl. Math, 17 (2) (2024), 1168-1182 1176

(H10,Ω) = {(ω1, {x2, x3}), (ω2, {x1, x2})},
(H11,Ω) = {(ω1, ∅), (ω2, {x3})},
(H12,Ω) = {(ω1, X), (ω2, {x3})},
(H13,Ω) = {(ω1, {x1}), (ω2, {x3})}, and

(H14,Ω) = {(ω1, {x2, x3}), (ω2, {x3})}.

Since x1 ∈ (H5,Ω), x2, x3 ̸∈ (H5,Ω) and x3 ∈ (H14,Ω), x2 ̸∈ (H14,Ω), then T (Σ) is
soft T0. On the other hand, neither of Σω1 nor Σω2 is T0.

Theorem 2. Let Σ = {Σω : ω ∈ Ω} be a family of crisp topologies on X. If Σω is a
T1-space for some ω ∈ Ω, then T (Σ) is a soft T1-space.

Proof. It is entirely analogous to the first part of the proof of Theorem 1.

The following example shows that the converse of Theorem 2 is not true in general. It
also refutes Theorem 3.5 in [18]:

Example 4. Let X = {x1, x2} and let Σω1 = {∅, {x1}, X} and Σω2 = {∅, {x2}, X} be
crisp topologies on X indexed by Ω = {ω1, ω2}. By using the Formula 1, the following soft
topology on X will be obtained:

T (Σ) = T ({Σω1 ,Σω2}) ={
Φ̃, (H1,Ω), (H2,Ω), (H3,Ω), (H4,Ω), (H5,Ω), (H6,Ω), (H7,Ω), X̃

}
,

where

(H1,Ω) = {(ω1, ∅), (ω2, X)},
(H2,Ω) = {(ω1, X), (ω2, ∅)},
(H3,Ω) = {(ω1, ∅), (ω2, {x2})},
(H4,Ω) = {(ω1, X), (ω2, {x2})},
(H5,Ω) = {(ω1, {x1}), (ω2, ∅)},
(H6,Ω) = {(ω1, {x1}), (ω2, X)}, and

(H7,Ω) = {(ω1, {x1}), (ω2, {x2})}.

Then (H4,Ω) and (H6,Ω) are soft open sets in T (Σ) such that x1 ∈ (H6,Ω), x2 /∈ (H6,Ω)
and x2 ∈ (H4,Ω), x1 /∈ (H4,Ω). Thus, T (Σ) is soft T1. On the other hand, neither of
Σω1 nor Σω2 is T1.

Theorem 3. Let Σ = {Σω : ω ∈ Ω} be a family of crisp topologies on X. Then Σω is a
T2-space for each ω ∈ Ω if and only if T (Σ) is a soft T2-space.

Proof. Assume that Σω is T2 for each ω ∈ Ω. Let x, y ∈ X with x ̸= y. Then,
for each ω, there exist open sets U(ω), V (ω) ∈ Σω such that x ∈ U(ω), y ∈ V (ω) and
U(ω) ∩ V (ω) = ∅. Set (U,Ω) = {(ω,U(ω)) : ω ∈ Ω} and (V,Ω) = {(ω, V (ω)) : ω ∈



Z. A. Ameen et al. / Eur. J. Pure Appl. Math, 17 (2) (2024), 1168-1182 1177

Ω}. Then, (U,Ω), (V,Ω) ∈ T (Σ) such that x ∈ (U,Ω), y ∈ (V,Ω) and (U,Ω)
⋂̃

(V,Ω) =
{(ω,U(ω) ∩ V (ω)) : ω ∈ Ω} = Φ̃. Hence, T (Σ) is soft T2.

Conversely, let x, y ∈ X with x ̸= y. Suppose that T (Σ) is soft T2. Then, there exist

soft open sets (G,Ω), (H,Ω) such that x ∈ (G,Ω), y ∈ (H,Ω) and (G,Ω)
⋂̃

(H,Ω) = Φ̃.
This means that for each ω ∈ Ω, x ∈ G(ω), y ∈ H(ω) and G(ω) ∩H(ω) = ∅. Thus, Σω is
T2 for each ω ∈ Ω.

Corollary 1. If (X,Σ,Ω) is a soft T2-space, then Σω is T2 for each ω ∈ Ω.

Proof. It is an immediate consequence of Lemma 4 and Theorem 3.

Notice that Theorem 3 and Corollary 1 generalize (part of) Theorem 4 in [29] and
Proposition 17 in [28], respectively.

Lemma 8. [29, Theorem 3] Let Σ be a (crisp) topology on a set X. A soft set (F,Ω) is
soft closed in T (Σ) if and only if (F,Ω) = {(ω, F (ω)) : F c(ω) ∈ Σ}.

Theorem 4. Let Σ = {Σω : ω ∈ Ω} be a family of crisp topologies on X. If T (Σ) is a
soft regular space, then Σω is a regular space for each ω ∈ Ω.

Proof. Let ω ∈ Ω. Take x ∈ X and F (ω) be a closed set in (X,Σω) such that
x /∈ F (ω). The Definition 6 and Lemma 3 tell us that soft regularity of T (Σ) guarantees
the equality of T (Σ) = T (Σ). Set (F,Ω) = {(ω, F (ω)) : F c(ω) ∈ Σ}. By Lemma 8,
(F,Ω) is soft closed in T (Σ) along with x /∈ (F,Ω). Since T (Σ) is soft regular, then there
exist soft open sets (U,Ω), (V,Ω) in T (Σ) such that x ∈ (U,Ω), (F,Ω)⊆̃(V,Ω) and Φ̃ =

(U,Ω)
⋂̃

(V,Ω) = {(ω,U(ω) ∩ V (ω)) : ω ∈ Ω}. This implies that x ∈ U(ω), F (ω) ⊆ V (ω)
and U(ω) ∩ V (ω) = ∅ for each ω ∈ Ω. Since U(ω), V (ω) ∈ Σω, then Σω is regular for each
ω ∈ Ω.

Corollary 2. If (X,Σ,Ω) is a soft regular space, then Σω is regular for each ω ∈ Ω.

Proof. It can be concluded from Lemma 4 and Theorem 4.

Remark 1. We shall mention that it is observed in Remark 3.23 (2’) [23] that if (X,Σ,Ω)
is a soft T3 space, then Σω is T3 for each ω ∈ Ω. This conclusion is more general than
Corollary 2, but it cannot be followed from any of our results due to Example 4.

The examples given below disprove the reverse of theorem 4:

Example 5. Let X = {x}, let Ω = {ω1, ω2}, and let Σ = {Σω1 ,Σω2}, where Σω1 = Σω2 =
{∅, X}. One can check that each Σωi is trivially a regular space. On the other hand,
the soft topology T (Σ) = {Φ̃, (F1,Ω), (F2,Ω), X̃} is not soft regular, where (F1,Ω) =
{(ω1, ∅), (ω2, X)} and (F2,Ω) = {(ω1, X), (ω2, ∅)}. Indeed, x /∈ (Fi,Ω)c for each i, but no
soft open sets in T (Σ) can separate them.



Z. A. Ameen et al. / Eur. J. Pure Appl. Math, 17 (2) (2024), 1168-1182 1178

Example 6. Consider (R,U), the natural topology on R. Let Ω = {ω1, ω2}. It is
well known that the natural topology is T3 space. We will prove that T (U) is not T3,
where T (U) = T (B(β̄)) which is defined in Lemma 5. Pick the point 0 and (F,Ω) =
{(ω1, [−1, 1]), (ω2, [2, 3])}. Hence, 0 ̸∈ (F,Ω) and (F,Ω) is a closed soft set. Let (W1,Ω)
and (W2,Ω) be any two basic open soft sets such that 0 ∈ (W1,Ω) and (F,Ω)⊆̃(W2,Ω).
Then, (W1,Ω) = {(ω1, (a, b)), (ω2, (c, d)} such that 0 ∈ (a, b), 0 ∈ (c, d) and (W2,Ω) =
{(ω1, (r, t)), (ω2, (s, k)}. Since 0 ∈ F (ω1) = [−1, 1] ⊂ W2(ω1), then 0 ∈ (r, t) which implies

that (W1,Ω)
⋂̃

(W2,Ω) ̸= Φ̃.

Theorem 5. Let Σ = {Σω : ω ∈ Ω} be a family of crisp topologies on X. Then Σω is a
normal space for each ω ∈ Ω if and only if T (Σ) is a soft normal space.

Proof. Let Σω be normal for each ω ∈ Ω. Suppose (A,Ω), (B,Ω) are disjoint soft closed
sets in T (Σ). Then (A,Ω) = {(ω,A(ω)) : Ac(ω) ∈ Σω, ω ∈ Ω} and (B,Ω) = {(ω,B(ω)) :

Bc(ω) ∈ Σω, ω ∈ Ω}. Therefore Φ̃ = (A,Ω)
⋂̃

(B,Ω) = {(ω,A(ω) ∩B(ω)) : Ac(ω), Bc(ω) ∈
Σω, ω ∈ Ω}. We obtain that A(ω) ∩ B(ω) = ∅. Since Σω is a normal space for each
ω ∈ Ω, there exist open sets G(ω), H(ω) such that A(ω) ⊆ G(ω), B(ω) ⊆ H(ω) and
G(ω) ∩H(ω) = ∅. Set (G,Ω) = {(ω,G(ω)) : ω ∈ Ω} and (H,Ω) = {(ω,H(ω)) : ω ∈ Ω}.
Then (G,Ω), (H,Ω) ∈ T (Σ) such that (A,Ω)⊆̃(G,Ω) and (B,Ω)⊆̃(H,Ω). Furthermore,

(G,Ω)
⋂̃

(H,Ω) = {(ω,G(ω) ∩H(ω)) : ω ∈ Ω} = Φ̃. This shows that T (Σ) is soft normal.
Conversely, for each ω ∈ Ω, we let C(ω), D(ω) be disjoint closed sets in Σω. By Lemma

8, (C,Ω) = {(ω,C(ω)) : Cc(ω) ∈ Σω, ω ∈ Ω} and (D,Ω) = {(ω,D(ω)) : Dc(ω) ∈ Σω, ω ∈
Ω} are soft closed sets in T (Σ) and (C,Ω)

⋂̃
(D,Ω) = {(ω,C(ω) ∩ D(ω)) : ω ∈ Ω} = Φ̃.

Since T (Σ) is soft normal, then there exist disjoint soft open sets (U,Ω), (V,Ω) such that
(C,Ω)⊆̃(U,Ω) and (D,Ω)⊆̃(V,Ω). This implies that C(ω) ⊆ U(ω), D(ω) ⊆ V (ω) and
U(ω) ∩ V (ω) = ∅ for each ω ∈ Ω. Thus, Σω is normal for each ω ∈ Ω.

We shall remark that if (X,Σ,Ω) is a soft normal space, then Σω need not be a normal
space for each ω ∈ Ω.

Example 7. Let X = {x1, x2, x3} and Ω = {ω1, ω2}. Suppose

Σ =
{

Φ̃, (H1,Ω), (H2,Ω), · · · , (H9,Ω), X̃
}
,

where

(H1,Ω) = {(ω1, ∅), (ω2, {x3})},
(H2,Ω) = {(ω1, {x3}), (ω2, ∅)},
(H3,Ω) = {(ω1, {x2, x3}), (ω2, ∅)},
(H4,Ω) = {(ω1, X), (ω2, ∅)},
(H5,Ω) = {(ω1, {x1, x3}), (ω2, ∅)},
(H6,Ω) = {(ω1, {x3}), (ω2, {x3})},
(H7,Ω) = {(ω1, {x2, x3}), (ω2, {x3})},
(H8,Ω) = {(ω1, X), (ω2, {x3})}, and



Z. A. Ameen et al. / Eur. J. Pure Appl. Math, 17 (2) (2024), 1168-1182 1179

(H9,Ω) = {(ω1, {x1, x3}), (ω2, {x3})}.

Then Σ is a soft normal space as each pair of non-null soft closed sets intersects each
other. On the other hand, Σω1 = {X, ∅, {x3}, {x2, x3}, {x1, x3}} is not normal. Indeed,
x3 /∈ {x1} and {x1} is a closed set in Σω1. Let U, V ∈ Σω1 such that x3 ∈ U and {x1} ⊆ V .
Hence, by the definition of Σω1, we know that {x3} ⊆ U and {x1, x3} ⊆ V . Therefore,
U ∩ V ̸= ∅.

The following example demonstrates that T (Σ) needs not to be soft normal if Σω is
not normal for some ω ∈ Ω.

Example 8. Let X = {x1, x2, x3} and Ω = {ω1, ω2}. Take Σω1 = {∅, X} and Σω2 =
{∅, {x1}, {x1, x2}, {x1, x3}, X}. One can easily verify that Σω1 is normal. However, Σω2

is not normal since {x2}, {x3} are disjoint closed sets in Σω2 that cannot be separated by
two disjoint open sets. The soft topology:

T (Σ) =
{

Φ̃, (F1,Ω), (F2,Ω), (F3,Ω), (F4,Ω), (F5,Ω), (F6,Ω), (F7,Ω), (F8,Ω), X̃
}
,

where

(F1,Ω) = {(ω1, ∅), (ω2, {x1})},
(F2,Ω) = {(ω1, ∅), (ω2, {x1, x2})},
(F3,Ω) = {(ω1, ∅), (ω2, {x1, x3})},
(F4,Ω) = {(ω1, ∅), (ω2, X)},
(F5,Ω) = {(ω1, X), (ω2, ∅)},
(F6,Ω) = {(ω1, X), (ω2, {x1})},
(F7,Ω) = {(ω1, X), (ω2, {x1, x2})}, and

(F8,Ω) = {(ω1, X), (ω2, {x1, x3})},

is not a soft normal space. Indeed, since {(ω1, ∅), (ω2, {x3})} and {(ω1, X), (ω2, {x2})} are
disjoint soft closed sets in T (Σ) but there are no disjoint soft open sets can separate them.

Remark 2. Notice that we can provide a more general proof sketch that proves the above
claim, which is another proof to the part one of Theorem 5. If Σ = {Σω : ω ∈ Ω} is not
a normal topology for some ω̄ ∈ Ω, then there are closed sets A,B in Σω̄ which cannot
be separated by any open sets. The soft sets (F ω̄

A ,Ω), (FB
ω̄ ,Ω) are closed and disjoint in

T (Σ). If there exist soft open sets (G,Ω), (H,Ω) in T (Σ) such that (F ω̄
A ,Ω)⊆̃(G,Ω),

(FB
ω̄ ,Ω)⊆̃(H,Ω) and (G,Ω)

⋂̃
(H,Ω) = Φ̃, then A ⊆ G(ω̄), B ⊆ H(ω̄) and G(ω̄) ∩H(ω̄) =

∅, which is a contradiction.

6. Conclusion

This study develops a methodical understanding of the connections between a system of
crisp topologies and the soft topology produced by it. The procedure is carried out with the



REFERENCES 1180

help of two formulas. If we start from an original soft topology, then it produces a system
of crisp topologies. With our formulas, we can generate two different new soft topologies.
We have discussed the relationships between these soft topologies. Moreover, we show that
the resulting soft topology via Formula 1 is always finer than the original one, while the soft
topology generated by Formula 2 is incomparable. We see that two different original soft
topologies may generate a single soft topology by either of the formulas. Furthermore, we
study the preservation of separation axioms between the system of crisp topologies and the
soft topology generated by it. More precisely, we show that Hausdorffness and normality
behave better in transforming to soft topologies and conversely. On the other hand, other
separation axioms act differently. If one of the crisp topologies is respectively T0, T1, then
it guarantees that the resulting soft topology is soft T0, T1. The converse may not be true.
All of the crisp topologies are regular when the soft topology generated by them is a soft
regular.

References

[1] Dina Abuzaid and Samer Al Ghour. Three new soft separation axioms in soft topo-
logical spaces. AIMS Mathematics, 9(2):4632–4648, 2024.

[2] Samer Al Ghour and Zanyar A Ameen. Maximal soft compact and maximal soft con-
nected topologies. Applied Computational Intelligence and Soft Computing, 2022:Ar-
ticle ID 9860015, 7 pages.

[3] Jose Carlos R Alcantud. Soft open bases and a novel construction of soft topologies
from bases for topologies. Mathematics, 8(5):672, 2020.

[4] José Carlos R Alcantud. The relationship between fuzzy soft and soft topologies.
International Journal of Fuzzy Systems, 24(3):1653–1668, 2022.

[5] M Irfan Ali, Feng Feng, Xiaoyan Liu, Won Keun Min, and Muhammad Shabir. On
some new operations in soft set theory. Computers & Mathematics with Applications,
57(9):1547–1553, 2009.

[6] Mesfer H Alqahtani and Zanyar A Ameen. Soft nodec spaces. AIMS Mathematics,
9:3289–3302, 2024.

[7] Zanyar Ameen. A non-continuous soft mapping that preserves some structural soft
sets. Journal of Intelligent & Fuzzy Systems, 42(6):5839–5845, 2022.

[8] Zanyar A Ameen and Samer Al Ghour. Extensions of soft topologies. Filomat,
36(15):5279–5287, 2022.

[9] Zanyar A Ameen and Samer Al Ghour. Cluster soft sets and cluster soft topologies.
Computational and Applied Mathematics, 42(8):337, 2023.



REFERENCES 1181

[10] Zanyar A Ameen, Tareq M Al-shami, and Baracan A Asaad. Further properties of
soft somewhere dense continuous functions and soft Baire spaces. J. Math. Comput.
Sci, 32:54–63, 2024.

[11] Zanyar A Ameen and Mesfer H Alqahtani. Congruence representations via soft ideals
in soft topological spaces. Axioms, 12(11):1015, 2023.

[12] Zanyar A Ameen and Mesfer H Alqahtani. Some classes of soft functions defined by
soft open sets modulo soft sets of the first category. Mathematics, 11(20):4368, 2023.

[13] Baravan A Asaad. Results on soft extremally disconnectedness of soft topological
spaces. J. Math. Computer Sci, 17:448–464, 2017.

[14] Abdülkadir Aygünoğlu and Halis Aygün. Some notes on soft topological spaces.
Neural computing and Applications, 21(1):113–119, 2012.

[15] Naim Çağman, Serkan Karataş, and Serdar Enginoglu. Soft topology. Computers &
Mathematics with Applications, 62(1):351–358, 2011.

[16] Orhan Dalkılıç and Naime Demirtaş. Algorithms for covid-19 outbreak using soft set
theory: estimation and application. Soft Computing, 27:3203–3211, 2023.

[17] Mostafa K El-Bably, Radwan Abu-Gdairi, and Mostafa A El-Gayar. Medical diagnosis
for the problem of chikungunya disease using soft rough sets. AIMS Mathematics,
8(4):9082–9105, 2023.

[18] Orhan Göçür and Abdullah Kopuzlu. On soft separation axioms. Ann. Fuzzy Math.
Inform, 9(5):817–822, 2015.

[19] Saeid Jafari, AA El-Atik, Raja Mohammad Latif, and MK El-Bably. Soft topological
spaces induced via soft relations. WSEAS Trans. Math, 20:1–8, 2021.

[20] Fucai Lin. Soft connected spaces and soft paracompact spaces. International Journal
of Mathematical and Computational Sciences, 7(2):277–283, 2013.

[21] Juthika Mahanta and Pramod Kumar Das. On soft topological space via semiopen
and semiclosed soft sets. Kyungpook Math J., 54:221–236, 2014.

[22] Pabitra Kumar Maji, Ranjit Biswas, and A Ranjan Roy. Soft set theory. Computers
& Mathematics with Applications, 45(4-5):555–562, 2003.

[23] Won Keun Min. A note on soft topological spaces. Computers & Mathematics with
Applications, 62(9):3524–3528, 2011.

[24] Dmitriy Molodtsov. Soft set theory—first results. Computers & Mathematics with
Applications, 37(4-5):19–31, 1999.

[25] Sk Nazmul and SK Samanta. Neighbourhood properties of soft topological spaces.
Ann. Fuzzy Math. Inform, 6(1):1–15, 2013.



REFERENCES 1182

[26] Zdzis law Pawlak. Rough sets. International journal of computer & information
sciences, 11(5):341–356, 1982.

[27] Daowu Pei and Duoqian Miao. From soft sets to information systems. In 2005 IEEE
international conference on granular computing, volume 2, pages 617–621. IEEE,
2005.

[28] Muhammad Shabir and Munazza Naz. On soft topological spaces. Computers &
Mathematics with Applications, 61(7):1786–1799, 2011.

[29] Ma lgorzata Terepeta. On separating axioms and similarity of soft topological spaces.
Soft Computing, 23(3):1049–1057, 2019.

[30] Saziye Yuksel, Tugbahan Dizman, Gulnur Yildizdan, and Unal Sert. Application of
soft sets to diagnose the prostate cancer risk. Journal of Inequalities and Applications,
2013(1):1–11, 2013.

[31] Yunus Yumak and Aynur Keskin Kaymarkci. Soft β-open sets and their applications.
Journal of New Theory, (4):80–89, 2015.

[32] L.A. Zadeh. Fuzzy sets. Information and Control, 8(3):338–353, 1965.

[33] İdris Zorlutuna, M Akdag, WK Min, and S Atmaca. Remarks on soft topological
spaces. Annals of fuzzy Mathematics and Informatics, 3(2):171–185, 2012.


	Introduction
	Preliminaries
	Methods of generating soft topologies and their relationships
	Non-uniqueness of soft topology T() associated with 
	Separation axioms preservation between  and T()
	Conclusion