EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS
Vol. 17, No. 2, 2024, 1352-1368
ISSN 1307-5543 – ejpam.com
Published by New York Business Global

Sets Related to Openness and Continuity
Decompositions in Primal Topological Spaces

Hanan Al-Saadi1,∗, Muna Al-Hodieb2

1 Department of Mathematics, Faculty of Sciences, Umm Al-Qura University,
Makkah 21955, Saudi Arabia
2 Department of Mathematics, College of Sciences, Qassim University, Buraidah,
Saudi Arabia

Abstract. This paper introduces and investigates several new classes of sets called P-α-open
sets, P-semiopen sets, P-preopen sets, and P-β-open sets within the framework of primal topo-
logical spaces. Their properties and relationships with other open set generalizations are studied
through examples. Additionally, the concepts of PR-sets and PRα -sets are defined and their char-
acteristics examined. Also, the notions of P-α-continuous, P-semicontinuous, P-precontinuous
and P-β-continuous mappings are initiated and their features and main characterizations de-
termined. A new class of sets called Ψ̃P -sets is also introduced in primal topological spaces
using the ΨP -operator. Their properties and relationships between Ψ̃P -sets, α-open, semi-open,
and pre-open are investigated. Theorems on arbitrary unions and finite intersections of Ψ̃P are
discussed.

2020 Mathematics Subject Classifications: 54A05, 54B99, 94A60

Key Words and Phrases: Primal topological spaces, P-open set, PR-sets, ΨP -sets, and Ψ̃P -
sets, P-continuous

1. Introduction and Preliminaries

Topology is one of the important scientific fields in mathematics and physics. It can
be used in many different areas of mathematics, including algebra, Riemann integration,
Perron integration, operations research, probability theory, game theory, smoothness
of functions, and measurement theory. Also, it is relevant to various fields in physics
such as condensed matter physics, quantum field theory, physical cosmology, mechanical
engineering, and materials science. Some applications of these mathematical concepts
influence the mechanical properties of solids, as well as the electrical and mechanical
characteristics that rely on the arrangement and network structures of molecules and
primary units in materials.

∗Corresponding author.
DOI: https://doi.org/10.29020/nybg.ejpam.v17i2.5171

Email addresses: hsssaadi@uqu.edu.sa (H. Al-Saadi), mmhdieb@qu.edu.sa (M. Al-Hodieb)

https://www.ejpam.com 1352 © 2024 EJPAM All rights reserved.



H. Al-Saadi, M. Al-Hodieb / Eur. J. Pure Appl. Math, 17 (2) (2024), 1352-1368 1353

Over the past years, open-set generalizations have been discussed by many researchers.
The initial concept of semi-open sets was introduced by Levine [1] in 1963. Nj̊astad [2]
introduced several classes of almost open sets in 1965; specifically, they looked into
the structure of α-open sets and provided several applications. Pre-open sets and pre-
continuous functions were presented and investigated by Mashhour et al. in 1982, [3]
presented and investigated the concepts of pre-open and pre-continuous functions. The
new notions of β-open sets, β-continuous mappings, and β-open mappings were first
presented by Abd El-Monsef et al. [4] in 1983.

Topology’s application in social science and science has led to the development of
many new ideas in addition to traditional structures. Kuratowski presented the concept
of the ideal derived from a filter. The concept of an ideal can be thought of as the dual
counterpart of a filter.

Likewise, among the new constructs in topology is the notion of a grill, which was
first defined by Choquet in 1947 [5] and Thron [6] introduced proximity structures within
the domain of grills. In 1977, Chattopadhyay and Thron [7] expanded the concepts of
closure spaces in conjunction with grills. Furthermore, Chattopadhyay and colleagues
[8] expanded the concept of grills to investigate merotopic spaces. Since that time, the
grill structure has found extensive application within the field of topology. Roy and
Mukherjee [9–11] conducted the initial attempt to identify the topological characteris-
tics associated with grills. The authors in [12, 13] defined operators on grill topological
space. Then, several variations on operators appeared from other researchers. Follow-
ing that, topologists have defined new concepts related to grill topological space, their
subsets, and continuity [14–18]. It is worth noting that the literature concerning grill
structures is relatively limited compared to filter, ideal, and other topics; additionally,
interdisciplinary applications of grill structures are scarce.

Njastad was the first to define the topology’s compatibility with an ideal I [19]. In
1990, Jankovic and Hamlett [20, 21] obtained other characteristics of ideal topological
spaces and Ψ-operator, for the ideal topological space (X, δ, I), local function of L ⊆ X
is defined as: L⋆(I) (or simply L⋆) = {x ∈ X : U ∩L /∈ I, U ∈ δ(x)}, where δ(x) = {U ∈
δ : x ∈ U}, whereas Ψ-operator is defined as Ψ(L) = X − (X − L)⋆. The Ψ-operator
was used in 2007 by Modak and Bandyopadhyay [22] to define the notion of generalized
open sets. In 2012, Al-Omari and Takashi [23] studied features of grill topological space
and a different operator denoted by Ψ-operators where Ψ(L) = X− Φ(X− L). Ψ̃G was
introduced and studied by Al-Omari and Takashi in [24], they also used the ΨG-operator
to define a new class of open sets.

Recently, the notion of primal topological space was introduced by Acharjee et al.
[25, 26] as the dual structure of the grill and the authors obtained many fundamental
properties of it. In 2023, Al-Omari, Acharjee, and Özkoç [26] defined and studied oper-
ator Ψ by using primal topological spaces as Ψ(L) = X− (X− L)♢. The authors in [27]
expand the class of primal lower pleasant functions to the setting of reflexive smooth
Banach spaces. Furthermore, generalized primal topological spaces are a new category
of generalized topology that Al-Saadi and Al-Malki recently introduced with the concept
of the primal [28]. In [29], soft spaces were also investigated, and these concepts were



H. Al-Saadi, M. Al-Hodieb / Eur. J. Pure Appl. Math, 17 (2) (2024), 1352-1368 1354

applied in primal topological spaces.

In this paper, new classes of weaker sets are defined as some of the weak continuity
on primal topological spaces, and some of their basic properties are investigated. Addi-
tionally, we investigated the relationship between them, and we provided examples of the
opposite of relationships that were not satisfying. We define some new classes of func-
tions and use these functions to introduce several interesting decomposition theorems of
continuity. Finally, we use the ΨP -operator to introduce and study Ψ̃P -sets as a new
class of sets in the structure of primal topological spaces, and we obtain their features.
A counter-example to the theorems based on arbitrary unions and finite intersections is
discussed. Moreover, we study the relationships between Ψ̃P -sets and their analogous
topological concepts.

Assume that (X, δ) be a topological space (TS, for short), and cl(L), int(L), respec-
tively, will be used to refer to the interior of L in (X, δ) and the closure of L in (X, δ).
The family of all open neighborhoods of a point x ∈ X is denoted by N (x).

Definition 1.1. ([5]) A family G of 2X is called a grill on X if G satisfies the following
conditions:

(a) ϕ /∈ G,
(b) If L ∈ G and L ⊆ E, then E ∈ G,
(c) If L ∪ E ∈ G, then L ∈ G or E ∈ G.

In [5], define an operator Φ : 2X → 2X for a grill G on a TS (X, δ), and for any
L ∈ 2X, Φ(L) = {x ∈ X : U ∩ L ∈ G, ∀U ∈ N (x)}.

Then, the author defined another operator, Ψ : 2X → 2X, as Ψ(L) = L ∪ Φ(L) for
L ⊆ X, is a Kuratowski closure operator, defining a distinct topology δG on X that is,
δ ⊆ δG .

Definition 1.2. Suppose that (X, δ) is a TS. Hence, a subset L of X can be defined as:

(a) α-open ([2]), if L ⊆ int(cl(int(L))),

(b) semi-open([1]), if L ⊆ cl(int(L)),

(c) pre-open ([3]), if L ⊆ int(cl(L)),

(d) β-open ([4]) or semi-pre-open ([30]), if L ⊆ cl(int(cl(L))),

(e) t-set ([31]), if int(L) = int(cl(L)),

(f) R-set ([31]), if L = L1 ∩ L2, where L1 is an open set and L2 is a t-set,

(g) tα-set ([32]), if int(L) = int(cl(int(L))),

(h) Rα-set ([32]), if L = L1 ∩ L2, where L1 is an open set and L2 is a tα-set.

For any collection of α-open (resp. semi-open, pre-open, and β-open) sets is denoted
by δα (resp. SO(X), PO(X), and βO(X)).



H. Al-Saadi, M. Al-Hodieb / Eur. J. Pure Appl. Math, 17 (2) (2024), 1352-1368 1355

Definition 1.3. ([25, 26]) For a collection P ⊆ 2X on X ̸= ϕ. We define a primal on
X as:

(a) X /∈ P,

(b) If L ∈ P and E ⊆ L, thus E ∈ P,

(c) If L ∩ E ∈ P, then L ∈ P or E ∈ P.

Definition 1.4. ([25, 26]) The TS (X, δ) with a primal P defined on X as (X, δ,P) is
called a primal topological space (PTS, for short).

Definition 1.5. ([25, 26]) Assume that the PTS is (X, δ,P). Defined an operator
(.)♢ : 2X → 2X as L♢(X, δ,P) = {x ∈ X : (∀U ∈ N (x))(Lc ∪ Uc ∈ P)} for each subset L
of X the primal by our needs, will be used L♢

P or L♢(X, δ,P) to refer to this operator.

Definition 1.6. ([25, 26]) Consider a map cl♢ : 2X → 2X in a PTS (X, δ,P), defined
as cl♢(L) = L ∪ L♢, where L is any subset of X.

Definition 1.7. ([25, 26]) In a PTS (X, δ,P), the collection δ♢ = {L ⊆ X : cl♢(Lc) =
Lc} is characterized as a topology on X that is generated by primal P and topology δ.
The primal topology on X is the term for it and we can write δ♢P instead of δ♢.

Clearly, δ ⊆ δ♢ for any primal P on a topological (X, δ). We will use δ♢-int(L) to
refer to the interior of L relative to δ♢.

Theorem 1.8. ([25, 26]) If (X, δ,P) is PTS. Consequently, the primal topology δ♢ is
finer than δ.

Theorem 1.9. ([25, 26]) Considering a PTS (X, δ,P), the following is true for any
two subsets L and E of X:
(a) If Lc ∈ δ, then L♢ ⊆ L,

(b) ϕ♢ = ϕ,

(c) cl(L♢) = L♢,

(d) (L♢)♢ ⊆ L♢,

(e) If L ⊆ E, then L♢ ⊆ E♢,

(f) L♢ ∪ E♢ = (L ∪ E)♢,

(g) (L ∩ E)♢ ⊆ L♢ ∩ E♢.

Lemma 1.10. ([26]) In a PTS (X, δ,P), if Lc /∈ P, then L♢ = ϕ.

Theorem 1.11. ([25, 26]) Assume that (X, δ,P) is a PTS. Then, the family BP =
{T ∩ P : T ∈ δ and P /∈ P} is a base for the primal topology δ♢ on X.

Definition 1.12. ([26]) Assume that (X, δ,P) is a PTS. An operator cl♢P : 2X → 2X is

defined as cl♢P(L) = {x ∈ X : (∃ U ∈ δ(x))((U − L)c /∈ P)} for every L ⊆ X.



H. Al-Saadi, M. Al-Hodieb / Eur. J. Pure Appl. Math, 17 (2) (2024), 1352-1368 1356

The theorem below demonstrates several characterizations of the operator cl♢P .

Theorem 1.13. ([26]) Assume that (X, δ,P) is a PTS. Consequently, these charac-
teristics are true:

(a) If L ⊆ X , then ΨP(L) = X− (X− L)♢,

(b) If L ⊆ X, then ΨP(L) is open,

(c) If L ⊆ E, then ΨP(L) ⊆ ΨP(E),

(d) If L,E ⊆ X, then ΨP(L ∩ E) = ΨP(L) ∩ΨP(E),

(e) If U ∈ δ♢, then U ⊆ ΨP(U),
(f) If L ⊆ X, then ΨP(L) ⊆ ΨP(ΨP(L)),

(g) If L ⊆ X, then L ∩ΨP(L) = int♢(L).

Corollary 1.14. ([26]) Assume that (X, δ,P) is a PTS. Then, U ⊆ ΨP(U) for each
open set U ∈ δ.

Theorem 1.15. ([26]) Consider (X, δ,P) as a PTS and L ⊆ X. Then, the following
properties hold:

(a) ΨP(L) = ∪{U ∈ δ : (U − L)c /∈ P},
(b) ΨP(L) ⊇ ∪{U ∈ δ : (U − L)c ∪ (L− U)c /∈ P}.

2. New Classes of Sets in Primal Topological Spaces

This section aims to describe, introduce, and examine several classes of open sets in
primal topological spaces, as well as their fundamental characteristics and relationships.

Definition 2.1. Suppose that a PTS (X, δ,P). So, we may define a subset L of X as
follows:

(a) P-open ([25, 26]), if L ⊆ int(L♢
P),

(b) P-α-open, if L ⊆ int(cl♢(int(L))),

(c) P-semi-open, if L ⊆ cl♢(int(L)),

(d) P-pre-open, if L ⊆ int(cl♢(L)),

(e) P-β-open, if L ⊆ cl(int(cl♢(L))).

Theorem 2.2. In a PTS (X, δ,P), the next characteristics are true:

(a) Each P-α-open set is α-open,

(b) Each P-semi-open set is semi-open,

(c) Each P-pre-open set is pre-open,

(d) Each P-β-open set is β-open.



H. Al-Saadi, M. Al-Hodieb / Eur. J. Pure Appl. Math, 17 (2) (2024), 1352-1368 1357

Proof. (a) Suppose that L be a P-α-open. Hence, L ⊂ int(cl♢(int(L))) = int(int(L)∪
(int(L))♢) ⊂ int(cl(int(L)) ∪ int(L)) ⊂ int(cl(int(L))). Thus, L is α-open.
(b) Suppose that L be a P-semi-open. Hence, L ⊆ cl♢(int(L)) = int(L) ∪ (int(L))♢ ⊆
int(L) ∪ cl(int(L)) (from Theorem 1.9) = cl(int(L)). Thus, L is semi-open.
(c) Suppose that L be a P-pre-open. Hence, L ⊆ int(cl♢(L)) = int(L ∪ L♢) ⊆
int(L ∪ cl(L)) = int(cl(L)). Therefore, L is a pre-open set.
(d) Suppose that L be a P-β-open set. Hence, L ⊆ cl(int(cl♢(L))) = cl(int(L ∪ L♢)) ⊂
cl(int(cl(L) ∪ L)) = cl(int(cl(L))). Therefore, L is a β-open set.

Remark 2.3. In general, the following examples demonstrate that the opposite of The-
orem 2.2 is not true.

Example 2.4. Assuming that X = {a1, a2, a3}, δ = {ϕ, {a1},X}, with the primal
P = {ϕ, {a1}, {a3}, {a1, a3}}. Thus,
(a) L = {a1, a3} is a α-open set that is not P-α-open, since L ⊆ int(cl(int(L))) = X,
but L ⊈ int(cl♢(int(L))) = {a1}.
(b) L = {a1, a3} is a semi-open set that is not P-semi-open, since L ⊆ cl(int(L)) = X,
but L ⊈ cl♢(int(L)) = {a1}.
(c) L = {a1, a3} is a pre-open set that is not P-pre-open, since L ⊆ int(cl(L)) = X, but
L ⊈ int(cl♢(L)) = {a1}.

Example 2.5. Assuming that X = {a1, a2, a3}, δ = {ϕ, {a1}, {a2, a3},X}, with the
primal P = {ϕ, {a1}, {a3}, {a1, a3}}. Thus, L = {a1, a3} is a β-open set, which is not
P-β-open since L ⊆ cl(int(cl(L))) = X, but L ⊈ cl(int(cl♢(L))) = {a1}.

Remark 2.6. Let (X, δ,P) be a PTS, then the concept of openness and P-openness are
independence.

Example 2.7. (a) Assuming that X = {a1, a2, a3}, δ = {ϕ, {a1}, {a3}, {a1, a3},X}, and
P = {ϕ, {a1}, {a2},
{a1, a2}}. Thus, (X, δ) is a TS. Moreover, P is a primal on X. Put U = {a1, a3} ∈ δ.
However, U♢

P = {a2, a3} in order that U is not P-open.
(b) Assuming that X = {a1, a2, a3}, δ = {ϕ,X} and P = {ϕ, {a1}, {a2}, {a1, a2}}. Thus,
(X, δ) is a TS. Moreover, P is a primal on X. Put L = {a2}. Thus, L♢

P = ϕ, so that L
is P-open. However, L is not open in (X, δ).

Theorem 2.8. If (X, δ,P) is a PTS, the next characteristics apply for L ⊆ X.
(a) L is P-α-open if and only if it is P-semi-open and P-pre-open,

(b) L is P-β-open, if L is P-semi-open,

(c) L is P-β-open, if L is P-pre-open,

(d) L is P-pre-open, if L is P-open.

Proof. (a) Necessity. It is clear.
Sufficiency. Let L be a P-semi-open and a P-pre-open. Hence, L ⊆ int(cl♢(L)) ⊆



H. Al-Saadi, M. Al-Hodieb / Eur. J. Pure Appl. Math, 17 (2) (2024), 1352-1368 1358

int(cl♢(cl♢(int(L)))) ⊆ int(cl♢(int(L))). Thus, L is P-α-open.
(b) Since L is P-semi-open and δ ⊆ δ♢, we already have L ⊆ cl♢(int(L)) ⊆ cl(int(L)) ⊆
cl(int(cl♢(L))). Thus, L is P-β-open.
(c) It is clear.
(d) Let L be a P-open. Thus, L ⊂ int(L♢

P) ⊂ int(L ∪ L♢
P) = int(cl♢(L)). Therefore, L

is P-pre-open.

Theorem 2.9. Each open set in a PTS is P-α-open.

Proof. If L is any open set, then L = int(L) ⊂ int((int(L))♢∪int(L)) = int(cl♢(int(L))).
Therefore, L is P-α-open.

Remark 2.10. The following figure represents several of the above-described sets, where
the opposite of the figure may not be as correct as the next.

Open P-α-Open

P-Open P-Pre-open P-Semi-open

P-β-Open

α-Open

β-Open

Pre-open Semi-open

Example 2.11. Consider X = {a1, a2, a3}, δ = {ϕ, {a1},X}, with the primal P =
{ϕ, {a1}, {a2},
{a3}, {a1, a2}, {a1, a3}, {a2, a3}}. Thus, L = {a1, a3} is a P-α-open that is not open,
since L ⊆ int(cl♢(int(L))) = X, but L /∈ δ.

Example 2.12. Consider X = {a1, a2, a3, a4}, δ = {ϕ, {a1}, {a1, a2}, {a1, a4}, {a1, a2, a4},X},
with the primal P = {ϕ, {a1}, {a2}, {a3}, {a1, a2}, {a2, a3}, {a1, a3}, {a1, a2, a3}}. Thus,
L = {a1, a3, a4} is a P-semi-open set that is not P-α-open, since L ⊆ cl♢(int(L)) = L,
but L ⊈ int(cl♢(int(L))) = {a1, a4}.



H. Al-Saadi, M. Al-Hodieb / Eur. J. Pure Appl. Math, 17 (2) (2024), 1352-1368 1359

Example 2.13. Consider X = {a1, a2, a3, a4}, δ = {ϕ, {a1}, {a1, a2}, {a1, a4}, {a1, a2, a4},X},
with the primal P = {ϕ, {a1}, {a2}, {a3}, {a1, a2}, {a2, a3}, {a1, a3}, {a1, a2, a3}}. Thus,
L = {a1, a3, a4} is a P-β-open set that is not P-pre-open, since L ⊆ cl(int(cl♢(L))) = X,
but L ⊈ int(cl♢(L)) = {a1, a4}.

Example 2.14. Consider X = {a1, a2, a3}, δ = {ϕ,X}, with the primal P = {ϕ, {a1}, {a2}, {a3}, {a1, a2},
{a2, a3}}. Thus,

(a) L = {a3} is a set that is P-β-open but not P-semi-open, since L ⊆ cl(int(cl♢(L))) =
X, but L ⊈ cl♢(int(L)) = ϕ.

(b) L = {a3} is a set that is P-pre-open but not P-α-open, since L ⊆ int(cl♢(L)) = X,
but L ⊈ int(cl♢(int(L))) = ϕ.

(c) L = {a1, a3} is a set that is P-pre-open but not P-open, since L ⊆ int(cl♢(L)) = X,
but L /∈ P.

Example 2.15. Consider X = {a1, a2, a3}, δ = {ϕ,X}, with the primal set given by
P = {ϕ, {a1}, {a2},
{a3}, {a1, a2}, {a1, a3}}. Thus, L = {a3} is a set that is P-open but not P-semi-open,
since L ⊆ int(L♢

P) = X, but L ⊈ cl♢(int(L)) = ϕ.

Definition 2.16. A subset L of a PTS (X, δ,P) is called P-dense in X if cl♢(L) = X.

Theorem 2.17. Assume that (X, δ,P) be PTS. Thus, for a subset L of X, the next is
true:

(a) If P = 2X \ {X}, L is P-β-open iff L is semi-open,

(b) Let P = {N ⊆ X \ N is the primal dense for all nowhere dense set} and L ⊆ X.
Then, L is P-β-open iff L is β-open.

Proof.
(a) When P = 2X \ {X}, thus L♢

P = ϕ for any L ⊂ X. Consequently, we have
cl(int(cl♢(L))) = cl(int(L♢ ∪ L)) = cl(int(L)). Thus, P-β-openness and semi-openness
are equivalent.
(b) By Theorem 2.2, every P-β-open set is β-open. If P = N , then it is well-known that
L♢ = cl(int(cl(L))). Therefore, if L is β-open, we obtain L ⊂ cl(int(cl(L))) = L♢ =
cl♢(L), and hence L ⊂ cl(int(cl(L))) = cl(int[cl(int(cl(L)))]) = cl(int(cl♢(L))).

3. PR-sets and PRα-sets

In this section, we focus on using the PTS of some defined sets, namely the PR-sets
and PRα-sets. Furthermore, their characterizations and main features are determined,
and their relationships with another set are investigated.

Definition 3.1. Assume that (X, δ,P) be PTS. Thus, L ⊆ X has the following defini-
tion:

(a) Primal t-set (briefly, Pt-set), if int(L) = int(cl♢(L)).



H. Al-Saadi, M. Al-Hodieb / Eur. J. Pure Appl. Math, 17 (2) (2024), 1352-1368 1360

(b) Primal tα-set (briefly, Ptα-set), if int(L) = int(cl♢(int(L))).

(c) Primal R-set (briefly, PR-set), if L = L1 ∩ L2, where L1 is open and L2 is Pt-set.

(d) Primal Rα-set (briefly, PRα-set), if L = L1∩L2, where L1 is open and L2 is Ptα-set.

Example 3.2. Let X = {a1, a2} and δ = {ϕ, {a1},X}. If P = {ϕ, {a1}}, then {a1} is Pt-
set, Ptα-set, PR-set, and PRα-set, since int({a1}) = int(cl♢({a1})) = int(cl♢(int({a1}))) =
{a1}.

Theorem 3.3. Assume that (X, δ,P) be PTS. Therefore,

(a) Each open set U is PR-set.

(b) Each Pt-set is PR-set.

Proof. (a) Put U = U ∩ X. Thus, int(cl♢(U)) = int(U).
(b) Let L be a Pt-set. If we assume that U = X ∈ δ, then L = U ∩ L, and hence L is
PR-set.

Remark 3.4. The opposite of Theorem 3.3 is untrue in all cases, as proved in the
following Examples.

Example 3.5. In Example 3.2, the set {a2} is PR-set. However, it is not an open set.

Example 3.6. Consider X = {a1, a2, a3}, δ = {ϕ, {a3},X} and P = {ϕ, {a1}, {a2}, {a1, a2}}.
Then, {a3} is PR-set but not Pt-set, since {a3} = int({a3}) ̸= int(cl♢({a3})) = X.

Proposition 3.7. Suppose that L and E are subsets of the space (X, δ,P). If L and E
are Pt-sets, then L ∩ E is a Pt-set.

Proof. Let L and E be Pt-sets. We have int(L∩E) ⊂ int(cl♢(L∩E)) ⊂ int(cl♢(L)∩
cl♢(E)) = int(cl♢(L))∩ int(cl♢(E)) = int(L)∩ int(E) = int(L∩E). Then int(L∩E) =
int(cl♢(L ∩ E)), and hence L ∩ E is a Pt-set.

The example below shows that the union of two Pt-sets need not be a Pt-sets.

Example 3.8. Consider X = {a1, a2, a3, a4}, δ = {ϕ, {a1, a3},X}, with the primal
P = {ϕ, {a1}, {a2}, {a3},
{a1, a2}, {a1, a3}, {a2, a3}, {a1, a2, a3}}. Then, L = {a1, a3} and E = {a1, a4} are Pt-
sets, since int({a1, a3}) = int(cl♢({a1, a3}) = {a1, a3} and int({a1, a4}) = int(cl♢({a1, a4}) =
ϕ, but L ∪ E = {a1, a3 a4} is not Pt-set.

Proposition 3.9. Assume that (X, δ,P) is a PTS. The following statements are
equivalent for a subset L of X:
(a) L is open,

(b) L is P-pre-open and PR-set.



H. Al-Saadi, M. Al-Hodieb / Eur. J. Pure Appl. Math, 17 (2) (2024), 1352-1368 1361

Proof. (a)⇒(b) : Consider L as open. Thus, L = int(L) ⊂ int(cl♢(L)), and L is
P-pre-open. Also by Theorem 3.3, L is PR-set.
(b) ⇒ (a): Consider L as a PR-set. So L = L1 ∩ L2, where L1 is open, and int(Q) =
int(cl(Q)). Thus, L ⊆ L1 = int(L1). Also, L is P-pre-open implies L ⊆ int(cl(L)) ⊂
int(cl♢(L2)) = int(L2) by assumption. Consequently, L ⊆ int(L1) ∩ int(L2) = int(L1 ∩
L2) = int(L), and so L is open.

Remark 3.10. Suppose that (X, δ,P) is a PTS. So, the concepts of P-pre-open sets
and PR-sets are independent.

Example 3.11. (a) Consider X = {a1, a2, a3, a4}, δ = {ϕ, {a1, a3},X}, with the primal
P = {ϕ, {a1}, {a2}, {a3},
{a1, a2}, {a1, a3}, {a2, a3}, {a1, a2, a3}}. Thus, L = {a1, a3, a4} is P-pre-open but not
PR-sets, since L ⊆ int(cl♢(L)) = X, but L = X∩L and {a1, a3} = int(L) ̸= int(cl♢(L)) =
X.
(b) In Example 2.13, {a1, a3, a4} is PR-sets but not P-pre-open.

Theorem 3.12. Suppose that (X, δ,P) is a PTS. So,

(a) Each open set is PRα-set,

(b) Each Ptα-set is PRα-set.

Proof. Obvious.

Example 3.13. (a) In Example 3.2, the set {a2} is PRα-set. However, it is not open.
(b) In Example 3.6, the set {a3} is PRα-set. However, it is not Ptα-set.

Proposition 3.14. If L1 and L2 are Ptα-sets, then L1 ∩ L2 is a Ptα-set.

Proof. Let L1 and L2 be Ptα-sets. Next, we have int(L1 ∩ L2) ⊂ int(cl♢(int(L1 ∩
L2))) ⊆ int[cl♢(int(L1))∩cl♢(int(L2))] = int(cl♢(int(L1)))∩int(cl♢(int(L2))) = int(L1)∩
int(L2) = int(L1 ∩L2). Then, int(L1 ∩L2) = int(cl♢(int(L1 ∩L2))). Therefore, L1 ∩L2

is a Ptα-set.

Proposition 3.15. Assume that (X, δ,P) is a PTS. The following statements are
equivalent for a subset L of X:
(a) L is open,

(b) L is P-α-open and PRα-set.

Proof. (a) ⇒ (b): Suppose that L is an open set. Thus, L = int(L) ⊆ cl♢(int(L))
and L = int(L) ⊆ int(cl♢(int(L))). Therefore, L is P-α-open. Also by Theorem 3.12, L
is PRα-set.
(b)⇒ (a): Let L be the PRα-set. So, L = L1 ∩ L2, where L1 is open and int(L2) =
int(cl♢(int(L2))). Thus, L ⊆ L1 = int(L1). Also, L is P-α-open implies L ⊆ int(cl♢(int(L))) ⊆
int(cl♢(int(L2))) = int(L2) by assumption. Thus, L ⊆ int(L1)∩int(L2) = int(L1∩L2) =
int(L), and L is open.



H. Al-Saadi, M. Al-Hodieb / Eur. J. Pure Appl. Math, 17 (2) (2024), 1352-1368 1362

Remark 3.16. The relationships between the above open sets are shown in the following
figure:

open set PR - set

Ptα -set

PRα -set

P-pre-open set

Pt -set

/

4. Decomposition of Generalized Continuity

In this section, we focus on defining some classes of primal continuous functions to
obtain decompositions of continuity.

Definition 4.1. A function F : (X, δ,P) → (Y, ς) is said to be P-α-continuous (resp.
P-semicontinuous, P-precontinuous, P-β-continuous) if the inverse image of each open
set in Y is P-α-open (resp. P-semi-open, P-pre-open, P-β-open) in (X, δ,P).

Theorem 4.2. Let F : (X, δ,P) → (Y, ς) be a function. Hence, F is a P-α-continuous
iff it is P-semicontinuous and P-precontinuous.

Proof. Clearly from Theorem 2.8.

Definition 4.3. Let F : (X, δ,P) → (Y, ς) be a function. Hence, F is said to be α-
continuous ([33]) (resp. semicontinuous ([1]), precontinuous ([3]), β-continuous ([4]))
if the inverse image of any open set of (Y, ς) is an α-open (resp. semi-open, pre-open,
β-open) in (X, δ,P).

Proposition 4.4. If a function F : (X, δ,P) → (Y, ς) is P-α-continuous (resp. P-
semicontinuous, P-precontinuous, P-β-continuous), thus F is α-continuous (resp. semi-
continuous, precontinuous, β-continuous).

Proof. Clearly from Theorem 2.2.

Example 4.5. Consider X = {a1, a2, a3, a4}, δ = {ϕ, {a1}, {a1, a2}, {a1, a4}, {a1, a2, a4},X},
with the primal P = {ϕ, {a1}, {a2}, {a3}, {a1, a2}, {a2, a3}, {a1, a3}, {a1, a2, a3}}. We
define a function F : (X, δ,P) → (X, δ) as follows: F(a1) = a1, F(a2) = a2, and
F(a3) = F(a4) = a3. Thus, F is not continuous since F−1({a1, a3}) = {a1, a4} is not
P-open. However, F is P-semicontinuous.

Definition 4.6. A function F : (X, δ,P) → (Y, ς) is said to be PR-continuous (resp.
PRα-continuous), if the inverse image of each open set in Y is PR-set (resp. PRα-set)
in (X, δ,P).



H. Al-Saadi, M. Al-Hodieb / Eur. J. Pure Appl. Math, 17 (2) (2024), 1352-1368 1363

Definition 4.7. A function F : (X, δ) → (Y, ς) is said to be R-continuous [31] (resp.
Rα-continuous [32]), if the inverse image of each open set in Y is R-set (resp. Rα-set)
in (X, δ).

Proposition 4.8. If a function F : (X, δ,P) → (Y, ς) is R-continuous (resp. Rα-
continuous), it is also PR-continuous (resp. PRα-continuous).

Proof. Straightforward.

Theorem 4.9. Let F : (X, δ,P) → (Y, ς) be a function. Hence, the following are
equivalent:

(a) F is continuous;

(b) F is a P-precontinuous and a PR-continuous;

(c) F is a P-α-continuous and a PRα-continuous.

Proof. It is a direct result of Propositions 3.9 and Propositions 3.15.

5. Ψ̃P-Sets

In this section, we describe a new class of sets in PTS that contain the class of all
open sets, using the ΨP -operator.

Definition 5.1. A subset L of a PTS (X, δ,P) is called Ψ̃P-set if L ⊆ cl(ΨP(L)).

The family of all Ψ̃P -sets in (X, δ,P) is denoted by Ψ̃P(X, δ).

Theorem 5.2. Suppose that (X, δ,P) is a PTS. If L ∈ δ, then L ∈ Ψ̃P(X, δ).

Proof. By Corollary 1.14, δ ⊂ Ψ̃P(X, δ) is obtained in the topological space (X, δ,P).

In general, the following example demonstrates that the opposite of Theorem 5.2 is
not true.

Example 5.3. Let X = {a1, a2, a3}, and δ = {ϕ, {a1}, {a2}, {a1, a2},X}, with the
primal P = {ϕ, {a1}, {a2}, {a1, a2}}. Now, ΨP({a3}) = X − {a1, a2}♢ = X − ϕ = X.
Thus, cl(ΨP({a3})) = X. Therefore, {a3} ⊆ cl(cl♢P({a3})), but {a3} is not open in δ.

Now, we show that any union of Ψ̃P -sets is a Ψ̃P(X, δ).

Proposition 5.4. Suppose that {Lα : α ∈ ∆} is a set of non-empty Ψ̃P-sets in a PTS
(X, δ,P), then ∪α∈∆Lα ∈ Ψ̃P(X, δ).

Proof. For every α ∈ ∆, Lα ⊆ cl(ΨP(Lα)) ⊆ cl(ΨP(∪α∈∆Lα)). This implies that
∪α∈∆Lα ⊆ cl(ΨP(∪α∈∆Lα)). Hence, ∪α∈∆Lα ∈ Ψ̃P(X, δ).

The example below demonstrates an intersection of two Ψ̃P -sets not necessarily a
Ψ̃P -set.



H. Al-Saadi, M. Al-Hodieb / Eur. J. Pure Appl. Math, 17 (2) (2024), 1352-1368 1364

Example 5.5. Assuming that X = {a1, a2, a3}, δ = {ϕ, {a1, a3},X}, with the primal
P = {ϕ, {a2}, {a3},
{a2, a3}}. Then L = {a1, a2} and E = {a1, a3} are Ψ̃P-sets, since ΨP({a1, a2}) =
X − {a3}♢ = X, {a1, a2} ⊆ cl(ΨP({a1, a2})) = X and ΨP({a1, a3}) = X − {a2}♢ =
X, {a1, a3} ⊆ cl(ΨP({a1, a3})) = X. Therefore, L ∩ E = {a1} is not Ψ̃P-set, since
ΨP({a1}) = X− {a2, a3}♢ = ϕ, {a1} ⊈ cl(ΨP({a1})) = ϕ.

We will demonstrate that the intersection of two Ψ̃P -sets are not often be a Ψ̃P -set,
we will show that the intersection of a δα with a Ψ̃P -set is a Ψ̃P -set.

Theorem 5.6. Suppose that we have a PTS (X, δ,P), and let L belong to the set
Ψ̃P(X, δ). Thus, if U is an element of δα, then it follows that the intersection of U with
L also belongs to the set Ψ̃P(X, δ).

Proof. We note that if G is open, for any L ⊆ X, G ∩ cl(L) ⊆ cl(G ∩ L). Let

U ∈ δα and L ∈ ˜
cl♢P(X, δ). Then by Theorem 1.15 and Corollary 1.14 we have U ∩ L ⊆

int(cl(int(U)))∩ cl(ΨP(L)) ⊆ int(cl(ΨP(U)))∩ cl(ΨP(L)) ⊆ cl[int(cl(ΨP(U)))∩ΨP(L)]
= cl[int(cl[ΨP(U) ∩ ΨP(L)])] = cl[ΨP(U) ∩ ΨP(L)] = cl[ΨP(U ∩ L)]. Hence, U ∩ L ∈
Ψ̃P(X, δ).

Corollary 5.7. Let (X, δ,P) be a PTS, and let L belong to Ψ̃P(X, δ). If U is an element
of δ, then their intersection U ∩ L also belongs to Ψ̃P(X, δ).

For any non-empty relative to an open set U ∩L, it holds that (U ∩L)∩D ∈ P when
U ∈ δ, then we refer to a set D as being relative P-dense in the set L.

Theorem 5.8. Let (X, δ,P) be a PTS. A set L /∈ Ψ̃P(X, δ) if and only if there exists
an element x in L such that there is a neighborhood Vx ∈ δ of x for which X − L is
relative to P-dense in Vx.

Proof. Suppose that a set L /∈ Ψ̃P(X, δ). We need to show the existence of x ∈ L
and a neighborhood Vx ∈ δ(x), then (X − L) is relative P-dense in Vx. Now, since
L ⊈ cl(ΨP(L)), and so there exists an element x ∈ X such that x is in L but not
in the closure of ΨP(L). Consequently, there is a neighborhood Vx ∈ δ(x) so that
Vx ∩ ΨP(L) = ϕ. Hence, Vx ∩ (X − (X − L)♢) = ϕ, and therefore, Vx ⊆ (X − L)♢.
Now, consider any non-empty open set U in Vx. Since Vx ⊆ (X − L)♢, it follows that
U ∩ (X− L) ∈ P. This demonstrates that (X− L) is relatively P-dense in Vx.

Definition 5.9. Let (X, δ,P) is a PTS. P is said to be primal anti-codense if δ−{ϕ} ⊆
P.

Theorem 5.10. In the PTS (X, δ,P), if P is characterized as a primal anti-codense,
then SO(X, δ♢) = Ψ̃P(X, δ).

Proof. First assume that L is an element of SO(X, δ♢). According to Theorem 1.15,
we have L ⊆ δ♢-cl(δ♢-int(L)) = δ♢-cl(ΨP(L)∩L). This implies that L ⊆ cl(ΨP(L)∩L) ⊆



H. Al-Saadi, M. Al-Hodieb / Eur. J. Pure Appl. Math, 17 (2) (2024), 1352-1368 1365

cl(ΨP(L)). Consequently, we can conclude that L belongs to Ψ̃P(X, δ). Therefore,
SO(X, δ♢) ⊆ Ψ̃P(X, δ).
Conversely, assume that L is an element of Ψ̃P(X, δ), and let x be an element of L. Con-
sider a basic neighborhood U1 of x in (X, δ♢). The neighborhood U1 can be represented
as U − G, where U ∈ δ and G /∈ P. This implies that x is in U , and consequently,
L ⊆ cl(ΨP(L)). Also, U ∈ δ(x), which means U ∩ΨP(L) ̸= ϕ. Now, let y ∈ U ∩ΨP(L).
Then, there is exists a neighborhood WY of y such that WY − L /∈ P (by definition of
ΨP(L)). Now, assume that U ∩WY = V. Consider G1 = V−L /∈ P. Since V ≠ ϕ, V ∈ δ,
and V − G1 ⊆ L, it follows that V ⊆ U . Consequently, M = V − (G1 ∪ G) ⊆ L and
M = V − (G1∪G) ̸= ϕ, since P is a primal anti-codense, then M ⊆ L∩ (U −G). Hence,
we have shown that L includes a nonempty δ♢-open set M included in U −G. Choose
x ∈ L, we have that L ⊆ δ♢-cl(δ♢-int(L)). Therefore, L is an element of SO(X, δ♢).
Thus, we have shown that Ψ̃P(X, δ) ⊆ SO(X, δ♢). Thus, SO(X, δ♢) = Ψ̃P(X, δ).

Definition 5.11. A subset L of a PTS (X, δ,P) is called a ΨL-set if L ⊆ int(cl(ΨP(L))).

The set of all ΨL-sets in (X, δ,P) is represented as δL. By Definitions 5.1 and
Definitions 5.9, it can be deduced that δL is a subset of δL ⊆ Ψ̃P(X, δ). We demonstrate
that the collection δL forms a topology.

Theorem 5.12. Suppose that (X, δ,P) is a PTS. If P is primal anti-codense, then the
collection δL = {L ⊆ X : L ⊆ int(cl(ΨP(L)))} forms a topology on X.

Proof. We have show that both ϕ and X satisfy the conditions ϕ ⊆ int(cl(ΨP(ϕ)))
and X ⊆ int(cl(ΨP(X))), which means that ϕ and X belong to the collection δL.
Suppose that a family of sets {Lα : α ∈ ∆} ⊆ δL. For any α ∈ ∆, it holds that
ΨP(Lα) ⊆ ΨP(∪Lα). Consequently, Lα ⊆ int(cl(ΨP(Lα))) ⊆ int(cl(ΨP(∪Lα))) for
any α ∈ ∆. This implies that ∪Lα ⊆ int(cl(ΨP(∪Lα))). Thus, ∪Lα is an element of δL.
Let L and E be two sets in δL. As ΨP(L) is open in (X, δ), we can apply Theorem
1.15, which leads to the conclusion that L ∩ E ⊆ int(cl(ΨP(L))) ∩ int(cl(ΨP(E))) =
int(cl(ΨP(L) ∩ΨP(E))) = int(cl(ΨP(L ∩E))). Therefore, L ∩E ⊆ int(cl(ΨP(L ∩E)))
and L ∩ E ∈ δL. This is the end of the proof.

Proposition 5.13. Suppose that (X, δ,P) is a PTS. Then, ΨP(L) ̸= ϕ if and only if
L has a non-empty δ♢-interior.

Proof. First, suppose that ΨP(L) ̸= ϕ. However, according to Theorem 1.15, ΨP(L)
we can write ΨP(L) = ∪{U ∈ δ : (U − L)♢ /∈ P}. This implies that there exists a
non-empty set U ∈ δ for which (U − L)c /∈ P. Let (U − L)c = T , where T /∈ P. Now,
U −T ⊆ L, and since U −T is a δ♢-open set according to Theorem 1.11, we can conclude
that L includes a non-empty δ♢-interior.
Conversely, assume that L includes a non-empty δ♢-interior. This implies that U ∈ δ
and T /∈ P such that U − T ⊆ L. Consequently, U − L ⊆ T . Let H = U − L ⊆ T , and
thus H /∈ P. Hence, ∪{U ∈ δ : (U − L)c /∈ P} = ΨP(L) ̸= ϕ.



REFERENCES 1366

Corollary 5.14. Suppose that (X, δ,P) is a PTS. Then, {ax} ∈ Ψ̃P(X, δ) if and only
if {ax} ∈ δL.

Proof. First, suppose that {ax} ∈ Ψ̃P(X, δ). This means that {ax} is open in (X, δ♢)
through Proposition 5.13. Since {ax} ⊆ ΨP({ax}) and ΨP({ax}) is open in (X, δ), we
can conclude that {ax} ⊆ int(cl(ΨP({ax}))). Thus, we have shown that {ax} ∈ δL.
Conversely, suppose that {ax} ⊆ int(cl(ΨP({ax}))) and {ax} ⊆ (cl(ΨP({ax}))). There-
fore, it follows that {ax} ∈ Ψ̃P(X, δ).

6. Conclusions

New classes of sets, namely P-α-open, P-semi-open, P-pre-open, P-β-open, PR-sets,
and PRα-sets in PTSs, have been studied along with results about the relationships
between class sets in TSs and a new class of sets in PTSs. Moreover, the relationships
between these classes of subsets and some generalizations of open have been introduced.
Many theorems are discussed together with the counterexamples. In addition, examples
are provided in such a way as to illustrate the independence between openness and P-
openness, as well as the independence between P-pre-open and PR-sets. Furthermore,
the decomposition of continuity by using a new class of sets in PTSs has been obtained.
Finally, Ψ̃P -sets were introduced and investigated by defining intriguing generalized
open sets in PTS using the Ψ-operator, besides investigating some of their important
properties.
In future work, the same concepts presented in this article can be introduced in the
context of primal topological spaces [29, 34, 35] by the techniques in soft sets or rough
sets. Additionally, we hope to relate these classes of sets to some concepts in different
topological structures.

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