EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS
2025, Vol. 18, Issue 2, Article Number 5863
ISSN 1307-5543 – ejpam.com
Published by New York Business Global

Supra Soft Somewhat Open Sets: Characterizations and
Continuity

Alaa M. Abd El-latif 1,∗, Radwan Abu-Gdairi2, A. A. Azzam 3,4, F. A. Gharib1,
Khaled A. Aldwoah5

1 Department of Mathematics, College of Science, Northern Border University, Arar 91431,
Saudi Arabia
2 Mathematics Department, Faculty of Science, Zarqa University, Zarqa 13132, Jordan
3 Department of Mathematics, Faculty of Science and Humanities, Prince Sattam Bin
Abdulaziz University, Alkharj 11942, Saudi Arabia
4 Department of Mathematics, Faculty of Science, New Valley University, Elkharga 72511,
Egypt
5Department of Mathematics, Faculty of Science, Islamic University of Madinah, Medinah,
Saudi Arabia

Abstract. In this manuscript, we used the supra soft interior operator to define a new approach of
generalized sets named, supra soft somewhat (briefly, SS-sw-) open sets. We discuss its relationships
with the other generalizations and provide the necessary examples and counterexamples. After
that, we define new continuity inspired by this new approach, named SS-sw-continuous function.
We characterize several of its essential properties. We use the SS-sw-closure (interior) operators
to present several equivalent conditions for the new approach. Furthermore, we define a new type
of functions related to SS-sw-open sets, named SS-sw-open functions.

2020 Mathematics Subject Classifications: 54A05, 54C10, 03E72

Key Words and Phrases: Supra soft somewhat open sets, SS-sw-closure operator, SS-sw-
continuous functions, SS-sw-open functions

1. Introduction

In light of the broadest crisp (fuzzy) sets, Molodtsov [1] in 1999, outlined the concept
of soft sets. Maji et al. [2], introduced more operations to soft theory. In 2001, Ahmad
and Kharal [3], defined the concept of soft continuity.

Shabir and Naz [4] defined the notions of soft topological space (STS, for short), which
investigated by Aygunoüglu and Aygün in [5]. In 2012, Zorlutuna et al. [6] presented

∗Corresponding author.
DOI: https://doi.org/10.29020/nybg.ejpam.v18i2.5863

Email addresses: alaa.ali@nbu.edu.sa alaa 8560@yahoo.com (Alaa M. Abd El-latif),
rgdairi@zu.edu.jo (Radwan Abu-Gdairi), aa.azzam@psau.edu.sa (A. A. Azzam),
fatouh.gharib@nbu.edu.sa (F. A. Gharib), aldwoah@yahoo.com (Khaled A. Aldwoah)

https://www.ejpam.com 1 Copyright: © 2025 The Author(s). (CC BY-NC 4.0)



Abd El-latif et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5863 2 of 18

more properties to STS. Later, several types of broader soft open sets and generalized soft
continuity are explored in [7–11].

It was first explained in [12] what soft ideal and soft local functions are. After then,
the soft semi-local functions approach was defined by several authors [13, 14] by using the
definition of soft semi-open sets. To define new soft ideal rough topological spaces, Abd
El-latif [15] employed the soft ideal for this purpose.

By utilizing the soft ideal notion, numerous soft open weaker classes have been ex-
panded in [16–21]. Subsequently, new approaches based on soft ideals were presented for
the soft separation axioms [22, 23], soft connectedness [24], and soft semi-compactness
[25].

El-Sheikh et al. [26] presented the definition of supra soft topological space (SSTS,
for short) in 2014. They also introduced many types of wider soft sets and soft continuity
in SSTS. Later, several valuable papers have been presented related to SS-locally closed
sets [27], SS-b-open sets [28], SS-δi-open sets [29, 30], SS-(strongly) generalized closed sets
[31, 32], SS-separation axioms [33, 34], SS-regular open sets [35], the Baire categories of
soft sets [36] and SS-sd-sets [37]. Recently, several soft topological spaces are introduced
to SSTS in [39–43].

Ameen et al. [44], defined the notion of soft somewhat open (sw-open) sets. In this
regards, Al-shami [45] applied this class to medical application. Also, he and others [46]
used this notion to defined new categories of connectedness and compactness.

The supra soft interior operator was utilized in this manuscript to construct a novel
generalized set approach known as SS-sw-open sets. We examined the key features of this
new approach. The relevant examples and counterexamples are given, and its connections
with the other generalizations are discussed. In addition, we applied this novel concept
for soft continuity. Furthermore, we presented a number of analogous conditions for our
novel methods using the SS-sw-closure (interior) operators.

2. Preliminaries

Definition 1. [1] Let χ be the initial universe set and η be the set of parameters. Then,
a pair (K, η) is called a soft set, which is defined by

Kη = {K(ϑ) : ϑ ∈ η, K : η → P (χ)}.

the category of all soft sets will be represented by S(χ)η. Also, the absolute (null) soft set
will represented by χ̃ (φ̃), where χ̃(ϑ) = χ and φ̃(ϑ) = φ, for all ϑ ∈ η.

Definition 2. [4] The class σ ⊆ S(χ)η is called a soft topology on χ if σ contains χ̃, φ̃
and closed under finite soft intersection and arbitrary soft union.

The triplet (χ, σ, η) is referred to as an STS over χ. Also, for any soft set (G, η), if
(G, η) ∈ σ, then (G, η) is called soft open set and its soft complements (Gc̃, η) is called soft
closed set.

Definition 3. [4, 6] Let (χ, σ, η) be an STS and (K, η) ∈ S(χ)η, then



Abd El-latif et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5863 3 of 18

(1) int(K, η) = ⊔{(O, η) : (O, η) ∈ σ and (O, η)⊆̃(K, η)}.

(2) cl(K, η) = ⊓{(H, η) : (H, η) ∈ σc and (K, η)⊆̃(H, η)}.

Definition 4. [3] Let ψsw : (χ1, σ1, η1) → (χ2, σ2, η2) be a function, where s : χ1 → χ2

and w : η1 → η2. Then

(1) The image of (K, η1) under ψsw, represented by ψsw(K, η1) = (ψsw(K), s(η1)), is a
soft set in S(χ2)η2 such that

ψsw(K)(θ) =

{
⊔θ∈s−1(θ)⊓η1 w(K(ϑ)), s−1(θ) ⊓K ̸= φ,

φ, otherwise.
for all θ ∈ η2.

(2) The pre-image of (H, η2) under ψsw, represented by ψ−1
sw (H, η2) = (ψ−1

sw (H), s−1(η2)),
is a soft set in S(χ1)η1 such that

ψ−1
sw (H)(ϑ) =

{
w−1(H(s(ϑ))), s(ϑ) ∈ η2,
φ, otherwise.

for all ϑ ∈ η1.

If s and w are surjective (injective) together, then ψsw is surjective (injective).

Theorem 5. [3] For the soft function ψsw : (χ1, σ1, η1) → (χ2, σ2, η2), the following
statements hold.

(1) ψ−1
sw ((N

c̃, η2)) = (ψ−1
sw (N, η2))

c̃ ∀ (N, η2) ∈ S(χ2)η2.

(2) ψsw(ψ
−1
sw ((N, η2)))⊆̃(N, η2) ∀ (N, η2) ∈ S(χ2)η2.

(3) (M,η1)⊆̃ψ−1
sw (ψsw((M,η1))) ∀ (M,η1) ∈ S(χ1)η1.

(4) ψsw(χ̃1)⊆̃χ̃2.

Definition 6. [26] The collection ρ ⊆ S(χ)η is called a supra soft topology (or SSTS) on
χ if it contains χ̃, φ̃ and closed under arbitrary soft union.
For any soft set (G, η), if (G, η) ∈ ρ, then (G, η) is called supra soft open (shortly, SS-
open) set or and its soft complements (Gc̃, η) is called SS-closed. Also, if σ ⊂ ρ, then ρ is
called an SSTS associated with σ.

Definition 7. [26] (χ, ρ, η) be an SSTS and (K, η) ∈ S(χ)η, then the SS-interior (closure),
denoted by ints(K, η) (cls(K, η)) where:

(1) ints(K, η) = ⊔{(O, η) : (O, η) ∈ ρ and (O, η)⊆̃(K, η)}.

(2) cls(K, η) = ⊓{(H, η) : (H, η) ∈ ρc and (K, η)⊆̃(H, η)}.

Definition 8. [26] If ψ−1
sw (G, η2) ∈ ρ1 ∀ (G, η2) ∈ σ2, then the soft function ψsw :

(χ1, σ1, η1) → (χ2, σ2, η2) with ρ1 as an associated SSTS with σ1 will called SS-continuous.

Definition 9. [26, 37, 38] A soft subset (G, η) of an SSTS (χ, ρ, η) is called



Abd El-latif et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5863 4 of 18

(1) SS-semi-open set if (G, η)⊆̃cls(ints(G, η)).

(2) SS-β-open set if (G, η)⊆̃cls(ints(cls(G, η))).

(3) SS-α-open set if (G, η)⊆̃ints(cls(ints(G, η))).

(4) SS-dense set if cls(G, η) = χ̃.

(5) SS-co-dense set if ints(G, η) = φ̃.

(6) SS-regular open set if ints(cls(G, η)) = (G, η).

(7) SS-sd-set if there is φ̃ ̸= (O, η) ∈ ρ such that (O, η)⊆̃cls[(O, η)⊓̃(K, η)].

The categories of SS-semi-open (respectively, β-open, α-open, regular-open, sd-) sets shall
be indicated by SOSs(χ)η (respectively, βOSs(χ)η, αOS

s(χ)η, ROS
s(χ)η, SD

s(χ)η).

Definition 10. [26, 37, 38] A soft function ψsw : (χ1, σ1, η1) → (χ2, σ2, η2) with ρ1 as
an associated SSTS with σ1 is referred to as

(1) SS-semi-cts if ψ−1
sw (G, η2) ∈ SOSs(χ1)η1 ∀ (G, η2) ∈ σ2.

(2) SS-β-cts if ψ−1
sw (G, η2) ∈ βOSs(χ1)η1 ∀ (G, η2) ∈ σ2.

(3) SS-α-cts ψ−1
sw (G, η2) ∈ αOSs(χ1)η1 ∀ (G, η2) ∈ σ2.

(4) SS-regular cts if ψ−1
sw (G, η2) ∈ ROSs(χ1)η1 ∀ (G, η2) ∈ σ2.

(5) SS-sd-cts if ψ−1
sw (G, η2) ∈ SDs(χ1)η1 ∀ (G, η2) ∈ σ2.

Theorem 11. [26] A soft subset (G, η) of an SSTS (χ, ρ, η) is SS-semi-open set if and
only if cls(G, η) = cls(ints(G, η)).

3. Supra soft sw-open sets and relationships

In this section, we present a new generalization of soft open sets in SSTS named SS-
sw-open sets. The relationships with other different types of SS-open sets are discussed.
With the confirmations of the counterexamples, we show that, this new class forms an
SSTS and fail to form an STS.

Definition 12. A soft subset (K, η) of an SSTS (χ, ρ, η) is said to be SS-sw-open set if it
is null or its SS-interior points is non-null.
The soft complement of an SS-sw-open set is called SS-sw-closed. The class of all SS-sw-
open (respectively, SS-sw-closed) sets will denoted by SWOs(χ)η (respectively, SWCs(χ)η).

Proposition 13. For an SSTS (χ, ρ, η) we have that:

(1) (K, η) ∈ S(χ)η is SS-sw-closed if it is the absolute soft set or it is not SS-dense set.



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(2) A non-null soft set (K, η) is SS-sw-open if and only if there is φ̃ ̸= (O, η) ∈ ρ such
that (O, η)⊑̃(K, η).

(3) A proper soft set (K, η) is SS-sw-closed if and only if there is χ̃ ̸= (C, η) ∈ ρc such
that (K, η)⊑̃(C, η).

Proof. Obvious from Definition 12.

Corollary 14. A non-null soft subset (H, η) of an SSTS (χ, ρ, η) is SS-sw-open if and
only if it is a neighborhood for each soft point in χ̃.

Proof. It is follows from Proposition 13.

Proposition 15. Every soft superset (subset) of an SS-sw-open (SS-sw-closed) set is SS-
sw-open (SS-sw-closed).

Proof. It is immediately from Definition 12.

Remark 16. The next example will confirm that, in general the above proposition is not
conversely.

Example 17. Assume that χ = {x1, x2, x3, x4}. Let η = {ϑ1, ϑ2} be the set of parameters.
Let (Ji, η), i = 1, 2, ..., 5, be soft sets over χ, where
J1(ϑ1) = {x1, x2}, J1(ϑ2) = {x1, x3, x4},
J2(ϑ1) = {x1}, J2(ϑ2) = φ,
J3(ϑ1) = {x1, x2}, J3(ϑ2) = {x3, x4},
J4(ϑ1) = {x3, x4}, J4(ϑ2) = {x1, x2},
J5(ϑ1) = χ, J5(ϑ2) = {x1, x2}.
Then, ρ = {χ̃, φ̃, (Ji, η), i = 1, 2, ..., 5} defines an SSTS on U . Hence, the soft set (J4, η),
is an SS-sw-open set, since
ints(J4, η)) ̸= φ̃. However, we have that (B, η)⊆̃(J4, η), where
B(ϑ1) = {x3, x4}, B(ϑ2) = φ,
is not SS-sw-open set, since ints(B, η)) = φ̃.
Also, for the soft sets (J3, η), (T, η), where
T (η1) = {x1, x2, x3}, T (η2) = {x1, x3, x4}.
We have that (J3, η)⊆̃(T, η), and (J3, η) is an SS-sw-closed set whereas (T, η) is not SS-
sw-closed.

Lemma 18. A soft subset (J, η) of an SSTS (χ, ρ, η) is SS-sw-open set if and only if
ints(J, η) is SS-sw-open set.

Proof. It is immediately from Definition 12.

Lemma 19. If (T, η)∩̃(S, η) = φ̃ for some (T, η) ∈ SWOs(χ)η and (S, η) ∈ S(χ)η, then
(S, η) ∈ SWCs(χ)η.

Proof. Assume that, (T, η)∩̃(S, η) = φ̃ such that (T, η) ∈ SWOs(χ)η and (S, η) ∈
S(χ)η. Then,



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(S, η)⊆̃(T c̃, η), (T c̃, η) ∈ SWCs(χ)η.

Given Proposition 15, (S, η) ∈ SWCs(χ)η.

Theorem 20. If {(Gı, η), ı ∈ I} is a family of SS-sw-open subsets of an SSTS (χ, ρ, η),
then

(1)
⋃̃

ı∈I(Gı, η) ∈ SWOs(χ)η.

(2)
⋂̃

ı∈I(G
c̃
ı , η) ∈ SWCs(χ)η.

Proof.

(1) Let {(Gı, η), ı ∈ I} is a family of SS-sw-open sets. Then, ints(Gı, η) ̸= φ̃ for each ı ∈ I.
Hence, φ̃ ̸=

⋃̃
ı∈Iint

s(Gı, η)⊆̃ints[
⋃̃

ı∈I(Gı, η)]. Therefore,
⋃̃

ı∈I(Gı, η) ∈ SWOs(χ)η.

(2) It is clear from (1).

Remark 21. If {(Wı, η), ı = 1, 2, ...., n} is a finite family of SS-sw-open subsets of an SSTS

(χ, ρ, η), then
⋂̃n

ı=1(Wı, η) ̸∈ SWOs(χ)η generally, as the example that follows illustrates.

Example 22. Suppose that ρ = {R̃, φ̃, (J1, η), (J2, η), (J3, η)} is an SSTS defined on the
set of real numbers R and the set of parameters η = {ϑ1, ϑ2} where
J1(ϑ1) = [5, 6], J1(ϑ2) = [7, 8],
J2(ϑ1) = [6, 7], J2(ϑ2) = [8, 9],
J3(ϑ1) = [5, 7], J3(ϑ2) = [7, 9].
Hence, the soft sets (J1, η) and (J2, η) are SS-sw-open sets, but their soft intersection
(J1, η)∩̃(J2, η) = {(η1, {6}), (η2, {8})} is not SS-sw-open set.

Corollary 23. If a non-null soft subset (G, η) of an SSTS (χ, ρ, η) is SS-semi-open set,
then ints(G, η) ̸= φ̃.

Proof. Suppose contrary that, ints(G, η) = φ̃ for an SS-semi-open set (G, η). Accord-
ing to Theorem 11, cls(G, η) = cls(ints(G, η)) = φ̃. If follows that, (G, η) = φ̃, which is a
contradiction.

Note 24. According to Corollary 23, if a soft subset (G, η) of an SSTS (χ, ρ, η) is SS-
semi-open set, then it is an SS-sw-open, but not conversely. In Example 17, the soft set
(N, η) where :
N(η1) = {x1, x3, x4}, N(η2) = {x1, x4} is an SS-sw-open set but not SS-semi-open.

Proposition 25. If soft subset (G, η) of an SSTS (χ, ρ, η) is SS-sw-open, then it is an
SS-sd-set.

Proof. Suppose contrary that, (G, η) is not SS-sd-set, then ints(cls(G, η)) = φ̃. Since
ints(G, η)⊆̃ints(cls(G, η)) = φ̃, ints(G, η) = φ̃, which is a contradiction. The converse of
this result is not generally accurate, refer to Example 22, the soft set {(η1, {6}), (η2, {8})}
is an SS-sd-set but not SS-sw-open.



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Remark 26. The classes of SS-β-open sets and SS-sw-open sets are independent, as shall
demonstrated in the upcoming examples.

Examples 27. (1) Let R be the set of real numbers, η = {ϑ1, ϑ2} and let

ρ = {R̃, φ̃, (A, η), (B, η), (C, η)}, where :

A(ϑ1) = [3, 5], A(ϑ2) = [5, 7].

B(ϑ1) = [4, 5], B(ϑ2) = [6, 7].

C(ϑ1) = [3, 4], C(ϑ2) = [5, 6].

Then, (T, η) = {(ϑ1, {4}), (ϑ2, {6})}

is an SS-β-subset of R̃ but not SS-sw-open.

(2) Let χ = {r1, r2, r3, r4}, η = {ϑ1, ϑ2} and let

ρ = {χ̃, φ̃, (I1, η), (I2, η), (I3, η), (I4, η), (I5, η), (I6, η), (I7, η)}, where :

I1(ϑ1) = {r1}, I1(ϑ2) = φ.

I2(ϑ1) = {r1, r2}, I2(ϑ2) = {r1}.

I3(ϑ1) = {r1, r2}, I3(ϑ2) = {r3, r4}.

I4(ϑ1) = {r3, r4}, I4(ϑ2) = {r1, r2}.

I5(ϑ1) = {r1, r3, r4}, I5(ϑ2) = {r1, r2}.

I6(ϑ1) = U, I6(ϑ2) = {r1, r2}.

I7(ϑ1) = {r1, r2}, I7(ϑ2) = {r1, r3, r4}.

Then, (Z, η) = {(ϑ1, {r2, r3, r4}), (ϑ2, χ)}

is an SS-sw-open set, but it is not SS-β-open.

Corollary 28. We can summarize the above relationships with the help of [Corollary 3.19,
[37]], in the subsequent ramifications for an SSTS (χ, ρ, η), which cannot be reversed .

ROSs(χ)η −→OSs(χ)η −→ αOSs(χ)η −→ SOSs(χ)η −→ βOSs(χ)η −→ SDs(χ)η
↘ ̸↕ ↗

SWOs(χ)η

Figure 1. The relationships among SS-sw-open sets and other generalizations.



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4. Soft continuity (openness) inspired by supra soft sw-open sets

In this section, we introduce new types of soft continuity related to SS-sw-open sets,
named SS-sw-cts functions. We characterize many of its essential properties. Also, we have
studied its relationships with previous similar types of generalizations. We used SS-sw-
closure (interior) operators to present several equivalent conditions of our new approach.
Furthermore, we define a new type of functions inspired by SS-sw-open sets, named SS-
sw-open functions.

Definition 29. A soft function ψsw : (χ1, σ1, η1) → (χ2, σ2, η2) with ρ1 as an associated
SSTS with σ1 is said to be an SS-sw-cts if ψ−1

sw (G, η2) ∈ SWOs(χ1)η1 ∀ (G, η2) ∈ σ2.

Note 30. According to Figure 1, we have the following diagram.

SS-regular-cts −→SS-cts −→ SS-α-cts −→ SS-semi-cts −→ SS-β-cts −→ SS-sd-cts
↘ ̸↕ ↗

SS-sw-cts

Figure 2. The relationships between some generalizations of SS-continuity

The next examples show that, the implications in Figure 2 are not reversible.

Examples 31. (1) Let χ1 = {r1, r2, r3, r4}, χ2 = {t1, t2, t3, t4}, η1 = {ϑ1, ϑ2} and η2 =
{θ1, θ2}.

Define s : χ1 → χ2 and w : η1 → η2 as follows :

s(r1) = t1, s(r2) = t4, s(r3) = t2, s(r4) = t3, w(ϑ1) = θ1, w(ϑ2) = θ2.

Let σ1 = {χ̃1, φ̃, (A, η1)} be an STS over χ1, where

A(ϑ1) = {r1}, A(ϑ2) = φ.

Let ρ1 = {χ̃1, φ̃, (Qi, η1), i = 1, 2, .., 5}

is an associated SSTS with σ1, where :

Q1(ϑ1) = χ1, Q1(ϑ2) = {r1, r2}.

Q2(ϑ1) = {r3, r4}, Q2(ϑ2) = {r1, r2}.

Q3(ϑ1) = {r1, r2}, Q3(ϑ2) = {r3, r4}.

Q4(ϑ1) = {r1}, Q4(ϑ2) = φ.

Q5(ϑ1) = {r1, r2}, Q5(ϑ2) = {r1, r3, r4}.

Let σ2 = {χ̃2, φ̃, (P,Θ2)} be an STS over χ2 , where :

P (θ1) = {t1, t2, t3}, P (θ2) = {t1, t3}.



Abd El-latif et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5863 9 of 18

Then,
ψ−1
sw (P,Θ2) = {(ϑ1, {r1, r3, r4}), (ϑ2, {r1, r4})}

is an SS-sw-subset of χ̃1, but not SS-semi-open. Therefore, ψsw is an SS-sw-cts, but
not SS-semi-cts.

(2) Let R be the set of real numbers, η1 = {ϑ1, ϑ2} and η2 = {θ1, θ2}.

Let s : R→ R and w : η1 → η2 be the identity functions.

Let σ1 = {R̃, φ̃, (A, η1)} be an STS over R, and let

ρ1 in Examples 27 (1) be an associated SSTS with σ1.

Let σ2 = {R̃, φ̃, (H, η2)} be an STS over R ,where :

H(θ1) = {4}, H(θ2) = {6}.

Then,
ψ−1
sw (H, η2) = {(ϑ1, {4}), (ϑ2, {6})}

is an SS-sd-subset of R̃ but not SS-sw-open. Therefore, ψsw is an SS-sd-cts but not
SS-sw-cts.

(3) In (2), we have ψ−1
sw (H, η2) = {(ϑ1, {4}), (ϑ2, {6})} is an SS-β-subset of R̃ but not

SS-sw-open. Therefore, ψsw is an SS-β-cts but not SS-sw-cts.

(4) Let χ1 = {r1, r2, r3, r4}, χ2 = {t1, t2, t3, t4}, η1 = {ϑ1, ϑ2} and η2 = {θ1, θ2}.

Define s : χ1 → χ2 and w : η1 → η2 as follows :

s(r1) = t4, s(r2) = t3, s(r3) = t1, s(r4) = t2, w(ϑ1) = θ1, w(ϑ2) = θ2.

Let σ1 = {χ̃1, φ̃, (I2, η1)} be an STS over χ1 and

ρ1 in Examples 27 (2) be an associated SSTS with σ1.

Let σ2 = {χ̃2, φ̃, (S, η2)} be a STS over χ2 where,

S(θ1) = {t1, t2, t3}, S(θ2) = χ2,

Then,
ψ−1
sd ((S, η2)) = {(ϑ1, {r2, r3, r4}), (ϑ2, χ1)}

is an SS-sw-open set, but it is not SS-β-open. Hence, ψsd is an SS-sw-cts, but it is
not SS-β-cts .

Definition 32. Let (G, η) be a soft subset of an SSTS (χ, ρ, η), then

(1) intssw(G, η) = ⊔{(O, η) : (O, η) ∈ SWOs(χ)η and (O, η)⊆̃(G, η)}.

(2) clssw(G, η) = ⊓{(H, η) : (H, η) ∈ SWCs(χ)η and (G, η)⊆̃(H, η)}.



Abd El-latif et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5863 10 of 18

(3) (G, η) is called SS-sw-co-dense if intsw(G, η) = φ̃.

(4) (G, η) is called SS-sw-dense if clsw(G, η) = χ̃.

Theorem 33. Let ψsw : (χ1, σ1, η1) → (χ2, σ2, η2) be a soft function with ρ1 as an
associated SSTS with σ1; then, the subsequent statements are equivalent:

(1) ψsw is an SS-sw-cts.

(2) For each (E, η2) ∈ σc2, ψ
−1
sw (E, η2) ∈ SWCs(χ1)η1.

(3) clssw(ψ
−1
sw (E, η2))⊆̃ψ−1

sw (cl(E, η2)) ∀ (E, η2)⊆̃χ̃2.

(4) ψsw(cl
s
sw(G, η1))⊆̃cl(ψsw(G, η1)) ∀ (G, η1)⊆̃χ̃1.

(5) ψ−1
sw (int(E, η2))⊆̃intssw(ψ−1

sw (E, η2)) ∀ (E, η2)⊆̃χ̃2.

Proof.

(1) ⇒ (2) Let (E, η2) ∈ σc2; then, (E
c̃, η2) ∈ σ2. Given (1), ψ−1

sw (E
c̃, η2) = [ψ−1

sw (E, η2)]
c̃ ∈

SWOs(χ1)η1 . Hence,
ψ−1
sw (E, η2) ∈ SWCs(χ1)η1 .

(2) ⇒ (3) Since cl(E, η2) ∈ σc2 for each (E, η2)⊆̃χ̃2,

ψ−1
sw (cl(E, η2)) ∈ SWCs(χ1)η1 , given (2), which implies

clssw(ψ
−1
sw (E, η2))⊆̃clssw(ψ−1

sw (cl(E, η2))) = ψ−1
sw (cl(E, η2)).

(3) ⇒ (4) Given that, ψsw(G, η1)⊆̃χ̃2 for each (G, η1)⊆̃χ̃1, and applying (3), we have

clssw(ψ
−1
sw (ψsw(G, η1)))⊆̃ψ−1

sw (cl(ψsw(G, η1))).

Hence,

ψsw[cl
s
sw(ψ

−1
sw (ψsw(G, η1)))]⊆̃ψsw[ψ

−1
sw (cl(ψsw(G, η1)))]⊆̃cl(ψsw(G, η1)), from Theorem 5 (2).

Therefore,

ψsw(cl
s
sw(G, η1))⊆̃cl(ψsw(G, η1)), from Theorem 5 (3).

(4) ⇒ (5) Since ψ−1
sw (E

c̃, η2)⊆̃χ̃1 for each (E c̃, η2)⊆̃χ̃2. Applying (4),

ψsw[cl
s
sw[ψ

−1
sw (E

c̃, η2)]]⊆̃cl(ψsw[ψ
−1
sw (E

c̃, η2)])⊆̃cl(E c̃, η2) = [int(E, η2)]
c̃.

It follows that,

ψ−1
sw [ψsw(cl

s
sw[ψ

−1
sw (E

c̃, η2)])]⊆̃ψ−1
sw [[int(E, η2)]

c̃] = [ψ−1
sw (int(E, η2))]

c̃.



Abd El-latif et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5863 11 of 18

Hence,

clssw[(ψ
−1
sw (E, η2))]

c̃⊆̃[ψ−1
sw (int(E, η2))]

c̃, from Theorem 5 (3).

Therefore,

ψ−1
sw (int(E, η2))⊆̃[clssw[(ψ

−1
sw (E, η2))]

c̃]c̃ = intssw(ψ
−1
sw (E, η2)).

(5) ⇒ (1) Since (E, η2) = int(E, η2) for each (E, η2) ∈ σ2. Then,

ψ−1
sw (E, η2)⊆̃intssw(ψ−1

sw (E, η2)), from (5),

and so ,
intssw(ψ

−1
sw (E, η2)) = ψ−1

sw (E, η2) ̸= φ̃.

It follow that,
ψ−1
sw (E, η2) ∈ SWOs(χ1)η1 .

Thus, ψsw is an SS-sw-cts.

Theorem 34. Let ψsw : (χ1, σ1, η1) → (χ2, σ2, η2) be a soft function with ρ1 as an
associated SSTS with σ1; then the subsequent statements are equivalent:

(1) ψsw is an SS-sw-cts.

(2) There exists φ̃ ̸= (O, η1) ∈ ρ1 such that (O, η1)⊑̃ψ−1
sw (E, η2), for each (E, η2) ∈ σ2

with ψ−1
sw (E, η2) ̸= φ̃.

(3) There exists χ̃1 ̸= (C, η1) ∈ ρc1 such that ψ−1
sw (E, η2)⊑̃(C, η1), for each (E, η2) ∈ σc2

with ψ−1
sw (E, η2) ̸= χ̃1.

(4) ψsw(G, η1) is SS-dense over ψsw(χ1), for each (G, η1) SS-dense over χ1.

Proof.

(1) ⇒ (2) Immediate from Proposition 13 (2).

(2) ⇒ (3) Let (E, η2) ∈ σc2 with ψ−1
sw (E, η2) ̸= χ̃1. It follows that, (E c̃, η2) ∈ σ2 with

ψ−1
sw (E

c̃, η2) ̸= φ̃. Given (2), there exists φ̃ ̸= (O, η1) ∈ ρ1 such that (O, η1)⊑̃ψ−1
sw (E

c̃, η2).
Hence, ψ−1

sw (E, η2)⊑̃(Oc̃, η1), χ̃1 ̸= (Oc̃, η1) ∈ ρc1.

(3) ⇒ (4) Assume conversely, ψsw(G, η1) is not SS-dense over ψsw(χ1), for some (G, η1)
SS-dense over χ1. Then, there exists χ̃2 ̸= (E, η2) ∈ σc2 such that ψsw(G, η1)⊑̃(E, η2)⊑̃ψsw(χ1),
and so (G, η1)⊑̃ψ−1

sw (E, η2). Given (3), there exists χ̃1 ̸= (C, η1) ∈ ρc1 such that
(G, η1)⊑̃ψ−1

sw (E, η2)⊑̃(C, η1) ̸= χ̃1, which contradicts our assumption.



Abd El-latif et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5863 12 of 18

(4) ⇒ (1) Let (E, η2) ∈ σ2 with ψ−1
sw (E, η2) ̸= φ̃. Assume contrary that ψ−1

sw (E, η2) is
not SS-sw-cts. Then, int(ψ−1

sw (E, η2)) = φ̃, which follows cl(ψ−1
sw (E

c̃, η2)) = χ̃1. This
means that ψ−1

sw (E
c̃, η2) is SS-dense over χ1. Given (4), ψsw[ψ

−1
sw (E

c̃, η2)] is an SS-
dense over ψsw(χ1), and so (E, η2) = φ̃, which contradicts our assumption. Thus,
ψsw is an SS-sw-cts.

Proposition 35. Let ψsw : (χ1, σ1, η1) → (χ2, σ2, η2) be an one to one soft function with
ρ1 as an associated SSTS with σ1; then the subsequent statements are equivalent:

(1) ψsw is an SS-sw-cts.

(2) ψsw(G, η1) is soft co-dense over χ2, for each SS-sw-co-dense subset (G, η1) of χ̃1.

Proof.

(1) ⇒ (2) Assume conversely that, ψsw(G, η1) is not soft co-dense set over χ2 for any
SS-sw-co-dense subset (G, η1) of χ̃1. It follows that, int[ψsw(G, η1)] ̸= φ̃. Given (1),
ψ−1
sw [int[ψsw(G, η1)]] ∈ SWOs(χ1)η1 . Since ψsw is one to one,

φ̃ ̸= intsw[ψ
−1
sw [int[ψsw(G, η1)]]]⊑̃intsw[ψ−1

sw [ψsw(G, η1)]] = intsw(G, η1).

Hence, (G, η1) is not SS-sw-co-dense set, which is a contradiction.

(2) ⇒ (1) Let φ̃ ̸= (G, η1) ∈ σ2. Assume conversely that, ψ−1
sw (G, η1) ̸∈ SWOs(χ1)η1 ,

then intsw[ψ
−1
sw (G, η1)] = φ̃. By condition and given ψsw is one to one, we get

φ̃ = int(ψsw[intsw[ψ
−1
sw (G, η1)]])⊑̃int(ψsw[ψ

−1
sw (G, η1)]) = int(G, η1), which is a con-

tradiction.

Definition 36. A soft function ψsw : (χ1, σ1, η1) → (χ2, σ2, η2) with ρ2 as an associated
SSTS with σ2 is said to be an SS-sw-open if ψsw(G, η1) ∈ SWOs(χ2)η2 ∀ (G, η1) ∈ σ1.

Proposition 37. A soft function ψsw : (χ1, σ1, η1) → (χ2, σ2, η2) with ρ2 as an associated
SSTS with σ2 is SS-sw-open if and only if for each φ̃ ̸= (G, η1) ∈ σ1, there exists φ̃ ̸=
(H, η2) ∈ SWOs(χ2)η2 such that (H, η2)⊑̃ψsw(G, η1).

Proof. Immediate from Definition 36.

Theorem 38. Let ψsw : (χ1, σ1, η1) → (χ2, σ2, η2) be a soft function with ρ2 as an
associated SSTS with σ2; then the subsequent statements are equivalent:

(1) ψsw is an SS-sw-open.

(2) ψsw(int(W, η1))⊆̃intssw(ψsw(W, η1)), for each (W, η1)⊆̃χ̃1.

(3) ψ−1
sw (cl

s
sw(Z, η2))⊑̃cl(ψ−1

sw (Z, η2)), for each (Z, η2)⊆̃χ̃2.

Proof.

(1) ⇒ (2) Since int(W, η1)⊆̃(W, η1), ψsw(int(W, η1))⊆̃ψsw((W, η1)). Given (1),



Abd El-latif et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5863 13 of 18

intssw[ψsw(int(W, η1))] = ψsw(int(W, η1))⊆̃intssw[ψsw((W, η1))].

(2) ⇒ (1) Assume that φ̃ ̸= (W, η1) ∈ σ1. Then,

ψsw(int(W, η1)) = ψsw(W, η1)⊆̃intssw[ψsw(W, η1)].

However,

intssw[ψsw(W, η1)]⊆̃ψsw(W, η1), and therefore

intssw[ψsw(W, η1)] = ψsw(W, η1).

Thus,

(W, η1) ∈ SWOs(χ1)Θ1 ; hence, ψsw is SS-sw-open.

(2) ⇒ (3) Since ψ−1
sw (Z

c̃, η2)⊆̃χ̃1 for each (Z, η2)⊆̃χ̃2. Applying (2),

ψsw(int(ψ
−1
sw (Z

c̃, η2)))⊆̃intssw(ψsw(ψ
−1
sw (Z

c̃, η2)))⊆̃intssw(Z c̃, η2) = [clssw(Z, η2)]
c̃,

from Theorem 5 (3).

So,

cl[(ψ−1
sw (Z, η2))]

c̃ =
int(ψ−1

sw (Z
c̃, η2))⊆̃ψ−1

sw [ψsw(int(ψ
−1
sw (Z

c̃, η2)))]⊆̃ψ−1
sw [[cl

s
sw(Z, η2)]

c̃].

Hence,

ψ−1
sw (cl

s
sw(Z, η2))⊑̃cl(ψ−1

sw (Z, η2)).

(3) ⇒ (2) By a similar technique.

Theorem 39. Let ψsw : (χ1, σ1, η1) → (χ2, σ2, η2) be an one to one soft function with
ρ2 as an associated SSTS with σ2; then the subsequent statements are equivalent:

(1) ψsw is an SS-sw-open.

(2) There exists χ̃2 ̸= (B, η2) ∈ ρc2 such that ψsw(A, η1)⊑̃(B, η2), for each (A, η1) ∈ σc1
with ψsw(A, η1) ̸= χ̃2.

Proof.

(1) ⇒ (2) Let (A, η1) ∈ σc1 with ψsw(A, η1) ̸= χ̃2. It follows that, (Ac̃, η1) ∈ σ1. Given
(1), there is φ̃ ̸= (B, η2) ∈ ρ2 such that (B, η2)⊑̃ψsw(A

c̃, η1). That is,
ψsw(A, η1)⊑̃(Bc̃, η2), ψsw(χ̃1) ̸= (Bc̃, η2) ∈ ρc2.



Abd El-latif et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5863 14 of 18

(2) ⇒ (1) Let φ̃ ̸= (G, η1) ∈ σ1. It follows, χ̃1 ̸= (Gc̃, η1) ∈ σc1. Applying the condi-
tion, there exists χ̃2 ̸= (H, η2) ∈ ρc2 such that ψsw(G

c̃, η1)⊑̃(H, η2). This implies,
(H c̃, η2)⊑̃ψsw(G, η1), φ̃ ̸= (H c̃, η2) ∈ ρ2. Hence, ψsw(G, η1) ∈ SWOs(χ2)η2 . There-
fore, ψsw is an SS-sw-open.

Theorem 40. Let ψsw : (χ1, σ1, η1) → (χ2, σ2, η2) be a soft function with ρ2 as an
associated SSTS with σ2; then the subsequent statements are equivalent:

(1) ψsw is an SS-sw-open.

(2) ψ−1
sw (K, η2) is soft dense over χ1, for each (K, η2) SS-sw-dense set over χ2.

Proof.

(1) ⇒ (2) Assume conversely that, ψ−1
sw (K, η2) is not soft dense set over χ1 for arbitrary

SS-sw-dense subset (K, η2) of χ̃2. It follows that, there is χ̃1 ̸= (V, η1) ∈ σc1 such
that ψ−1

sw (K, η2)⊑̃(V, η1). It follows that,

ψsw(V
c̃, η1)⊑̃(K c̃, η2) (1)

Since φ̃ ̸= (V c̃, η1) ∈ σ1, given(1) there exists φ̃ ̸= (H, η2) ∈ SWOs(χ2)η2 such that

(H, η2)⊑̃ψsw(V
c̃, η1), from Proposition 37. (2)

From Eqs (1) and (2), (K, η2)⊑̃(H c̃, η2), (H
c̃, η2) ∈ SWCs(χ2)η2 . That is, (K, η2)

is not SS-sw-dense set, which contradicts our assumption. Therefore, ψ−1
sw (K, η2) is

soft dense set over χ1.

(2) ⇒ (1) Let φ̃ ̸= (G, η1) ∈ σ1. If ψsw(G, η1) ̸∈ SWOs(χ2)η2 , then int[ψsw(G, η1)] = φ̃,
and hence cl[ψsw(G

c̃, η1)] = χ̃2. It follows that, ψsw(G
c̃, η1) is an SS-sw-dense set

over χ2. By assumption,
cl[ψ−1

sw [ψsw(G
c̃, η1)]] = χ̃1 (3)

However, we have (G, η1)⊑̃ψ−1
sw [ψsw(G, η1) from Theorem 5. Hence, ψ−1

sw [ψsw(G
c̃, η1)⊑̃(Gc̃, η1).

Since (G,c̃ η1) ∈ σc1,

cl[ψ−1
sw [ψsw(G

c̃, η1)]⊑̃cl[(Gc̃, η1)] = (Gc̃, η1) (4)

From Eqs (3) and (4), χ̃1⊑̃(Gc̃, η1), which follows (G, η1) = φ̃, which is a contradic-
tion. Hence, ψsw(G, η1) ∈ SWOs(χ2)η2 , and therefore ψsw is an SS-sw-open.

5. Conclusion

In this paper, we used the supra soft interior operator to define a new approach of gen-
eralized sets named, SS-sw-open sets. We studied the essential characterizations of this
new approach. We discuss its relationships with the other generalizations and provide the
necessary examples and counterexamples. Furthermore, we applied this new notion to soft



Abd El-latif et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5863 15 of 18

continuity. Especially, we presented the notions of SS-sw-cts and Ss-sw-open functions.
Moreover, we used the SS-sw-closure (interior) operators to present several equivalent con-
ditions for our new approaches.
We plan to extend the previously mentioned concepts by basing them on the soft ideal [32].
Furthermore, by employing the aforementioned methods, additional topological character-
istics like separation axioms, compactness and connectedness will be presented, and this
will be the focus of our upcoming work. Lastly, using the presented generalizations, the
enhancement of the accuracy measures for subsets in information systems will be taken
into consideration.

6. Conflict of Interest

The author declares no conflicts of interest.

7. Acknowledgments

The authors extend their appreciation to the Deanship of Scientific Research at North-
ern Border University, Arar, KSA for funding this research work through the project
number ”NBU-FFR-2025-2727-01”. Also, this study is supported via funding from Prince
Sattam bin Abdulaziz University project number (PSAU/2025/R/1446) and this research
is funded by Zarqa University Jordan.

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