Symmetric Extensions of Separation Axioms via GP-Operators and Their Applications

Abstract

This study is motivated by the need to investigate the largest possible collection of classical separation axioms within a newly introduced triple structure of generalized primal topological spaces, and to understand how primal collections influence these familiar notions. The purpose of the paper is to extend several classical concepts by introducing new classes of separation axioms, including $(\mathfrak{g},\mathcal{P})$-$D_i$, $(\mathfrak{g},\mathcal{P})$-$T_i$, and $(\mathfrak{g},\mathcal{P})$-$R_i$ for $i=0,1,2$. Within the same framework, we also define $(\mathfrak{g},\mathcal{P})$-$G_{\delta }$ sets and $(\mathfrak{g},\mathcal{P})$-$F_{\sigma }$ sets, which naturally lead to new symmetric variants of separation axioms such as $(\mathfrak{g},\mathcal{P})$-$R_{\delta }$, $(\mathfrak{g},\mathcal{P})$-weakly regular, $(\mathfrak{g},\mathcal{P})$-$R_{D_{\delta }}$, and $(\mathfrak{g},\mathcal{P})$-$R_{D^{★}}$. The main contribution of this work lies in establishing the relationships among these newly introduced axioms and demonstrating how primal collections affect their behavior. Several illustrative examples based on simple graphs are provided to highlight the structure and significance of the results. Overall, the findings offer a broader perspective on separation phenomena in generalized primal settings and deepen the understanding of symmetry within these spaces.

1. Introduction

Separation axioms in the theory of topology are mostly formulated to recognize non-homeomorphic topologies. If there are two topological spaces such that one satisfies a separation axiom but not the other, these two topologies are not homeomorphic. These axioms have received great interest from topologists for a long time. A topological space that satisfies $T_i$ is called a $T_i$-space for short.

Recently, in [1], the author offered a special kind of structure called generalized primal topological spaces. This structure possesses all the properties of a generalized topology in the sense of a new topological structure called “primal collection.” Then, Al-Saadi and Al-Malki [2] studied some types of weak open sets based on an operator employing primal, which satisfied Kuraski’s closure axioms. The concept of “continuity” has gained a lot of attention for its importance in topology. Several types of generalized primal continuous functions are described in [3].

In this study, we continue our work to clarify the notion of “separation axioms” in this new structure. This paper contains five sections, presented as follows.

Section 1 contains two sub-sections. The first deals with previous studies on which this scientific paper was based. The second includes a recall of the basic concepts and results. Section 2 presents several types of separation axioms, which are based on the concept of $D_{(\mathfrak{g},\mathcal{P})}$-set. Next, we study the relationship between them and the $T_i$ axioms for $i=0,1,2.$ We also introduce $R_i$ for the $i=0,1$ axioms, which are based on the concept of closure. In Section 3, we present the definition of $(\mathfrak{g},\mathcal{P})$-$G_{\delta }$-set and $(\mathfrak{g},\mathcal{P})$-$F_{\sigma }$-set, which leads us to new and multiple types of axioms. In Section 4, we present the fundamental theorem and propositions that are connected between the $\mathtt{GP}$-continuous function and the $\mathtt{GP}$-homeomorphism function and the separation axioms. Finally, we discuss the conclusions of this research.

1.1. Literature Review

Over the years, various classical mathematical structures have appeared. The mathematical construction “grill” [4] was introduced as a useful tool for investigating topological concepts with significant applications in the field of topology. Building on this, Acharjee [5] introduced the dual structure, called a “primal”, in 2022. A collection $\mathcal{P}$ of $P\left(X\right)$ is named a primal over X when the following requirements hold for all $A,B\subseteq X$: (i) X never belongs to $\mathcal{P}$; (ii) if $A\in \mathcal{P}$ and $B\subseteq A$, then $B\in \mathcal{P}$; (iii) if $A\cap B\in \mathcal{P}$, then either A or B belongs to $\mathcal{P}$. A primal topological space ($\mathbf{PT}$ space) is defined as the triple $(X,\mathcal{T},\mathcal{P})$.

In 1997, Császár [6] established the notion of generalized topological spaces, ($\mathbf{GT}$ spaces), where a family $\mathfrak{g}$ of subsets of X satisfies the following: (i) ∅ (the empty set) always belongs to $\mathfrak{g}$; (ii) any union of members of $\mathfrak{g}$ is also a member of $\mathfrak{g}$. $(X,\mathfrak{g})$ is considered a generalized topological space. According to [7], each element in X is called a $\mathfrak{g}$-open set, and $\mathfrak{g}$-closed refers to its complement. Respectively, $c_{\mathfrak{g}}\left(A\right)$ and $i_{\mathfrak{g}}\left(A\right)$ denote the closure and interior of A.

Various studies have investigated the topological properties of generalized spaces. In particular, separation axioms have been explored extensively. For example, ref. [8] introduced early forms of generalized separation axioms, ref. [9] examined their extensions in different contexts, ref. [10] focused on applications of these axioms in specific topological structures, and ref. [11] analyzed neighborhood structures under these axioms. Separation axioms have also been extended by substituting the concept of open sets with more generalized expressions [12,13]. Several forms of generalized continuity have been studied: contra generalized continuity [14], almost generalized continuity [15], and strong generalized continuity [16].

Although both $\mathbf{PT}$ and $\mathbf{GT}$ have been studied extensively, their integration has not been systematically explored. Generalized primal topological spaces ($\mathbf{GPT}$ spaces), introduced by [1], combine the structural constraints of $\mathbf{PT}$ with the flexibility of $\mathbf{GT}$, resulting in a unified framework where classical concepts such as separation axioms and continuity can be independently developed and analyzed. This integration allows us to investigate how primal collections influence generalized notions, filling a gap in the literature. In particular, three previous studies [1,2,3] addressed foundational aspects of $\mathbf{GPT}$ spaces: ref. [2] studied a collection of new generalized structures, and ref. [3] focused on continuous functions and their generalizations in $\mathbf{GPT}$ spaces.

While these studies have significantly advanced the understanding of classical and generalized structures, they do not fully integrate the effects of primal collections. Therefore, $\mathbf{GPT}$ spaces offer a novel framework where separation axioms can be independently developed and analyzed. This work aims to fill this gap and highlight the influence of primal collections on classical topological notions.

1.2. Preliminaries

First, let us recall some basics about $\mathbf{GPT}$ spaces.

Definition 1

([1]). A generalized primal topological space, or ($\mathbf{GPT}$ space), is a generalized topology $\mathfrak{g}$ and a primal $\mathcal{P}$ defined on $X.$ The indication $(X,\mathfrak{g},\mathcal{P})$ symbolizes the $\mathbf{GPT}$ space.

Definition 2

([1]). Let $(X,\mathfrak{g},\mathcal{P})$ be a $\mathbf{GPT}$ space with $A\subseteq X,$ and we have the following:

 (i)

This structure’s members are called $(\mathfrak{g},\mathcal{P})$-open sets. The complement of $(\mathfrak{g},\mathcal{P})$-open sets is called $(\mathfrak{g},\mathcal{P})$-closed sets.

 (ii)

$C_{(\mathfrak{g},\mathcal{P})}\left(X\right)$ symbolize the family of $(\mathfrak{g},\mathcal{P})$-closed sets.

 (iii)

$c{l}_{(\mathfrak{g},\mathcal{P})}\left(A\right)$ symbolize the closure of $A.$ Hence, $c{l}_{(\mathfrak{g},\mathcal{P})}\left(A\right)$ is given as the intersection of all $(\mathfrak{g},\mathcal{P})$-closed sets which contain A.

 (iv)

A $\mathbf{GPT}$ space is named strong if $X=G_{(\mathfrak{g},\mathcal{P})},$ where $G_{(\mathfrak{g},\mathcal{P})}=\cup \{U:U\in \mathfrak{g}\}.$

 (v)

An operator $\psi :X\to 2^{2^X}$ is called a generalized primal neighborhood of x, if $x\in U$ for each $U\in \psi \left(x\right).$

 (vi)

An operator ${(·)}^{\diamond }:2^X\to 2^X$ is defined by

$$A^{\diamond }(X,\mathfrak{g},\mathcal{P})=\{x\in X:A^c\cup U^c\in \mathcal{P},\forall U\in \psi \left(x\right)\}.$$

Definition 3

([3]). Let X and $X^{‵}$ be $\mathbf{GPT}$ spaces. A function $f:X\to X^{‵}$ is known as $\mathtt{GP}$-continuous if for each $({\mathfrak{g}}^{‵},{\mathcal{P}}^{‵})$-open set $U,$$f^{-1}\left(U\right)$ is $(\mathfrak{g},\mathcal{P})$-open.

Also, when f is $\mathtt{GP}$-continuous, we get

$$f(X\setminus G_{(\mathfrak{g},\mathcal{P})})\subseteq (X^{‵}\setminus G_{({\mathfrak{g}}^{‵},{\mathcal{P}}^{‵})}).$$

Moreover, $\mathtt{GP}$-continuity of f leads to $f\left(G_{(\mathfrak{g},\mathcal{P})}\right)\subseteq G_{({\mathfrak{g}}^{‵},{\mathcal{P}}^{‵})},$ then

$$f^{-1}(X^{‵}\setminus G_{({\mathfrak{g}}^{‵},{\mathcal{P}}^{‵})})=(X\setminus G_{(\mathfrak{g},\mathcal{P})})\mathit{and}{f}^{-1}\left(G_{({\mathfrak{g}}^{‵},{\mathcal{P}}^{‵})}\right)=G_{(\mathfrak{g},\mathcal{P})}.$$

Definition 4

([3]). A function $f:X\to X^{‵}$ is known as $\mathtt{GP}$-closed if for each $(\mathfrak{g},\mathcal{P})$-closed set U in $X,$ the image $f\left(U\right)$ is $({\mathfrak{g}}^{‵},{\mathcal{P}}^{‵})$-closed in $X^{‵}.$

It follows from the above definition that f is $\mathtt{GP}$-closed iff

$$c{l}_{({\mathfrak{g}}^{‵},{\mathcal{P}}^{‵})}\left(f\left(A\right)\right)\subseteq f\left(c{l}_{(\mathfrak{g},\mathcal{P})}\left(A\right)\right),$$

for all $A\subseteq X.$

2. Separation Axioms

Sarsak [17] studied the separation axioms: $\mathfrak{g}$-$D_0$, $\mathfrak{g}$-$D_1$, $\mathfrak{g}$-$D_2$, $\mathfrak{g}$-$T_0$, $\mathfrak{g}$-$T_1$, $\mathfrak{g}$-$T_2$, $\mathfrak{g}$-$R_0$, and $\mathfrak{g}$-$R_1$ in generalized topological spaces. In this section, we study the alternative definitions of these kinds of separation axioms in a generalized primal topological space. Also, we investigate the main features of them.

Definition 5.

Let $(X,\mathfrak{g},\mathcal{P})$ be a $\mathbf{GPT}$ space with $A\subseteq X.$ A is known as a $D_{(\mathfrak{g},\mathcal{P})}$-set when we have $(\mathfrak{g},\mathcal{P})$-open sets U and V satisfies $U\ne X$ with $A=U\setminus V$.

Note that every $(\mathfrak{g},\mathcal{P})$-open set $A\ne X$ is $D_{(\mathfrak{g},\mathcal{P})}$-open. Moreover, a $\mathbf{GPT}$ space X is known as $(\mathfrak{g},\mathcal{P})$-$D_0$ if for each $x,y\in X$ with $x\ne y,$ there exists a $D_{(\mathfrak{g},\mathcal{P})}$-set U of X that satisfies $x\in U,$$y\notin U$ or vice versa. However, since a $D_{(\mathfrak{g},\mathcal{P})}$-set is always contained in $G_{(\mathfrak{g},\mathcal{P})},$ a $\mathbf{GPT}$ space in which $X\setminus G_{(\mathfrak{g},\mathcal{P})}$ contains more than one point trivially does not have the property $(\mathfrak{g},\mathcal{P})$-$D_0$.

Example 1.

Consider

$$X=\mathbb{N},\mathfrak{g}=\{\varnothing ,\{1\},\{2\},\{1,2\left\}\right\},\mathcal{P}=P\left(X\right)\setminus \left\{X\right\}.$$

We have that $(X,\mathfrak{g},\mathcal{P})$ is a $\mathbf{GPT}$ space. Then, the family of $(\mathfrak{g},\mathcal{P})$-open sets coincides with $\mathfrak{g}$, and hence

$$G_{(\mathfrak{g},\mathcal{P})}=\{1,2\}.$$

Consequently,

$$X\setminus G_{(\mathfrak{g},\mathcal{P})}=\mathbb{N}\setminus \{1,2\},$$

which is an infinite set. Since every $D_{(\mathfrak{g},\mathcal{P})}$-set is of the form $U\setminus V$, where U is a $(\mathfrak{g},\mathcal{P})$-open set with $U\ne X$, it follows that every $D_{(\mathfrak{g},\mathcal{P})}$-set is contained in $G_{(\mathfrak{g},\mathcal{P})}$.

Hence, no $D_{(\mathfrak{g},\mathcal{P})}$-set can separate any two distinct points of $X\setminus G_{(\mathfrak{g},\mathcal{P})}$. Therefore, the space $(X,\mathfrak{g},\mathcal{P})$ does not satisfy the $(\mathfrak{g},\mathcal{P})$-$D_0$ property.

Hence, we present the notion of $(\mathfrak{g},\mathcal{P})$-$D_i$ for $i=0,1,2$ as follows:

Definition 6.

Let $(X,\mathfrak{g},\mathcal{P})$ be a $\mathbf{GPT}$ space; X is known as

 (i)

$(\mathfrak{g},\mathcal{P})$-$D_1$ if for each $x,y\in G_{(\mathfrak{g},\mathcal{P})}:x\ne y,$ there exist $D_{(\mathfrak{g},\mathcal{P})}$-sets U and V satisfying $x\in U,y\notin U$ and $y\in V,x\notin V.$

 (ii)

$(\mathfrak{g},\mathcal{P})$-$D_2$ if for each $x,y\in G_{(\mathfrak{g},\mathcal{P})}:x\ne y,$ there exist disjoint $D_{(\mathfrak{g},\mathcal{P})}$-sets $U,V$ satisfying $x\in U$ and $y\in V.$

Definition 7.

Let $(X,\mathfrak{g},\mathcal{P})$ be a $\mathbf{GPT}$ space. Then, X is known as

 (i)

$(\mathfrak{g},\mathcal{P})$-$T_0$ if for each $x,y\in G_{(\mathfrak{g},\mathcal{P})}:x\ne y,$ there exists $(\mathfrak{g},\mathcal{P})$-open set containing precisely one element of x and y;

 (ii)

$(\mathfrak{g},\mathcal{P})$-$T_1$ if each $x,y\in G_{(\mathfrak{g},\mathcal{P})}:x\ne y,$ there are $(\mathfrak{g},\mathcal{P})$-open sets U and V satisfying $x\in U$ and $y\notin U$ and $y\in V$ and $x\notin V$;

 (iii)

$(\mathfrak{g},\mathcal{P})$-$T_2$ if each $x,y\in G_{(\mathfrak{g},\mathcal{P})}:x\ne y,$ there are disjoint $(\mathfrak{g},\mathcal{P})$-open sets U and V satisfying $x\in U$ and $y\in V$;

 (iv)

$(\mathfrak{g},\mathcal{P})$-regular if for each $(\mathfrak{g},\mathcal{P})$-closed set F and $x\notin F,$ there are disjoint $(\mathfrak{g},\mathcal{P})$-open sets U and V satisfying $x\in U$ and $F\cap G_{(\mathfrak{g},\mathcal{P})}\subseteq V$;

 (v)

$(\mathfrak{g},\mathcal{P})$-normal if for all $(\mathfrak{g},\mathcal{P})$-closed sets A and B satisfying $A\cap B\cap G_{(\mathfrak{g},\mathcal{P})}=\varnothing ,$ there are disjoint $(\mathfrak{g},\mathcal{P})$-open sets U and V satisfying $A\cap G_{(\mathfrak{g},\mathcal{P})}\subseteq U$ and $B\cap G_{(\mathfrak{g},\mathcal{P})}\subseteq V.$

Remark 1.

Let $(X,\mathfrak{g},\mathcal{P})$ be a $\mathbf{GPT}$ space. When

 (i)

X is $(\mathfrak{g},\mathcal{P})$-$T_k,$ it is $(\mathfrak{g},\mathcal{P})$-$T_{k-1},$$k=1,2$;

 (ii)

X is $(\mathfrak{g},\mathcal{P})$-$T_k,$ it is $(\mathfrak{g},\mathcal{P})$-$D_k,$$k=0,1,2$;

 (iii)

X is $(\mathfrak{g},\mathcal{P})$-$D_k,$ it is $(\mathfrak{g},\mathcal{P})$-$D_{k-1},$$k=1,2.$

Theorem 1.

Let $(X,\mathfrak{g},\mathcal{P})$ be a $\mathbf{GPT}$ space.

 (i)

X is $(\mathfrak{g},\mathcal{P})$-$D_0$ iff X has the $(\mathfrak{g},\mathcal{P})$-$T_0$ property;

 (ii)

X is $(\mathfrak{g},\mathcal{P})$-$D_1$ iff X has the $(\mathfrak{g},\mathcal{P})$-$D_2$ property.

Proof. 

(i) Consider X as $(\mathfrak{g},\mathcal{P})$-$D_0$ and $x,y\in G_{(\mathfrak{g},\mathcal{P})}:$$x\ne y.$ Thus, there is a $D_{(\mathfrak{g},\mathcal{P})}$-set G satisfying $x\in G,$$y\notin G.$ Thus, there are $(\mathfrak{g},\mathcal{P})$-open sets $U_1$ and $U_2$ satisfying $G=U_1\setminus U_2.$ Thus, $x\in U_1$ and $x\notin U_2.$ If $y\in U_2,$ then $U_2$ is the required set. If $y\notin U_2$, then $y\notin U_1$ and $U_1$ satisfies the matter.

• The opposite direction follows from Remark 1.
• (ii) Consider X as $(\mathfrak{g},\mathcal{P})$-$D_1$ and $x,y\in G_{(\mathfrak{g},\mathcal{P})}$ with $x\ne y$. Then, there exist $D_{(\mathfrak{g},\mathcal{P})}$-sets $G_1$ and $G_2$ satisfies $x\in G_1,y\notin G_1,$ and $x\notin G_2,$$y\in G_2.$ Consider

$$G_1=V_1\setminus V_2\mathit{and}{G}_2=V_3\setminus V_4,$$

where $V_1,V_2,V_3$ and $V_4$ are $D_{(\mathfrak{g},\mathcal{P})}$-sets. Now, $x\notin G_2$ implies $x\notin V_3$ or $x\in V_3$ and $x\in V_4.$ Now, if $x\notin V_3$, from $y\notin G_1$ we get the next two cases:

(1)

$y\notin V_1.$ From $x\in V_1\setminus V_2,$ we get $x\in V_1\setminus (V_2\cup V_3).$ From $y\in V_3\setminus V_4,$ we get $y\in V_3\setminus (V_1\cup V_4).$ Also,

$$(V_1\setminus (V_2\cup V_3))\cap (V_3\setminus (V_1\cup V_4))=\varnothing .$$

(2)

$y\in V_1$ and $y\in V_2.$ We get $x\in V_1\setminus V_2$ and $y\in V_2$ and

$$(V_1\setminus V_2)\cap V_2=\varnothing .$$

Moreover, if $x\in V_3$ and $x\in V_4$, we get $y\in V_3\setminus V_4$ and $x\in V_4$ and

$$(V_3\setminus V_4)\cap V_4=\varnothing .$$

The opposite direction follows from Remark 1. □

Corollary 1.

Let $(X,\mathfrak{g},\mathcal{P})$ be a $\mathbf{GPT}$ space. When $X,$ is $(\mathfrak{g},\mathcal{P})$-$D_1,$ then it is $(\mathfrak{g},\mathcal{P})$-$T_0.$

Theorem 2.

Let $(X,\mathfrak{g},\mathcal{P})$ be a $\mathbf{GPT}$ space. Then, X is $(\mathfrak{g},\mathcal{P})$-$T_0$ iff for every $x,y\in G_{(\mathfrak{g},\mathcal{P})}$ with $x\ne y,$ we have

$$c{l}_{(\mathfrak{g},\mathcal{P})}\left(\left\{x\right\}\right)\ne c{l}_{(\mathfrak{g},\mathcal{P})}\left(\left\{y\right\}\right).$$

Proof. 

Let $\mathbf{GPT}$ space be $(\mathfrak{g},\mathcal{P})$-$T_0.$ Hence, for every $x,y\in G_{(\mathfrak{g},\mathcal{P})}$ with $x\ne y,\exists$$(\mathfrak{g},\mathcal{P})$-open set U satisfies $x\in U,$$y\notin U.$ Thus, $(X\setminus U)$ is $(\mathfrak{g},\mathcal{P})$-closed, with $x\notin (X\setminus U)$, which implies $y\in (X\setminus U).$ Hence,

$$c{l}_{(\mathfrak{g},\mathcal{P})}\left(\left\{y\right\}\right)\subseteq (X\setminus U)$$

Note that $x\notin c{l}_{(\mathfrak{g},\mathcal{P})}\left(\left\{y\right\}\right).$ Therefore, $c{l}_{(\mathfrak{g},\mathcal{P})}\left(\left\{x\right\}\right)\ne c{l}_{(\mathfrak{g},\mathcal{P})}\left(\left\{y\right\}\right).$ □

Theorem 3.

Let $(X,\mathfrak{g},\mathcal{P})$ be a $\mathbf{GPT}$ space. X is $(\mathfrak{g},\mathcal{P})$-$T_1$ iff for every $x\in G_{(\mathfrak{g},\mathcal{P})},$$\left\{x\right\}\cup (X\setminus G_{(\mathfrak{g},\mathcal{P})})$ is a $(\mathfrak{g},\mathcal{P})$-closed set.

Proof. 

Consider $\left\{x\right\}$ as $(\mathfrak{g},\mathcal{P})$-closed. Also, for each $x\in G_{(\mathfrak{g},\mathcal{P})}$ consider $x,y\in G_{(\mathfrak{g},\mathcal{P})}:x\ne y.$ Thus, the complements of $\left\{x\right\}$ and $\left\{y\right\}$ are $(\mathfrak{g},\mathcal{P})$-open sets with $x\in (X\setminus \{y\left\}\right),$$y\notin (X\setminus \{y\left\}\right)$ and $x\notin (X\setminus \{x\left\}\right),$$y\in (X\setminus \{x\left\}\right).$ Therefore, the $\mathbf{GPT}$ space is $(\mathfrak{g},\mathcal{P})$-$T_1.$ □

Definition 8.

Let $(X,\mathfrak{g},\mathcal{P})$ be a $\mathbf{GPT}$ space. A point $x\in G_{(\mathfrak{g},\mathcal{P})}$ is called $(\mathfrak{g},\mathcal{P})$-neat point if $x\in U\in \mathfrak{g},$ leads to $U=G_{(\mathfrak{g},\mathcal{P})}.$

Remark 2.

Let $(X,\mathfrak{g},\mathcal{P})$ be a $\mathbf{GPT}$ space. If X is $(\mathfrak{g},\mathcal{P})$-$T_0$ and has $(\mathfrak{g},\mathcal{P})$-neat point, then it is unique.

Theorem 4.

Let $(X,\mathfrak{g},\mathcal{P})$ be a $\mathbf{GPT}$ space. If X is $(\mathfrak{g},\mathcal{P})$-$T_0$ and has no $(\mathfrak{g},\mathcal{P})$-neat point, then it is $(\mathfrak{g},\mathcal{P})$-$D_1.$

In Figure 1, we present the relationship among the above notions in a simple way.

Figure 1. The relationship among the separation axioms in $\mathbf{GPT}$ spaces.

Example 2.

Consider the infinite $\mathbf{GPT}$ space defined as follows:

$$X=\mathbb{N}=\{1,2,3,\dots \},\mathcal{P}=\{A\subseteq \mathbb{N}\mid 1\notin A\},\mathfrak{g}=\{\varnothing \}\cup \{U\subseteq \mathbb{N}:1\in U\}\cup \left\{\mathbb{N}\right\}.$$

 (i)

We show that for each pair of distinct elements $x\ne y$ in $G_{(\mathfrak{g},\mathcal{P})}$, there exists a $(\mathfrak{g},\mathcal{P})$-open set that contains exactly one of them. This proves that the space satisfies the $(\mathfrak{g},\mathcal{P})$-$T_0$ separation axiom.

Consider any two distinct points $x,y\in \mathbb{N}$.

Case 1: One of the points is 1. Assume without loss of generality that $x=1$ and $y\ne 1$. Since any subset of $\mathbb{N}$ containing 1 is $(\mathfrak{g},\mathcal{P})$-open and satisfies

$$1\in \left\{1\right\},y\notin \left\{1\right\}.$$

Thus, 1 is distinguished from any other point in X.

Case 2: Both points are greater than 1. Suppose $x=n>1$ and $y=m>1$ with $n\ne m$. The set $\{1,n\}$ is $(\mathfrak{g},\mathcal{P})$-open because it contains the element 1. Moreover,

$$n\in \{1,n\},m\notin \{1,n\}.$$

Hence, n is distinguished from m by a $(\mathfrak{g},\mathcal{P})$-open set.

Since in both cases we can find a $(\mathfrak{g},\mathcal{P})$-open set containing exactly one of x and y, the space satisfies the $(\mathfrak{g},\mathcal{P})$-$T_0$ property.

 (ii)

The space is not $(\mathfrak{g},\mathcal{P})$-$T_1$. Since every nonempty $(\mathfrak{g},\mathcal{P})$-open set must contain the element 1, no $(\mathfrak{g},\mathcal{P})$-open set can isolate any $n>1$ from 1. Thus, the $(\mathfrak{g},\mathcal{P})$-$T_1$ requirement fails for pairs $(1,n)$. Consequently, the space cannot satisfy the stronger $(\mathfrak{g},\mathcal{P})$-$T_2$ property.

 (iii)

Since the space represents $(\mathfrak{g},\mathcal{P})$-$T_0$, it also represents the $(\mathfrak{g},\mathcal{P})$-$D_0$-property, Theorem 1.

However, no pair of disjoint $(\mathfrak{g},\mathcal{P})$-open sets can separate all distinct elements of the space. Therefore, the space does not satisfy the $(\mathfrak{g},\mathcal{P})$-$D_1$ property, and consequently, it also does not satisfy the $(\mathfrak{g},\mathcal{P})$-$D_2$ property.

Definition 9.

Let $(X,\mathfrak{g},\mathcal{P})$ be a $\mathbf{GPT}$ space. Then, X is known as

 (i)

$(\mathfrak{g},\mathcal{P})$-$R_0$ when $x\in U\in \mathfrak{g}$ implies

$$c{l}_{(\mathfrak{g},\mathcal{P})}\left(\left\{x\right\}\right)\cap G_{(\mathfrak{g},\mathcal{P})}\subseteq U.$$

 (ii)

$(\mathfrak{g},\mathcal{P})$-$R_1$ when for every $x,y\in G_{(\mathfrak{g},\mathcal{P})}$ satisfies

$$c{l}_{(\mathfrak{g},\mathcal{P})}\left(\left\{x\right\}\right)\ne c{l}_{(\mathfrak{g},\mathcal{P})}\left(\left\{y\right\}\right),$$

there exist disjoint $(\mathfrak{g},\mathcal{P})$-open sets U and V satisfying $x\in U,$$y\in V.$

In a $\mathbf{GPT}$ space that is not strong, it is impossible for the closure of any set to be contained in $(\mathfrak{g},\mathcal{P})$-open set, as is required in definitions of $(\mathfrak{g},\mathcal{P})$-$R_0$ and $(\mathfrak{g},\mathcal{P})$-$R_1.$ Then, a $\mathbf{GPT},$ that is not strong is not a $(\mathfrak{g},\mathcal{P})$-$R_0$ unless $G_{(\mathfrak{g},\mathcal{P})}$= ∅. Similarly, it is not a $(\mathfrak{g},\mathcal{P})$-$R_1.$

Theorem 5.

Let $(X,\mathfrak{g},\mathcal{P})$ be a $\mathbf{GPT}$ space. Then,

 (i)

X is $(\mathfrak{g},\mathcal{P})$-$T_1$ iff it is $(\mathfrak{g},\mathcal{P})$-$T_0$ and $(\mathfrak{g},\mathcal{P})$-$R_0$;

 (ii)

When X is $(\mathfrak{g},\mathcal{P})$-$R_1,$ then it is $(\mathfrak{g},\mathcal{P})$-$R_0.$

Proof. 

The result is obtained as a direct consequence of Definition 7 together with Theorems 2 and 3. □

Definition 10.

Let $(X,\mathfrak{g},\mathcal{P})$ be a $\mathbf{GPT}$ space. Then, X is called $(\mathfrak{g},\mathcal{P})$-symmetric if for each $x,y\in G_{(\mathfrak{g},\mathcal{P})}$ satisfying $x\in c{l}_{(\mathfrak{g},\mathcal{P})}\left(\left\{y\right\}\right),$ we get $y\in c{l}_{(\mathfrak{g},\mathcal{P})}\left(\left\{x\right\}\right).$

Theorem 6.

Let $(X,\mathfrak{g},\mathcal{P})$ be a $\mathbf{GPT}$ space. Then, X is $(\mathfrak{g},\mathcal{P})$-$R_0$ iff it is $(\mathfrak{g},\mathcal{P})$-symmetric.

Theorem 7.

Let $(X,\mathfrak{g},\mathcal{P})$ be a $\mathbf{GPT}$ space. Then, X is $(\mathfrak{g},\mathcal{P})$-$R_1$ iff for every $x,y\in G_{(\mathfrak{g},\mathcal{P})},$ with $y\notin c{l}_{(\mathfrak{g},\mathcal{P})}\left(\left\{x\right\}\right),$ there exist disjoint $(\mathfrak{g},\mathcal{P})$-open sets $U,V$ satisfying $x\in U,$$y\in V.$

Corollary 2.

Let $(X,\mathfrak{g},\mathcal{P})$ be a $\mathbf{GPT}$ space. Then, X is $(\mathfrak{g},\mathcal{P})$-$R_1$ iff for every $x,y\in G_{(\mathfrak{g},\mathcal{P})}$ satisfying $c{l}_{(\mathfrak{g},\mathcal{P})}\left(\left\{x\right\}\right)\ne c{l}_{(\mathfrak{g},\mathcal{P})}\left(\left\{y\right\}\right),$ there exist disjoint $(\mathfrak{g},\mathcal{P})$-open sets U and V satisfying

$$c{l}_{(\mathfrak{g},\mathcal{P})}\left(\left\{x\right\}\right)\cap G_{(\mathfrak{g},\mathcal{P})}\subseteq U$$

and

$$c{l}_{(\mathfrak{g},\mathcal{P})}\left(\left\{y\right\}\right)\cap G_{(\mathfrak{g},\mathcal{P})}\subseteq V.$$

Corollary 3.

In a $(\mathfrak{g},\mathcal{P})$-$R_0$$\mathbf{GPT}$ space X, the following are identical:

 (i)

X is $(\mathfrak{g},\mathcal{P})$-$T_0$.

 (ii)

X is $(\mathfrak{g},\mathcal{P})$-$T_1$.

 (iii)

X is $(\mathfrak{g},\mathcal{P})$-$D_1$.

Corollary 4.

In a $\mathbf{GPT}$ space X, the following are identical:

 (i)

X is $(\mathfrak{g},\mathcal{P})$-$T_2$.

 (ii)

X is $(\mathfrak{g},\mathcal{P})$-$T_1$ and $(\mathfrak{g},\mathcal{P})$-$R_1.$

 (iii)

X is $(\mathfrak{g},\mathcal{P})$-$T_0$ and $(\mathfrak{g},\mathcal{P})$-$R_1.$

Corollary 5.

In a $(\mathfrak{g},\mathcal{P})$-$R_1$$\mathbf{GPT}$ space X, the following are identical:

 (i)

X is $(\mathfrak{g},\mathcal{P})$-$T_2$.

 (ii)

X is $(\mathfrak{g},\mathcal{P})$-$T_1$.

 (iii)

X is $(\mathfrak{g},\mathcal{P})$-$T_0$.

Definition 11.

Let $(X,\mathfrak{g},\mathcal{P})$ be a $\mathbf{GPT}$ space with $A\subseteq G_{(\mathfrak{g},\mathcal{P})}.$ The $(\mathfrak{g},\mathcal{P})$-kernel of $A,$ denoted by $(\mathfrak{g},\mathcal{P})$-$ker\left(A\right),$ is given by

$$(\mathfrak{g},\mathcal{P})-ker\left(A\right)=\cap \{U\in \mathfrak{g}:A\subseteq U\}.$$

Lemma 1.

Let $(X,\mathfrak{g},\mathcal{P})$ be a $\mathbf{GPT}$ space with $A\subseteq G_{(\mathfrak{g},\mathcal{P})}.$ Thus,

$$(\mathfrak{g},\mathcal{P})-ker\left(A\right)=\{x\in G_{(\mathfrak{g},\mathcal{P})}:c{l}_{(\mathfrak{g},\mathcal{P})}\left(\left\{x\right\}\right)\cap A\ne \varnothing \}.$$

Lemma 2.

Let $(X,\mathfrak{g},\mathcal{P})$ be a $\mathbf{GPT}$ space with $x\in G_{(\mathfrak{g},\mathcal{P})}.$ Thus,

$$y\in (\mathfrak{g},\mathcal{P})-ker\left(\left\{x\right\}\right)⟺x\in c{l}_{(\mathfrak{g},\mathcal{P})}\left(\left\{y\right\}\right).$$

Lemma 3.

Let $(X,\mathfrak{g},\mathcal{P})$ be a $\mathbf{GPT}$ space with $x,y\in G_{(\mathfrak{g},\mathcal{P})}.$ Then,

$$(\mathfrak{g},\mathcal{P})-ker\left(\left\{x\right\}\right)\ne (\mathfrak{g},\mathcal{P})-ker\left(\left\{y\right\}\right)⟺c{l}_{(\mathfrak{g},\mathcal{P})}\left(\left\{x\right\}\right)\ne c{l}_{(\mathfrak{g},\mathcal{P})}\left(\left\{y\right\}\right).$$

Corollary 6.

Let $(X,\mathfrak{g},\mathcal{P})$ be a $\mathbf{GPT}$ space. Then, X is $(\mathfrak{g},\mathcal{P})$-$R_1$ iff for every $x,y\in G_{(\mathfrak{g},\mathcal{P})}$ satisfying

$$(\mathfrak{g},\mathcal{P})-ker\left(\right\{x\left\}\right)\ne (\mathfrak{g},\mathcal{P})-ker\left(\right\{y\left\}\right),$$

there are disjoint $(\mathfrak{g},\mathcal{P})$-open sets $U,V$ satisfying $x\in U,$$y\in V.$

Theorem 8.

Let $(X,\mathfrak{g},\mathcal{P})$ be a $\mathbf{GPT}$ space. Then, X is $(\mathfrak{g},\mathcal{P})$-$R_0$ iff every $x,y\in G_{(\mathfrak{g},\mathcal{P})}$ satisfying

$$c{l}_{(\mathfrak{g},\mathcal{P})}\left(\left\{x\right\}\right)\ne c{l}_{(\mathfrak{g},\mathcal{P})}\left(\left\{y\right\}\right)$$

leads to

$$c{l}_{(\mathfrak{g},\mathcal{P})}\left(\left\{x\right\}\right)\cap c{l}_{(\mathfrak{g},\mathcal{P})}\left(\left\{y\right\}\right)\cap G_{(\mathfrak{g},\mathcal{P})}=\varnothing .$$

The following counterexamples demonstrate $\mathbf{GPT}$-spaces where the $(\mathfrak{g},\mathcal{P})$-$R_0$ and $(\mathfrak{g},\mathcal{P})$-$R_1$ properties fail, highlighting the role of the $(\mathfrak{g},\mathcal{P})$-kernel in understanding closure and separability.

Example 3.

Consider the infinite $\mathbf{GPT}$-space defined by

$$X=\mathbb{N},\mathcal{P}=\{A\subseteq \mathbb{N}\mid 1\notin A\},\mathfrak{g}=\{\varnothing \}\cup \{U\subseteq \mathbb{N}:1\in U\}\cup \left\{\mathbb{N}\right\}.$$

The closures of singletons are

$${\mathrm{cl}}_{(\mathfrak{g},\mathcal{P})}\left(\left\{1\right\}\right)=\left\{1\right\},{\mathrm{cl}}_{(\mathfrak{g},\mathcal{P})}\left(\left\{n\right\}\right)=\mathbb{N}\mathrm{for}\mathrm{all}n>1.$$

The $(\mathfrak{g},\mathcal{P})$-kernels are

$$(\mathfrak{g},\mathcal{P})-\ker \left(\right\{1\left\}\right)=\mathbb{N},(\mathfrak{g},\mathcal{P})-\ker \left(\right\{n\left\}\right)=\mathbb{N}\mathrm{for}\mathrm{all}n>1.$$

$${\mathrm{cl}}_{(\mathfrak{g},\mathcal{P})}\left(\left\{1\right\}\right)\cap {\mathrm{cl}}_{(\mathfrak{g},\mathcal{P})}\left(\left\{2\right\}\right)\cap G_{(\mathfrak{g},\mathcal{P})}=\left\{1\right\}\cap \mathbb{N}=\left\{1\right\}\ne \varnothing .$$

Therefore, this space does not represent the $(\mathfrak{g},\mathcal{P})$-$R_0$ property.

• Also, since

$$(\mathfrak{g},\mathcal{P})-\ker \left(\right\{1\left\}\right)=\mathbb{N}=(\mathfrak{g},\mathcal{P})-\ker \left(\right\{n\left\}\right)\mathrm{for}\mathrm{all}n>1,$$

this space does not represent the $(\mathfrak{g},\mathcal{P})$-$R_1$ property.

Theorem 9.

Let $(X,\mathfrak{g},\mathcal{P})$ be a $\mathbf{GPT}$ space. Then, X is $(\mathfrak{g},\mathcal{P})$-$R_0$ iff for each $x,y\in G_{(\mathfrak{g},\mathcal{P})},$ we get

$$(\mathfrak{g},\mathcal{P})-ker\left(\right\{x\left\}\right)\ne (\mathfrak{g},\mathcal{P})-ker\left(\right\{y\left\}\right)$$

leads to

$$(\mathfrak{g},\mathcal{P})-\mathit{ker}\left(\right\{x\left\}\right)\cap (\mathfrak{g},\mathcal{P})-\mathit{ker}\left(\right\{y\left\}\right)=\varnothing .$$

Theorem 10.

In a $\mathbf{GPT}$ space X, the following are identical:

 (i)

X is $(\mathfrak{g},\mathcal{P})$-$R_0.$

 (ii)

For every $\varnothing \ne A\subseteq X,$$G\in \mathfrak{g}$ satisfying $A\cap G\ne \varnothing ,$ there is a $(\mathfrak{g},\mathcal{P})$-closed set F satisfying

$$A\cap F\ne \varnothing \mathit{with}F\cap G_{(\mathfrak{g},\mathcal{P})}\subseteq G.$$

 (iii)

Every $G\in \mathfrak{g}$ has the property

$$G=\cup \{F\cap G_{(\mathfrak{g},\mathcal{P})}:F\mathit{is}(\mathfrak{g},\mathcal{P})-\mathit{closedand}F\cap G_{(\mathfrak{g},\mathcal{P})}\subseteq G\}.$$

 (iv)

Every $(\mathfrak{g},\mathcal{P})$-closed set F has the property

$$F\cap G_{(\mathfrak{g},\mathcal{P})}=(\mathfrak{g},\mathcal{P})-ker(F\cap G_{(\mathfrak{g},\mathcal{P})}).$$

 (v)

Every $x\in G_{(\mathfrak{g},\mathcal{P})}$ has the property

$$c{l}_{(\mathfrak{g},\mathcal{P})}\left\{x\right\}\cap G_{(\mathfrak{g},\mathcal{P})}\subseteq (\mathfrak{g},\mathcal{P})-ker\left(\left\{x\right\}\right).$$

Theorem 11.

In a $\mathbf{GPT}$ space, the following are identical:

 (i)

X is $(\mathfrak{g},\mathcal{P})$-$R_0$;

 (ii)

When F is $(\mathfrak{g},\mathcal{P})$-closed, $x\in F\cap G_{(\mathfrak{g},\mathcal{P})},$ we get $(\mathfrak{g},\mathcal{P})$-$ker\left(\right\{x\left\}\right)\subseteq F;$

 (iii)

When $x\in G_{(\mathfrak{g},\mathcal{P})},$ thus $(\mathfrak{g},\mathcal{P})$-$ker\left(\left\{x\right\}\right)\subseteq c{l}_{(\mathfrak{g},\mathcal{P})}\left(\left\{x\right\}\right).$

Theorem 12.

Let $(X,\mathfrak{g},\mathcal{P})$ be a $\mathbf{GPT}$ space. Then, X is $(\mathfrak{g},\mathcal{P})$-$R_0$ iff for each $x\in G_{(\mathfrak{g},\mathcal{P})},$ we get

$$c{l}_{(\mathfrak{g},\mathcal{P})}\left(\left\{x\right\}\right)\cap G_{(\mathfrak{g},\mathcal{P})}=(\mathfrak{g},\mathcal{P})-ker\left(\left\{x\right\}\right).$$

3. More Types of Separation Axioms

Following the same approach, some researchers present different types of separation axioms in generalized topology, for example, $R_{\delta }$, weakly regular, $R_{D_{\delta }},$ and $R_{D^{★}}$ in [18,19]. Here, we will study them on $\mathbf{GPT}$ spaces.

Definition 12.

Let $(X,\mathfrak{g},\mathcal{P})$ be a $\mathbf{GPT}$ space. A subset $A\subseteq G_{(\mathfrak{g},\mathcal{P})}$ is known as a $(\mathfrak{g},\mathcal{P})$-$G_{\delta }$-set when A represents a countable intersection of $(\mathfrak{g},\mathcal{P})$-open sets. Moreover, its complement is known as a $(\mathfrak{g},\mathcal{P})$-$F_{\sigma }$ set.

Definition 13.

Let $(X,\mathfrak{g},\mathcal{P})$ be a $\mathbf{GPT}$ space. A subset $A\subseteq G_{(\mathfrak{g},\mathcal{P})}$ is known as a regular $(\mathfrak{g},\mathcal{P})$-$G_{\delta }$-set if

$$A=\bigcap _{i=1}^{\infty }{F}_i\cap G_{(\mathfrak{g},\mathcal{P})}=\bigcap _{i=1}^{\infty }{i}_{\mathfrak{g}}\left(F_i\right),$$

where each $F_i$ in the sequence $\left\{F_i\right\}$ is a $(\mathfrak{g},\mathcal{P})$-closed set.Moreover, its complement known as a regular $(\mathfrak{g},\mathcal{P})$-$F_{\sigma }$-set, which means that A is a regular $(\mathfrak{g},\mathcal{P})$-$F_{\sigma }$-set if

$$A=\bigcup _{i=1}^{\infty }{O}_i\cup (X\setminus G_{(\mathfrak{g},\mathcal{P})})=\bigcup _{i=1}^{\infty }c{l}_{(\mathfrak{g},\mathcal{P})}\left(O_i\right),$$

where each $O_i$ is $(\mathfrak{g},\mathcal{P})$-open.

Definition 14.

Let $(X,\mathfrak{g},\mathcal{P})$ be a $\mathbf{GPT}$ space. A subset $A\subseteq G_{(\mathfrak{g},\mathcal{P})}$ is called a $(\mathfrak{g},\mathcal{P})$-$d_{\delta }$-open set if

$$A=\bigcup _i{F}_i\cap G_{(\mathfrak{g},\mathcal{P})},$$

where each $F_i$ is a regular $(\mathfrak{g},\mathcal{P})$-$F_{\sigma }$ set.

• Moreover, its complement is called $(\mathfrak{g},\mathcal{P})$-$d_{\delta }$-closed, which means that A is $(\mathfrak{g},\mathcal{P})$-$d_{\delta }$-closed when

$$A=\bigcap _i{G}_i\cup (X\setminus G_{(\mathfrak{g},\mathcal{P})}),$$

where $G_i$ is a regular $(\mathfrak{g},\mathcal{P})$-$G_{\delta }$-set.

Definition 15.

Let $(X,\mathfrak{g},\mathcal{P})$ be a $\mathbf{GPT}$ space. A subset A is known as $(\mathfrak{g},\mathcal{P})$-d-closed if

$$A=\bigcap _i\{F_i:F_i\mathit{is}(\mathfrak{g},\mathcal{P})-\mathit{closedsetand}{F}_i\cap G_{(\mathfrak{g},\mathcal{P})}\mathit{is}(\mathfrak{g},\mathcal{P})-G_{\delta }-\mathit{set}\}.$$

Lemma 4.

Let $(X,\mathfrak{g},\mathcal{P})$ be a $\mathbf{GPT}$ space with $A\subseteq X.$ The following are true:

 (i)

When A is a regular $(\mathfrak{g},\mathcal{P})$-$G_{\delta }$-set, thus $A\cup (X\setminus G_{(\mathfrak{g},\mathcal{P})})$, consider as $(\mathfrak{g},\mathcal{P})$-$d_{\delta }$-closed;

 (ii)

Each regular $(\mathfrak{g},\mathcal{P})$-$G_{\delta }$-set consider as a $(\mathfrak{g},\mathcal{P})$-$G_{\delta }$-set;

 (iii)

When A is a regular $(\mathfrak{g},\mathcal{P})$-$G_{\delta }$-set, thus $A\cup (X\setminus G_{(\mathfrak{g},\mathcal{P})})$, consider as $(\mathfrak{g},\mathcal{P})$-closed;

 (iv)

Each $(\mathfrak{g},\mathcal{P})$-closed set F satisfying $F\cap G_{(\mathfrak{g},\mathcal{P})}$, consider as a $(\mathfrak{g},\mathcal{P})$-$G_{\delta }$-set;

 (v)

Each $(\mathfrak{g},\mathcal{P})$-d-closed set consider as $(\mathfrak{g},\mathcal{P})$-closed;

 (vi)

Each $(\mathfrak{g},\mathcal{P})$-$d_{\delta }$-closed set consider as $(\mathfrak{g},\mathcal{P})$-d-closed.

Proof. 

The proof comes immediately from the definitions. □

Definition 16.

For a $\mathbf{GPT}$ space X with $A\subseteq X.$

 (i)

A collection $\mathfrak{S}$ contained in $\mathfrak{g}$ is called a $(\mathfrak{g},\mathcal{P})$-open complementary system if for every $S\in \mathfrak{S},$ there exists a sequence $\left\{S_n\right\}$ in $\mathfrak{S}$ satisfying

$$S=\bigcup _n(X\setminus S_n)\cap G_{(\mathfrak{g},\mathcal{P})}.$$

 (ii)

A subset A is known as strongly open $(\mathfrak{g},\mathcal{P})$-$F_{\sigma }$-set when we have a countable $(\mathfrak{g},\mathcal{P})$-open complementary system $\mathfrak{S}$ satisfying $A\in \mathfrak{S}$. Moreover, its complement is known as a strongly closed $(\mathfrak{g},\mathcal{P})$-$G_{\delta }$-set.

 (iii)

Any intersection of a strongly closed $(\mathfrak{g},\mathcal{P})$-$G_{\delta }$-set is a $(\mathfrak{g},\mathcal{P})$-$d^{★}$-closed set. Then, each strongly closed $(\mathfrak{g},\mathcal{P})$-$G_{\delta }$-set is considered as $(\mathfrak{g},\mathcal{P})$-$d^{★}$-closed.

 (iv)

Each $(\mathfrak{g},\mathcal{P})$-$d^{★}$-closed set is considered as $(\mathfrak{g},\mathcal{P})$-d-closed.

The following examples illustrate infinite $\mathbf{GPT}$-spaces where certain new separation axioms hold, highlighting the role of $(\mathfrak{g},\mathcal{P})$-closed sets, $(\mathfrak{g},\mathcal{P})$-$G_{\delta }$ sets, and $(\mathfrak{g},\mathcal{P})$-kernels.

Definition 17.

A $\mathbf{GPT}$ space is called weakly $(\mathfrak{g},\mathcal{P})$-$D_1$ if

$$\bigcap _{x\in G_{(\mathfrak{g},\mathcal{P})}}c{l}_{(\mathfrak{g},\mathcal{P})}\left(\left\{x\right\}\right)=X\setminus G_{(\mathfrak{g},\mathcal{P})}.$$

Theorem 13.

A $\mathbf{GPT}$ space X is weakly $(\mathfrak{g},\mathcal{P})$-$D_1$ iff X has no $(\mathfrak{g},\mathcal{P})$-neat points.

Proof. 

Consider X as a $\mathbf{GPT}$ space that is weakly $(\mathfrak{g},\mathcal{P})$-$D_1$ and X has a $(\mathfrak{g},\mathcal{P})$-neat point $y.$ Thus, $y\in c{l}_{(\mathfrak{g},\mathcal{P})}\left(\left\{x\right\}\right),$ for every $x\in G_{(\mathfrak{g},\mathcal{P})},$ is a contradiction.

• Conversely, let the $\mathbf{GPT}$ space have no $(\mathfrak{g},\mathcal{P})$-neat points and not be weakly $(\mathfrak{g},\mathcal{P})$-$D_1.$ Then, there exists a

$$y\in (\bigcap _{x\in G_{(\mathfrak{g},\mathcal{P})}}c{l}_{(\mathfrak{g},\mathcal{P})}\left(\left\{x\right\}\right)\cap G_{(\mathfrak{g},\mathcal{P})}).$$

Then, any $(\mathfrak{g},\mathcal{P})$-open set containing y must be $G_{(\mathfrak{g},\mathcal{P})}.$ Thus, y is $(\mathfrak{g},\mathcal{P})$-neat point of $X,$ is a contradiction. □

Theorem 14.

When a $\mathbf{GPT}$ space X is $(\mathfrak{g},\mathcal{P})$-$T_0$ and weakly $(\mathfrak{g},\mathcal{P})$-$D_1,$ then it is $(\mathfrak{g},\mathcal{P})$-$D_1.$

Theorem 15.

A $\mathbf{GPT}$ space is weakly $(\mathfrak{g},\mathcal{P})$-$D_1$ iff for each $x\in G_{(\mathfrak{g},\mathcal{P})}$

$$(\mathfrak{g},\mathcal{P})-ker\left(\left\{x\right\}\right)\ne G_{(\mathfrak{g},\mathcal{P})}$$

Proof. 

Consider X as weakly $(\mathfrak{g},\mathcal{P})$-$D_1$. Consider $y\in G_{(\mathfrak{g},\mathcal{P})}$ as satisfying

$$(\mathfrak{g},\mathcal{P})-\mathit{ker}\left(\left\{y\right\}\right)=G_{(\mathfrak{g},\mathcal{P})}.$$

According to Lemma 2, $y\in c{l}_{(\mathfrak{g},\mathcal{P})}\left(\left\{x\right\}\right),\forall x\in G_{(\mathfrak{g},\mathcal{P})},$ is a contradiction.

• Conversely, let $(\mathfrak{g},\mathcal{P})-\mathit{ker}\left(\left\{x\right\}\right)\ne G_{(\mathfrak{g},\mathcal{P})}$ for every x that belongs to $G_{(\mathfrak{g},\mathcal{P})}.$ If X is not weakly $(\mathfrak{g},\mathcal{P})$-$D_1$, then according to Theorem 13, it has a $(\mathfrak{g},\mathcal{P})$-neat point $y.$ Therefore,

$$(\mathfrak{g},\mathcal{P})-\mathit{ker}\left(\left\{x\right\}\right)=G_{(\mathfrak{g},\mathcal{P})},$$

is a contradiction. □

Definition 18.

A $\mathbf{GPT}$ space is called

 (i)

$(\mathfrak{g},\mathcal{P})$-${\pi }_0$ if for every $(\mathfrak{g},\mathcal{P})$-open set $V\ne \varphi ,$ there exists $(\mathfrak{g},\mathcal{P})$-closed set F satisfying

$$\varnothing \ne F\cap G_{(\mathfrak{g},\mathcal{P})}\subseteq V.$$

 (ii)

$(\mathfrak{g},\mathcal{P})$-weakly regular if it has a base $\{F\cap G_{(\mathfrak{g},\mathcal{P})}:F$ is regular $(\mathfrak{g},\mathcal{P})$-$F_{\sigma }$-set $\}.$

 (iii)

$(\mathfrak{g},\mathcal{P})$-$R_{D_{\delta }}$ if for every $(\mathfrak{g},\mathcal{P})$-open V and for every $x\in V,$ there exists a regular $(\mathfrak{g},\mathcal{P})$-$G_{\delta }$-set F satisfying

$$x\in F\subseteq V.$$

 (iv)

$(\mathfrak{g},\mathcal{P})$-$R_{d_{\delta }}$ if for all $(\mathfrak{g},\mathcal{P})$-open V and for all $x\in V,$ there exists a $(\mathfrak{g},\mathcal{P})$-$d_{\delta }$-closed set F satisfying

$$x\in F\cap G_{(\mathfrak{g},\mathcal{P})}\subseteq V.$$

 (v)

$(\mathfrak{g},\mathcal{P})$-$R_D$ if for all $x\in V\in \mathfrak{g},$ there exists a $(\mathfrak{g},\mathcal{P})$-closed set F satisfying $F\cap G_{(\mathfrak{g},\mathcal{P})}$ is a $(\mathfrak{g},\mathcal{P})$-$G_{\delta }$-set with

$$x\in F\cap G_{(\mathfrak{g},\mathcal{P})}\subseteq V.$$

 (vi)

$(\mathfrak{g},\mathcal{P})$-$R_d$ if for all $x\in V\in \mathfrak{g},$ there exists a $(\mathfrak{g},\mathcal{P})$-d-closed set F satisfying

$$x\in F\cap G_{(\mathfrak{g},\mathcal{P})}\subseteq V.$$

 (vii)

$(\mathfrak{g},\mathcal{P})$-$R_D^{★}$ if for all $(\mathfrak{g},\mathcal{P})$-open set V and for all $x\in V,$ there exists a strongly closed $(\mathfrak{g},\mathcal{P})$-$G_{\delta }$-set F satisfying

$$x\in F\cap G_{(\mathfrak{g},\mathcal{P})}\subseteq V.$$

 (viii)

$(\mathfrak{g},\mathcal{P})$-$R_d^{★}$ if for all $(\mathfrak{g},\mathcal{P})$-open set V and for all $x\in V,$ there exists a $(\mathfrak{g},\mathcal{P})$-$d^{★}$-closed set F satisfying

$$x\in F\cap G_{(\mathfrak{g},\mathcal{P})}\subseteq V.$$

Theorem 16.

When a $\mathbf{GPT}$ space X is $(\mathfrak{g},\mathcal{P})$-$R_{d_{\delta }},$ then it is $(\mathfrak{g},\mathcal{P})$-$R_1.$

Proof. 

Consider that $x,y\in G_{(\mathfrak{g},\mathcal{P})}$ satisfies $y\notin c{l}_{(\mathfrak{g},\mathcal{P})}\left(\left\{x\right\}\right).$ Thus,

$$y\in X\setminus c{l}_{(\mathfrak{g},\mathcal{P})}\left(\left\{x\right\}\right)\in \mathfrak{g}.$$

As the space is $(\mathfrak{g},\mathcal{P})$-$R_{d_{\delta }},$ thus $\exists (\mathfrak{g},\mathcal{P})$-$d_{\delta }$-closed set

$$H=\bigcap _{i}Oi\cup (X\setminus G_{(\mathfrak{g},\mathcal{P})})$$

satisfies

$$y\in H\cap G_{(\mathfrak{g},\mathcal{P})}\subseteq X\setminus c{l}_{(\mathfrak{g},\mathcal{P})}\left(\left\{x\right\}\right).$$

Thus, $y\in O_i$ and $x\notin O_i$ for some $i.$ Now

$$O_i=\bigcap _{n=1}^{\infty }{F}_n\cap G_{(\mathfrak{g},\mathcal{P})}=\cap i_{\mathfrak{g}}\left(F_n\right),$$

where $\left\{F_n\right\}$ is a sequence of $(\mathfrak{g},\mathcal{P})$-closed sets. Then, there exists $F_n$ such that $y\in F_n$ and $x\notin F_n.$ Then, $i_{\mathfrak{g}}\left(F_n\right)$ and $X\setminus F_n$ are required $(\mathfrak{g},\mathcal{P})$-open sets. □

Example 4.

Consider the $\mathbf{GPT}$ space defined as

$$X=\mathbb{N},\mathcal{P}=\{A\subseteq \mathbb{N}\mid 1\notin A\},\mathfrak{g}=\{\varnothing \}\cup \{U\subseteq \mathbb{N}:1\in U\}\cup \left\{\mathbb{N}\right\}.$$

For each $x\in X$, define a descending sequence of $(\mathfrak{g},\mathcal{P})$-open sets:

$$V_n\left(x\right)=\{1,x,x+1,x+2,\dots \},n\ge 1.$$

The corresponding $(\mathfrak{g},\mathcal{P})$-closures:

$${\mathrm{cl}}_{(\mathfrak{g},\mathcal{P})}\left(V_n\left(x\right)\right)=V_n\left(x\right),\bigcap _{n=1}^{\infty }{V}_n\left(x\right)=\{1,x,x+1,\dots \}\subseteq X$$

Define the $(\mathfrak{g},\mathcal{P})$-$d_{\delta }$-closed set associated with x:

$$F_x=\bigcup _{n=1}^{\infty }{\mathrm{cl}}_{(\mathfrak{g},\mathcal{P})}\left(V_n\left(x\right)\right)=\{1,x,x+1,\dots \}.$$

So, for any $(\mathfrak{g},\mathcal{P})$-open set U with $x\in U$, there exists $(\mathfrak{g},\mathcal{P})$-$d_{\delta }$-closed $F_x$ satisfying

$$x\in F_x\cap G_{(\mathfrak{g},\mathcal{P})}\subseteq U.$$

Hence, this space represents $(\mathfrak{g},\mathcal{P})$-$R_{d_{\delta }}$ and, by Theorem 16, is also $(\mathfrak{g},\mathcal{P})$-$R_1$.

Theorem 17.

A $(\mathfrak{g},\mathcal{P})$-weakly regular $\mathbf{GPT}$ space is $(\mathfrak{g},\mathcal{P})$-$R_0.$

Proof. 

Consider $x\in U\in \mathfrak{g}$ and $y\in G_{(\mathfrak{g},\mathcal{P})}\setminus U.$ Thus, $x\notin c{l}_{(\mathfrak{g},\mathcal{P})}\left(\left\{y\right\}\right).$ Hence, there exists a regular $(\mathfrak{g},\mathcal{P})$-$F_{\sigma }$-set F satisfying

$$x\in F\cap G_{(\mathfrak{g},\mathcal{P})}\subseteq X\setminus c{l}_{(\mathfrak{g},\mathcal{P})}\left(\left\{y\right\}\right).$$

Let

$$F=\cup O_n\cup (X\setminus G_{(\mathfrak{g},\mathcal{P})})=\bigcup _{n=1}^{\infty }c{l}_{(\mathfrak{g},\mathcal{P})}\left(O_n\right),$$

where $O_n\in \mathfrak{g}$ and $c{l}_{(\mathfrak{g},\mathcal{P})}\left(O_n\right)\subseteq F,$$x\in O_n$ for some $n.$ Then,

$$c{l}_{(\mathfrak{g},\mathcal{P})}\left(O_n\right)\cap G_{(\mathfrak{g},\mathcal{P})}\subseteq F\cap G_{(\mathfrak{g},\mathcal{P})}\subseteq X\setminus c{l}_{(\mathfrak{g},\mathcal{P})}\left(\left\{y\right\}\right).$$

Hence, $y\in X\setminus c{l}_{(\mathfrak{g},\mathcal{P})}\left(O_n\right).$ Therefore, $y\notin c{l}_{(\mathfrak{g},\mathcal{P})}\left(\left\{x\right\}\right).$ □

Theorem 18.

When a $\mathbf{GPT}$ space X is $(\mathfrak{g},\mathcal{P})$-regular, hence it is $(\mathfrak{g},\mathcal{P})$-$R_{D_{\delta }}.$

Proof. 

Consider X is as $(\mathfrak{g},\mathcal{P})$-regular with $x\in U\in \mathfrak{g}.$ Hence, $\exists (\mathfrak{g},\mathcal{P})$-open set $V_1$ satisfies

$$x\in V_1\subseteq c{l}_{(\mathfrak{g},\mathcal{P})}\left(V_1\right)\cap G_{(\mathfrak{g},\mathcal{P})}\subseteq U.$$

Again $(\mathfrak{g},\mathcal{P})$-regularity yields a $(\mathfrak{g},\mathcal{P})$-open set $V_2$ with

$$x\in V_2\subseteq c{l}_{(\mathfrak{g},\mathcal{P})}\left(V_2\right)\cap G_{(\mathfrak{g},\mathcal{P})}\subseteq V_1.$$

By induction we get a sequence of $(\mathfrak{g},\mathcal{P})$-open sets $\left\{V_n\right\}$ satisfying

$$V_{n+1}\subseteq c{l}_{(\mathfrak{g},\mathcal{P})}\left(V_{n+1}\right)\cap G_{(\mathfrak{g},\mathcal{P})}\subseteq V_n.$$

Then, the set

$$A_x=\bigcap _{n=1}^{\infty }{V}_n=\bigcap _{n=1}^{\infty }c{l}_{(\mathfrak{g},\mathcal{P})}\left(V_n\right)\cap G_{(\mathfrak{g},\mathcal{P})}$$

is a regular $(\mathfrak{g},\mathcal{P})$-$G_{\delta }$-set satisfying $x\in A_x\subseteq U.$ □

We can summarize the previous results in the following diagram (Figure 2).

Figure 2. The relationship among new kinds of separation axioms in a $\mathbf{GPT}$ space.

Example 5.

Consider the $\mathbf{GPT}$ space:

$$X=\mathbb{Z},\mathcal{P}=\{A\subseteq \mathbb{Z}\mid 0\notin A\},\mathfrak{g}=\{\varnothing \}\cup \{U\subseteq \mathbb{Z}:0\in U\}\cup \left\{\mathbb{Z}\right\}.$$

 (i)

Let the base of $(\mathfrak{g},\mathcal{P})$-weakly regular sets be

$$\mathcal{B}=\{F\cap G_{(\mathfrak{g},\mathcal{P})}:F=\{0,n,n+1,\dots \},n\in \mathbb{Z}\}.$$

Each F is a regular $(\mathfrak{g},\mathcal{P})$-$F_{\sigma }$-set (countable union of $(\mathfrak{g},\mathcal{P})$-closed sets). Every $(\mathfrak{g},\mathcal{P})$-open set U contains a set $F\cap G_{(\mathfrak{g},\mathcal{P})}\subseteq U$ from the base $\mathcal{B}$. Thus, this space represents $(\mathfrak{g},\mathcal{P})$-weakly regular and, by Theorem 17, is also $(\mathfrak{g},\mathcal{P})$-$R_0$, hence $(\mathfrak{g},\mathcal{P})$-${\pi }_0.$

 (ii)

Define the closures of singletons as

$${\mathrm{cl}}_{(\mathfrak{g},\mathcal{P})}\left(\left\{0\right\}\right)=\left\{0\right\},{\mathrm{cl}}_{(\mathfrak{g},\mathcal{P})}\left(\left\{n\right\}\right)=\{n,n+1,n+2,\dots \},\mathit{\text{for all}}n>0.$$

The corresponding $(\mathfrak{g},\mathcal{P})$-kernels are

$$(\mathfrak{g},\mathcal{P})-\ker \left(\right\{0\left\}\right)=\mathbb{Z},(\mathfrak{g},\mathcal{P})-\ker \left(\right\{n\left\}\right)=\{n,n+1,n+2,\dots \},\mathit{\text{for all}}n>0.$$

$(\mathfrak{g},\mathcal{P})$-$R_{d_{\delta }}$ does not hold for $n>1$, since no $(\mathfrak{g},\mathcal{P})$-$d_{\delta }$ closed set $F_n$ satisfies $n\in F_n\cap G_{(\mathfrak{g},\mathcal{P})}\subseteq U$ for arbitrary U containing n, showing that $(\mathfrak{g},\mathcal{P})$-weakly regular ⇏$(\mathfrak{g},\mathcal{P})$-$R_{d_{\delta }}$ in this space.

 (iii)

Similarly, $(\mathfrak{g},\mathcal{P})$-$R_1$ fails for $n>1$, showing that $(\mathfrak{g},\mathcal{P})$-weakly regular ⇏$R_1$ in this space.

It is worth mentioning that focusing the study on special classes of sets whose definitions rely on the primal collection often leads to results that are particularly interesting. This is exactly the approach we shall adopt next. To begin, we first define the notion of a ${\mathcal{P}}^{\diamond }$-open set within a GPT-space.

4. $\mathtt{GP}$-Homeomorphism Mapping

As we said previously, separation axioms in the theory of topology are very important for the rule that plays to recognize non-homeomorphic topologies. In this section, we present the fundamental theorem and propositions about the $\mathtt{GP}$-continuous functions and the $\mathtt{GP}$-homeomorphism functions.

Definition 19.

Assuming X and $X^{‵}$ are two $\mathbf{GPT}$ spaces and h is a function from X into $X^{‵}.$ Then, h is called $\mathtt{GP}$-homeomorphism if it is bijective, $\mathtt{GP}$-continuous, and its inverse is $\mathtt{GP}$-continuous too.

Theorem 19.

Let X and $X^{‵}$ be two $\mathbf{GPT}$ spaces. When a $\mathtt{GP}$-continuous function $h:X\to X^{‵}$ satisfies

$$h\left(G_{(\mathfrak{g},\mathcal{P})}\right)\subseteq G_{({\mathfrak{g}}^{‵},{\mathcal{P}}^{‵})},$$

the following are true:

 (i)

The inverse image or (for short $h^{-1}$) of $({\mathfrak{g}}^{‵},{\mathcal{P}}^{‵})$-$G_{\delta }$-sets is $(\mathfrak{g},\mathcal{P})$-$G_{\delta }$-set;

 (ii)

$h^{-1}$ of a strongly closed $({\mathfrak{g}}^{‵},{\mathcal{P}}^{‵})$-$G_{\delta }$-set is a strongly closed $(\mathfrak{g},\mathcal{P})$-$G_{\delta }$-set;

 (iii)

$h^{-1}$ of a $({\mathfrak{g}}^{‵},{\mathcal{P}}^{‵})$-$d^{★}$-closed set is $(\mathfrak{g},\mathcal{P})$-$d^{★}$-closed;

 (iv)

$h^{-1}$ of a regular $({\mathfrak{g}}^{‵},{\mathcal{P}}^{‵})$-$G_{\delta }$-set is regular $(\mathfrak{g},\mathcal{P})$-$G_{\delta }$-set;

 (v)

$h^{-1}$ of a $({\mathfrak{g}}^{‵},{\mathcal{P}}^{‵})$-$d_{\delta }$-closed set is $(\mathfrak{g},\mathcal{P})$-$d_{\delta }$-closed;

 (vi)

$h^{-1}$ of s $({\mathfrak{g}}^{‵},{\mathcal{P}}^{‵})$-d-closed is $(\mathfrak{g},\mathcal{P})$-d-closed.

Proof. 

(i) and (iii) are clear.

• (ii) Is true since

$$h^{-1}(\bigcap _k{B}_k\cup (X^{‵}\setminus G_{({\mathfrak{g}}^{‵},{\mathcal{P}}^{‵})}))=\bigcap _k{h}^{-1}\left(B_k\right)\cup (X\setminus G_{(\mathfrak{g},\mathcal{P})}).$$

 (iv) Consider a regular $({\mathfrak{g}}^{‵},{\mathcal{P}}^{‵})$-$G_{\delta }$-set H given by

$$H=\bigcap _{k=1}^{\infty }{F}_k\cap G_{({\mathfrak{g}}^{‵},{\mathcal{P}}^{‵})}=\bigcap _{k=1}^{\infty }{i}_{{\mathfrak{g}}^{‵}}\left(F_k\right)$$

Then,

$$\begin{array}{cc}\hfill h^{-1}\left(H\right)& =\bigcap _{k=1}^{\infty }{h}^{-1}\left(F_k\right)\cap G_{(\mathfrak{g},\mathcal{P})},\hfill \\ \hfill & =\bigcap _{k=1}^{\infty }{h}^{-1}\left(i_{{\mathfrak{g}}^{‵}}\left(F_k\right)\right).\hfill \end{array}$$

Now,

$$\begin{array}{cc}\hfill h^{-1}\left(i_{{\mathfrak{g}}^{‵}}\left(F_k\right)\right)& =h^{-1}(X^{‵}\setminus c{l}_{({\mathfrak{g}}^{‵},{\mathcal{P}}^{‵})}(X^{‵}\setminus F_k)),\hfill \\ \hfill & =X\setminus h^{-1}\left(c{l}_{({\mathfrak{g}}^{‵},{\mathcal{P}}^{‵})}(X^{‵}\setminus F_k)\right),\hfill \\ \hfill & \subseteq X\setminus c{l}_{(\mathfrak{g},\mathcal{P})}\left(h^{-1}(X^{‵}\setminus F_k)\right),\hfill \\ \hfill & =i_{\mathfrak{g}}\left(h^{-1}\left(F_k\right)\right).\hfill \end{array}$$

Since

$$i_{\mathfrak{g}}\left(h^{-1}\left(F_k\right)\right)\subseteq h^{-1}\left(F_k\right)\cap G_{(\mathfrak{g},\mathcal{P})},$$

the result follows.

• (v) Let

$$H=\bigcap _i{O}_i\cup (X^{‵}\setminus G_{({\mathfrak{g}}^{‵},{\mathcal{P}}^{‵})}),$$

where $O_i$ is regular $({\mathfrak{g}}^{‵},{\mathcal{P}}^{‵})$-$G_{\delta }$-set, be $({\mathfrak{g}}^{‵},{\mathcal{P}}^{‵})$-$d_{\delta }$-closed. Now, apply (iv). Then,

$$h^{-1}\left(H\right)=\bigcap _i{h}^{-1}\left(O_i\right)\cup (X\setminus G_{(\mathfrak{g},\mathcal{P})}).$$

(vi) Comes directly from (i). □

Theorem 20.

Let X and $X^{‵}$ be two $\mathbf{GPT}$ spaces. When X is finer than $X^{‵}$ and X is $(\mathfrak{g},\mathcal{P})$-$D_0$ (resp. $(\mathfrak{g},\mathcal{P})$-$D_1$, $(\mathfrak{g},\mathcal{P})$-$D_2$, $(\mathfrak{g},\mathcal{P})$-$T_0$, $(\mathfrak{g},\mathcal{P})$-$T_1$, $(\mathfrak{g},\mathcal{P})$-$T_2$), then $X^{‵}$ is $({\mathfrak{g}}^{‵},{\mathcal{P}}^{‵})$-$D_0$ (resp. $({\mathfrak{g}}^{‵},{\mathcal{P}}^{‵})$-$D_1$, $({\mathfrak{g}}^{‵},{\mathcal{P}}^{‵})$-$D_2$, $({\mathfrak{g}}^{‵},{\mathcal{P}}^{‵})$-$T_0$, $({\mathfrak{g}}^{‵},{\mathcal{P}}^{‵})$-$T_1$, $({\mathfrak{g}}^{‵},{\mathcal{P}}^{‵})$-$T_2$).

Theorem 21.

Let X and $X^{‵}$ be two $\mathbf{GPT}$ spaces. When a $\mathtt{GP}$-continuous function $h:X\to X^{‵}$ satisfies

$$G_{({\mathfrak{g}}^{‵},{\mathcal{P}}^{‵})}\subseteq h\left(G_{(\mathfrak{g},\mathcal{P})}\right),$$

then $h^{-1}\left(E\right)$ is $D_{(\mathfrak{g},\mathcal{P})}$-set for each $D_{({\mathfrak{g}}^{‵},{\mathcal{P}}^{‵})}$-set $E.$

Theorem 22.

Let X and $X^{‵}$ be two $\mathbf{GPT}$ spaces. When $X^{‵}$ is $({\mathfrak{g}}^{‵},{\mathcal{P}}^{‵})$-$D_1$ and $h:X\to X^{‵}$ is $\mathtt{GP}$-continuous such that h is injective on $G_{(\mathfrak{g},\mathcal{P})}$ and satisfies

$$h\left(G_{(\mathfrak{g},\mathcal{P})}\right)=G_{({\mathfrak{g}}^{‵},{\mathcal{P}}^{‵})},$$

then X is $(\mathfrak{g},\mathcal{P})$-$D_1.$

Theorem 23.

A $\mathbf{GPT}$ space X is $(\mathfrak{g},\mathcal{P})$-$D_1$ iff for all $x,y\in X$ with $x\ne y,$ there exists a $\mathbf{GPT}$ space $X^{‵}$ and $\mathtt{GP}$-continuous mapping $h:X\to X^{‵}$, where $X^{‵}$ is $({\mathfrak{g}}^{‵},{\mathcal{P}}^{‵})$-$D_1$ with

$$G_{({\mathfrak{g}}^{‵},{\mathcal{P}}^{‵})}\subseteq h\left(G_{(\mathfrak{g},\mathcal{P})}\right)\mathit{and}h\left(x\right)\ne h\left(y\right).$$

Theorem 24.

Let X and $X^{‵}$ be two $\mathbf{GPT}$ spaces. Assuming $h:X\to X^{‵}$ as a $\mathtt{GP}$-continuous and $\mathtt{GP}$-closed function satisfying

$$h^{-1}\left(G_{({\mathfrak{g}}^{‵},{\mathcal{P}}^{‵})}\right)=G_{(\mathfrak{g},\mathcal{P})}.$$

When X is $(\mathfrak{g},\mathcal{P})$-$R_0$, then $X^{‵}$ is $({\mathfrak{g}}^{‵},{\mathcal{P}}^{‵})$-$R_0$.

Proof. 

Consider $y\in V\in {\mathfrak{g}}^{‵}.$ Thus, there exists $x\in G_{(\mathfrak{g},\mathcal{P})}$ satisfying $h\left(x\right)=y.$ Then, $x\in h^{-1}\left(V\right).$ Since X is $(\mathfrak{g},\mathcal{P})$-$R_0,$

$$c{l}_{(\mathfrak{g},\mathcal{P})}\left(\left\{x\right\}\right)\cap G_{(\mathfrak{g},\mathcal{P})}\subseteq h^{-1}\left(V\right).$$

As h is $\mathtt{GP}$-closed,

$$\begin{array}{cc}\hfill c{l}_{({\mathfrak{g}}^{‵},{\mathcal{P}}^{‵})}\left(h\left(x\right)\right)& \subseteq h\left(c{l}_{(\mathfrak{g},\mathcal{P})}\left(\left\{x\right\}\right)\right),\hfill \\ \hfill & =h((c{l}_{(\mathfrak{g},\mathcal{P})}\left(\left\{x\right\}\right)\cap G_{(\mathfrak{g},\mathcal{P})})\cup (X\setminus G_{(\mathfrak{g},\mathcal{P})})),\hfill \\ \hfill & \subseteq h(c{l}_{(\mathfrak{g},\mathcal{P})}\left\{x\right\}\cap G_{(\mathfrak{g},\mathcal{P})})\cup (X^{‵}\setminus G_{({\mathfrak{g}}^{‵},{\mathcal{P}}^{‵})}).\hfill \end{array}$$

as $h^{-1}\left(G_{({\mathfrak{g}}^{‵},{\mathcal{P}}^{‵})}\right)=G_{(\mathfrak{g},\mathcal{P})}.$ Therefore,

$$\begin{array}{cc}\hfill c{l}_{({\mathfrak{g}}^{‵},{\mathcal{P}}^{‵})}\left(\left\{y\right\}\right)\cap G_{({\mathfrak{g}}^{‵},{\mathcal{P}}^{‵})}& \subseteq h(c{l}_{(\mathfrak{g},\mathcal{P})}\left(\left\{x\right\}\right)\cap G_{(\mathfrak{g},\mathcal{P})})\hfill \\ \hfill & \subseteq h{h}^{-1}\left(V\right)\hfill \\ & \subseteq V.\hfill \end{array}$$

Thus, $X^{‵}$ is $({\mathfrak{g}}^{‵},{\mathcal{P}}^{‵})$-$R_0$. □

Theorem 25.

Let X and $X^{‵}$ be two $\mathbf{GPT}$ spaces. When $h:X\to X^{‵}$, consider as $\mathtt{GP}$-homeomorphism and X as $(\mathfrak{g},\mathcal{P})$-$R_1$, thus $X^{‵}$ is $({\mathfrak{g}}^{‵},{\mathcal{P}}^{‵})$-$R_1.$

Proof. 

Consider $x,y\in G_{({\mathfrak{g}}^{‵},{\mathcal{P}}^{‵})}$ and $y\notin c{l}_{({\mathfrak{g}}^{‵},{\mathcal{P}}^{‵})}\left(\left\{x\right\}\right).$ Then,

$$y_0=h^{-1}\left(y\right)\notin h^{-1}(c{l}_{({\mathfrak{g}}^{‵},{\mathcal{P}}^{‵})}\left(\left\{x\right\}\right).$$

Since

$$c{l}_{({\mathfrak{g}}^{‵},{\mathcal{P}}^{‵})}\left(h^{-1}\left(x\right)\right)\subseteq h^{-1}(c{l}_{({\mathfrak{g}}^{‵},{\mathcal{P}}^{‵})}\left(\left\{x\right\}\right).$$

Therefore,

$$y_0\notin c{l}_{(\mathfrak{g},\mathcal{P})}\left(h^{-1}\left(x\right)\right)=c{l}_{(\mathfrak{g},\mathcal{P})}\left(\left\{x_0\right\}\right),$$

where $x_0=h^{-1}\left(x\right)\in G_{({\mathfrak{g}}^{‵},{\mathcal{P}}^{‵})}.$ As X is $(\mathfrak{g},\mathcal{P})$-$R_1,\exists (\mathfrak{g},\mathcal{P})$-open sets $U,V$ satisfying $x_0\in U$ and $y_0\in V.$ Then, $h\left(U\right),h\left(V\right)$ are disjoint $({\mathfrak{g}}^{‵},{\mathcal{P}}^{‵})$-open with $x\in h\left(U\right),y\in f\left(V\right).$ Thus, $X^{‵}$ is $({\mathfrak{g}}^{‵},{\mathcal{P}}^{‵})$-$R_1.$ □

Theorem 26.

Let X and $X^{‵}$ be two $\mathbf{GPT}$ spaces. When $h:X\to X^{‵}$ is $\mathtt{GP}$-closed injection function satisfying

$$h\left(G_{(\mathfrak{g},\mathcal{P})}\right)\subseteq G_{({\mathfrak{g}}^{‵},{\mathcal{P}}^{‵})}$$

implies

$$h(X\setminus G_{(\mathfrak{g},\mathcal{P})})\subseteq (X^{‵}\setminus G_{({\mathfrak{g}}^{‵},{\mathcal{P}}^{‵})}),$$

X is weakly $(\mathfrak{g},\mathcal{P})$-$D_1,$ thus $X^{‵}$ is weakly $({\mathfrak{g}}^{‵},{\mathcal{P}}^{‵})$-$D_1.$

Proof. 

Since X is weakly $(\mathfrak{g},\mathcal{P})$-$D_1,$ then

$$\bigcap _{x\in G_{(\mathfrak{g},\mathcal{P})}}c{l}_{(\mathfrak{g},\mathcal{P})}\left(\left\{x\right\}\right)=X\setminus G_{(\mathfrak{g},\mathcal{P})}.$$

Since h is an injection,

$$\begin{array}{cc}\hfill h\left(\bigcap _{x\in G_{(\mathfrak{g},\mathcal{P})}}c{l}_{(\mathfrak{g},\mathcal{P})}\left(\left\{x\right\}\right)\right)& =\bigcap _{x\in G_{(\mathfrak{g},\mathcal{P})}}h\left(c{l}_{(\mathfrak{g},\mathcal{P})}\left(\left\{x\right\}\right)\right),\hfill \\ \hfill & =h(X\setminus G_{(\mathfrak{g},\mathcal{P})}).\hfill \end{array}$$

As h is a $\mathtt{GP}$-closed function,

$$\begin{array}{cc}\hfill \bigcap _{y\in G_{({\mathfrak{g}}^{‵},{\mathcal{P}}^{‵})}}c{l}_{({\mathfrak{g}}^{‵},{\mathcal{P}}^{‵})}\left(\left\{y\right\}\right)& \subseteq \bigcap _{x\in G_{(\mathfrak{g},\mathcal{P})}}c{l}_{({\mathfrak{g}}^{‵},{\mathcal{P}}^{‵})}\left(h\left(\left\{x\right\}\right)\right),\hfill \\ \hfill & \subseteq \bigcap _{x\in G_{(\mathfrak{g},\mathcal{P})}}h\left(c{l}_{(\mathfrak{g},\mathcal{P})}\left(\left\{x\right\}\right)\right),\hfill \\ \hfill & =h(X\setminus G_{(\mathfrak{g},\mathcal{P})}),\hfill \\ \hfill & \subseteq X^{‵}\setminus G_{({\mathfrak{g}}^{‵},{\mathcal{P}}^{‵})}.\hfill \end{array}$$

Therefore, $X^{‵}$ is weakly $({\mathfrak{g}}^{‵},{\mathcal{P}}^{‵})$-$D_1.$ □

5. Discussion

In this section, we provide a brief discussion highlighting that certain classical axioms in $\mathbf{GPT}$ spaces may differ when restricted to a primal collection. Based on this observation, we introduce relevant definitions and preliminary concepts to clarify the structural differences. This framework will allow us to systematically analyze the consequences of these restrictions.

Definition 20.

In a $\mathbf{GPT}$ space X, a subset $A\subseteq X$ is called ${\mathcal{P}}^{\diamond }$-open if $A\subseteq i_{\mathfrak{g}}{\left(A\right)}^{\diamond }.$

Definition 21.

A $\mathbf{GPT}$ space X is said to be ${\mathcal{P}}^{\diamond }$-Hausdorff if for any two distinct points $x,y\in X$, there exist two disjoint ${\mathcal{P}}^{\diamond }$-open sets U and V such that $x\in U$ and $y\in V$.

It is important to note that the relationships between separation axioms may differ when we restrict attention to ${\mathcal{P}}^{\diamond }$-open sets. In particular, one property does not necessarily imply the other without specific assumptions on the primal collection.

Remark 3.

A $(\mathfrak{g},\mathcal{P})$-Hausdorff $\mathbf{GPT}$ space does not necessarily imply that it is ${\mathcal{P}}^{\diamond }$-Hausdorff, and vice versa.

To illustrate, consider a subset $A\subseteq X$. If $A^c\subseteq \mathcal{P}$, then $A^{\diamond }=\varnothing$ (see Theorem 3.1 [1]), meaning that no set other than the empty set is ${\mathcal{P}}^{\diamond }$-open. Consequently, it is impossible to find two disjoint ${\mathcal{P}}^{\diamond }$-open sets containing two distinct points.

Furthermore, consider $X=\{x_1,x_2\}$ with $\mathfrak{g}=\{\varnothing ,\left\{x_1\right\},X\}$ and primal collection $\mathcal{P}=\{\varnothing ,\left\{x_1\right\},\left\{x_2\right\}\}$. This example demonstrates that a ${\mathcal{P}}^{\diamond }$-Hausdorff space may fail to be $(\mathfrak{g},\mathcal{P})$-Hausdorff.

Within the framework developed in this paper, however, every $(\mathfrak{g},\mathcal{P})$-Hausdorff $\mathbf{GPT}$ space is indeed a ${\mathcal{P}}^{\diamond }$-Hausdorff space, while the converse does not always hold. To avoid ambiguity, we shall adopt the term $\mathcal{P}$-Hausdorff to refer to the new notion.

Definition 22.

A $\mathbf{GPT}$ space X is called $\mathcal{P}$-Hausdorff if for each pair of distinct points $x,y\in X$, there exist two $(\mathfrak{g},\mathcal{P})$-open sets $U_1$ and $U_2$ such that $x\in U_1$, $y\in U_2$, and $U_1\cap U_2\in \mathcal{P}$.

From Definition 22, it follows that every $(\mathfrak{g},\mathcal{P})$-Hausdorff $\mathbf{GPT}$ space is $\mathcal{P}$-Hausdorff, regardless of the specific primal collection.

Consider the infinite $\mathbf{GPT}$ space

$$X=\mathbb{N},g=\{\varnothing \}\cup \{U\subseteq \mathbb{N}:1\in U\}\cup \left\{\mathbb{N}\right\},\mathcal{P}=\{A\subseteq \mathbb{N}:1\notin A\}.$$

Observe that without considering the primal collection, the space fails to be Hausdorff because it is impossible to find disjoint $(\mathfrak{g},\mathcal{P})$-open sets separating points $n>1$. However, with the primal collection included, we can define ${\mathcal{P}}^{\diamond }$-open sets that separate each point of X, ensuring that the space is $\mathcal{P}$-Hausdorff. This example clearly shows that the primal collection has a direct effect on separation properties, even in infinite spaces.

6. Conclusions

In this paper, we introduced new classes of separation axioms within the framework of generalized primal topological ($\mathbf{GPT}$) spaces and analyzed their interrelations. These axioms extend classical concepts and provide a deeper understanding of the structural properties of $\mathbf{GPT}$ spaces. Our results demonstrate how primal collections influence generalized notions of separation and continuity, thereby enriching the theoretical foundation of $\mathbf{GPT}$ spaces.

For future research, several specific directions are suggested:

1.

Extension to higher-order $\mathbf{GPT}$ spaces: Investigate how the newly introduced separation axioms behave in more complex or higher-dimensional $\mathbf{GPT}$ spaces, possibly with additional algebraic or combinatorial structures.

2.

Applications in functional analysis and continuity: Study how generalized continuous functions and other types of function mappings behave under these new axioms, and whether new classes of continuous functions emerge.

3.

Connections with other generalized topologies: Explore relationships between $\mathbf{GPT}$ spaces and other generalized or hybrid topological structures, such as soft topologies, fuzzy topologies, or closure spaces, to identify common properties and potential unifying frameworks.

4.

Algorithmic and computational aspects: Develop algorithms to construct $\mathbf{GPT}$ spaces satisfying certain separation axioms, and investigate computational complexity of checking these properties in finite or discrete settings.

5.

Topological invariants and classification: Examine whether these new axioms lead to new invariants or classifications within $\mathbf{GPT}$ spaces, aiding in the systematic categorization of different types of generalized primal topologies.

These directions aim to deepen the understanding of $\mathbf{GPT}$ spaces and extend their theoretical and practical applications in mathematics.

In summary, this work provides a deeper insight into the structure and properties of $\mathbf{GPT}$ spaces through the introduction of new separation axioms. The results not only extend classical topological concepts but also offer a foundation for further theoretical and practical developments. Future studies may build on these findings to explore more complex $\mathbf{GPT}$ structures, applications, and connections with other generalized topologies.