## 1. Introduction

B. Greene [

1] mentioned in his book “The Elegance of the Universe” that the 11th dimension appears when the Heterotic-E matching constant is greater than 1 but not less than 1. In this case, it may be thought that the 11th dimension is a different topology. I think this may be a discrete topology. So, is a topology possible in which different topologies of different dimensions can exist? For topologies of different sizes to be measured, should each topology be the same and measurable? In this context, for a topological space in which different topologies can be written in different parameters, it should be looked at whether a set is created in which different sets can be written in different parameters. So, let us first analyze Molodtsov’s soft set [

2], which makes it possible to write different sets with different parameters.

Molodtsov [

2] defined a soft set and gave some properties about it. For this, he thought that there are many uncertainties to solve complicated problems such as in sociology, economics, engineering, medical science, environment problems, statistics, etc. There is no deal to solve them successfully. However, there are some theories such as vague sets theory [

3], fuzzy sets [

4], probability, intuitionistic fuzzy sets [

5], rough sets [

6], interval mathematics [

7], etc., but these studies have their own complexities.

Maji et al. [

8] defined soft subset, soft superset, soft equality and they gave some operators such as intersection, union of two soft sets, and complement of a soft set. They presented some properties about them. But some properties of them are false. So, Yang [

9] gave counterexample about some of them. Also, Ali et al. [

10] gave counter example about the others. Then they redefined operations of soft sets. Then the other researchers pointed out these false about soft operators and they gave some new definitions about soft operators of soft sets [

11,

12,

13,

14,

15,

16,

17,

18,

19]. These are so valuable studies.

Up to now, there are many studies on the soft sets and their operations, also, taking universal set of parameter

$E$ is finite and countable because of the definition of the soft set. There are many soft sets and their operators defined. As first, in the soft set defined by Molodtsov [

2] and developed by Maji et al. [

8] the union of any soft set and its complement need not be a universal set. So, this situation makes a lot of deficiencies like a problem about complement of any soft set. To overcome this problem, researchers see two different ways. One way is using fixed parameter set in soft sets like Shabir and Naz’s work [

20]. They defined a lot of concept about a soft set and its topology. They used same parameter sets in their soft sets. This situation limits the soft topology. As a result of the fixed parameter, a soft point could be defined with fixed parameter. As a result of this, all soft sets have same values in all their parameters [

21,

22,

23,

24]. Some researchers tried to overcome this situation by defining another soft point [

11,

18,

21,

25]. You can see the latest study examples based on Shabir and Naz’s work [

26,

27,

28,

29,

30,

31,

32]. The other way is to redefine soft set and its operations. As first, Çağman and Enginoğlu [

13] redefined soft set and its operations. This study is so valuable. But anyone could not take uncountable or infinite universal parameter set in practice. Because it is not pointed out what will we do with parameter sets between two soft sets exactly while using soft operators. You can see the latest study examples based on Çağman and Enginoglu’s work [

26,

30,

31,

32,

33,

34]. Çağman et al. [

35] defined a soft topology. They use different parameter sets in their soft sets. But all of them are finite because of the definition of soft operations. Then, Zhu and Wen [

19] redefined a soft set and gave operations of it. Also, they pointed out what will we do with parameters set between two soft sets while using soft operators. But it is so complicated and as a result, cause same problems. You can see the latest study examples based on Zhu and Wen’s work [

36,

37,

38]. A similar situation drew attention of Fatimah et al. in [

39]. They said that in their study all soft and hybrid soft sets used so far binary operation (either 0 or 1) or else real numbers between 0 and 1. So they defined a new soft set and it is called an N-Soft set. They used

n parameter in their study,

$n\in \mathbb{N},$ is natural numbers. Riaz et al. defined an N-Soft topology in [

30]. Therefore, we can say that they use set of parameters

$E$ as infinite and countable. They use initial universe

$X$ as finite and countable. But in fact, in real life or in space they do not have to be finite and countable.

In order to overcome all the problems mentioned about above Göçür O [

40] defined an amply soft set. He named this soft set as an amply soft set, together with its operations, in order to eliminate the complexity by selecting the ones that are suitable for a certain purpose among the previously defined soft sets and the operations between them and redefining otherwise. Amply soft sets use any kind of universal parameter set and initial universe (such as finite or infinite, countable or uncountable). Also, he introduced subset, superset, equality, empty set, whole set about amply soft sets. And he gave operations such as union, intersection, difference of two amply soft sets and complement of an amply soft set. Then he defined three different amply soft points such as amply soft whole point, amply soft point, monad point. He also gave examples related taking universal set as uncountable.

Göçür O. defined a new soft topology, and it is called as a PAS topology in [

40]. The PAS topology allows to write different elements of classical topologies in its each parameter sets. The classical topologies may be finite, infinite, countable or uncountable. This situation removes all of boundaries in a soft topology and cause it to spread over larger areas. A PAS topology is a special case of an AS topology. For this purpose, he defined a new soft topology, and it is called as an amply soft topology or briefly an AS topology. He gave parametric separation axioms which are different from

T_{i} separation axioms.

T_{i} questions the relationship between the elements of space itself while

P_{i} questions the strength of the connection between their parameters.

The main aim of this study is to define monad metrizable spaces. A monad metrizable space may have got any topological spaces and any different metric spaces in each different dimension. The distance in real space between these topologies is computed. First of all, I define passing points between different topologies, and define a monad metric. Then I give a monad metrizable space and a PAS metric space. I show that any PAS metric space is also a monad metrizable space. Some properties and some examples about them are also presented.

## 2. Materials and Methods

These terminologies are used hereafter in the paper: X denotes an initial universe, E denotes a universal set of parameters; $A,B,C$ are subsets of $E$.

**Definition** **1.** Ref [40]. Let $P\left(X\right)$ denote the power set of $X$. If $F:E\to P\left(X\right)$ is a mapping given bythen $F$ with $A$ is called as an amply soft set over $X$ and it is denoted by $F\ast A$. We can say an AS set instead of an amply soft set for briefness. **Example** **1.** Ref [40]. Let $X=\left\{{x}_{1},{x}_{2},{x}_{3},{x}_{4}\right\}$ be a universal set, $E=\left\{{e}_{1},{e}_{2},{e}_{3},{e}_{4},{e}_{5}\right\}$ a set of parameters and $A=\left\{{e}_{1},{e}_{2},{e}_{4},{e}_{5}\right\}$ a subset of $E.$ Let $F:A\to P\left(X\right)$ be the mapping given by $F\left({e}_{1}\right)=\left\{{x}_{1},{x}_{2}\right\},F\left({e}_{2}\right)=\left\{{x}_{2},{x}_{4}\right\},F\left({e}_{4}\right)=\left\{{x}_{3}\right\},F\left({e}_{5}\right)=\left\{{x}_{3}\right\}.$ Then we can show it looks like the following: **Definition** **2.** Ref [40]. Let $F\ast A$ and $G\ast B$ be two amply soft sets over $X.F\ast A$ is subset of $G\ast B,$ denoted by $F\ast A\tilde{\subseteq}G\ast B$,

if $F\left(e\right)\subseteq G\left(e\right),$ for all $e\in A$.

**Definition** **3.** Ref [40]. Let $F\ast A$ and $G\ast B$ be two amply soft sets over $X.F\ast A$ is superset of $G\ast B,$ denoted by $F\ast A\tilde{\supseteq}G\ast B,$ if $F\left(e\right)\supseteq G\left(e\right)$, for all $e\in B.$ **Definition** **4.** Ref [40]. Let $F\ast A$ and $G\ast B$ be two amply soft sets over $X$.

If $F\ast A$ is subset of $G\ast B$ and $G\ast B$ is subset of $F\ast A$ also, then $F\ast A$ and $G\ast B$ are said to be equal and denoted by $F\ast A\tilde{=}G\ast B$.

**Definition** **5.** Ref [40]. An amply soft set $F\ast A$ over $X$ is said to be an empty amply soft set denoted by $\tilde{\varnothing}$ if $F\left(e\right)=\varnothing $ for all $e\in E$.

**Definition** **6.** Ref [40]. An amply soft set $F\ast E$ over $X$ is said to be an absolute amply soft set denoted by $\tilde{X}$ if for all $e\in E,F\left(e\right)=X$.

**Definition** **7.** Ref [40]. The union of two amply soft sets of $F\ast A$ and $G\ast B$ over a common universe $X$ is the amply soft set $H\ast C$, where $C=A\cup B$ and for all $e\in E,$We can write $F\ast A\tilde{\cup}G\ast B\tilde{=}F\cup G\ast A\cup B\tilde{=}H\ast C$.

**Definition** **8.** Ref [40]. The intersection $H\ast C$ of two amply soft sets $F\ast A$ and $G\ast B$ over a common universe $X$ denoted by $F\ast A\tilde{\cap}G\ast B$ is defined as $C=A\cap B$ and for all $e\in E$,

We can write $F\ast A\tilde{\cap}G\ast B\tilde{=}F\cap G\ast A\cap B\tilde{=}H\ast C.$ **Definition** **9.** Ref [40]. The difference H∗A of two amply soft sets $F\ast A$ and $G\ast B$ over $X$ is denoted by $F\ast A\tilde{\backslash}G\ast B$ and it is defined asWe can write $F\ast A\tilde{\backslash}G\ast B\tilde{=}F\backslash G\ast A\tilde{=}H\ast A.$ **Definition** **10.** Ref [40]. Let $F\ast A$ be an amply soft set over $\tilde{X}$.

The complement of an amply soft set $F\ast A$ over $X$ is denoted by $\left(F\ast A\right)\tilde{\prime}\tilde{=}{F}^{\prime}\ast E$ where ${F}^{\prime}:E\to P\left(X\right)$ is a mapping defined as ${F}^{\prime}\left(e\right)=X-F\left(e\right)$ for all $e\in E.$ Note that, $F\prime :E\to P\left(X\right)$ is a mapping given by **Example** **2.** Ref [40]. Let $X=\left\{{h}_{1},{h}_{2},{h}_{3},{h}_{4},{h}_{5}\right\},E=\left\{{e}_{1},{e}_{2},{e}_{3},{e}_{4},{e}_{5}\right\}$ and its subsets $A=\left\{{e}_{1},{e}_{2},{e}_{3}\right\},B=\left\{{e}_{1},{e}_{3},{e}_{4}\right\},C=\left\{{e}_{1},{e}_{4}\right\}$ and $F\ast A,G\ast B,H\ast C$ are amply soft sets over $X$ defined as follows respectively,Then, **Proposition** **1.** Ref [40]. Let $F\ast A,G\ast B$ and $H\ast C$ be amply soft sets over $X;A,B,C\subseteq E$.

Then the following holds. **Definition** **11.** Ref [40]. Let $\left\{a\right\}\subset E$ and let $F\ast \{a$} be an amply soft set over $X,x\in X$.

If $F\ast \left\{a\right\}$ is defined as $F\left(a\right)=\left\{x\right\},$ then $F\ast \left\{a\right\}$ is called as a monad point and it is denoted by ${x}_{a}$.

**Definition** **12.** Ref [40]. Let $a\in A$ and $F\ast A$ be an amply soft set over $X,x\in X$.

We say that ${x}_{a}\tilde{\in}F\ast A$ read as monad point $x$ belongs to the amply soft set $F\ast A$ if $x\in F\left(a\right)$.

**Definition** **13.** Ref [40]. Let $a\in E$ and $F\ast \left\{a\right\}$ be an amply soft set over $X,x\in X$.

We say that ${x}_{a}\tilde{\notin}F\ast \left\{a\right\}$ if $x\notin F\left(a\right).$ **Definition** **14.** Ref [40]. Let $\tilde{\tau}$ be the collection of amply soft sets over $X$,

then $\tilde{\tau}$ is said to be an amply soft topology (or briefly AS topology) on $X$ if, $\varnothing ,X$ belong to $\tilde{\tau}$

The union of any number of amply soft sets in $\tilde{\tau}$ belongs to $\tilde{\tau}$

The intersection of any two amply soft sets in $\tilde{\tau}$ belongs to $\tilde{\tau}$.

The triplet $\left(\tilde{X},\tilde{\tau},E\right)$ is called as an amply soft topological space over $\tilde{X}$.

We will use AS topological space $\tilde{X}$ instead of amply soft topological space $\left(\tilde{X},\tilde{\tau},E\right)$ for shortly.

**Definition** **15.** Ref [40]. Let $\left(\tilde{X},\tilde{\tau},E\right)$ be an AS topological space, then the members of $\tilde{\tau}$ are said to be AS open sets in an AS topological space $\tilde{X}$.

**Definition** **16.** Ref [40]. Let $\left(\tilde{X},\tilde{\tau},E\right)$ be an AS topological space. An amply soft set $F\ast A$ over $\tilde{X}$ is said to be an AS closed set in an AS topological space $\tilde{X}$,

if its complement $\left(F\ast A\right)\tilde{\prime}$ belongs to $\tilde{\tau}$.

**Proposition** **2.** Let $\left(\tilde{X},\tilde{\tau},E\right)$ be an AS topological space. Then

$\tilde{\varnothing},\tilde{X}$ are AS closed sets in AS topological space $\tilde{X}$,

The intersection of any number of AS closed sets is an AS closed set in an AS topological space $\tilde{X}$,

The union of any two AS closed sets is an AS closed set in an AS topological space $\tilde{X}$.

**Proposition** **3.** Ref [40]. Let $\left(\tilde{X,}\tilde{\tau},E\right)$ be an AS topological space. Then the collection ${\tau}_{e}=\left\{F\left(e\right)\right|F\ast E\tilde{\in}\tilde{\tau}\}$ for each $e\in E$, defines topologies on $X$.

**Definition** **17.** Ref [40]. Let $\left(\tilde{X,}\tilde{\tau},E\right)$ be an AS topological space and $\tilde{\beta}\tilde{\subseteq}\tilde{\tau}$. If every element of $\tilde{\tau}$ can be written as any union of elements of $\tilde{\beta}$, then $\tilde{\beta}$ is called as an AS basis for the AS topology $\tilde{\tau}$. Then we can say that each element of $\tilde{\beta}$ is an AS basis element. **Definition** **18.** Ref [40]. Let $n\in \mathbb{N}$,

${e}_{n}\in E$.

Let $\left(X,{\tau}_{n}\right)$ be topological spaces over same initial universe $X$. Let $f$ be mapping from ${\tau}_{n}$ to ${\tau}_{{e}_{n}}$ for all $e\in E$.

Then $\beta =\left\{{\tau}_{{e}_{n}}\right\}$ is an AS basis for an AS topology $\tilde{\tau}$. We can say that it is called as an AS topology produced by classical topology and for shortly a PAS topology. Note that these separation axioms defined as parametric separations in below are different from ${T}_{i}$ separation axioms. ${T}_{i}$ questions the relationship between the elements of space itself while ${P}_{i}$ questions the strength of the connection between their parameters.

**Definition** **19.** Ref [40]. Let $\left(\tilde{X},\tilde{\tau},E\right)$ be an AS topological space and $a,b\in E$ such that $a\ne b$, if there exist $x\in X$ and AS open sets $F\ast A$ and $G\ast B$ such that ${x}_{a}\tilde{\in}F\ast A$ and ${x}_{b}\tilde{\notin}F\ast A$; or ${x}_{b}\tilde{\in}G\ast B$ and ${x}_{a}\tilde{\notin}G\ast B$, then $\left(\tilde{X},\tilde{\tau},E\right)$ is called as a ${P}_{0}$ space. **Definition** **20.** Ref [40]. Let $\left(\tilde{X},\tilde{\tau},E\right)$ be an AS topological space and $a,b\in E$ such that $a\ne b$. If there exist $x\in X$ and AS open sets $F\ast A$ and $G\ast B$ such that ${x}_{a}\tilde{\in}F\ast A$and ${x}_{b}\tilde{\notin}F\ast A$; and ${x}_{b}\tilde{\in}G\ast B$ and ${x}_{a}\tilde{\notin}G\ast B$, then $\left(\tilde{X},\tilde{\tau},E\right)$ is called as a ${P}_{1}$ space. **Theorem** **1.** Ref [40]. Let $\left(\tilde{X},\tilde{\tau},E\right)$ be a ${P}_{1}$ space, then it is also a ${P}_{0}$ space. **Definition** **21.** Ref [40]. Let $\left(\tilde{X},\tilde{\tau},E\right)$ be an AS topological space and $a,b\in E$ such that $a\ne b$. If there exist $x\in X$ and AS open sets $F\ast A$ and $G\ast B$ such that ${x}_{a}\tilde{\in}F\ast A$,${x}_{b}\tilde{\in}G\ast B$ and $F\ast A\tilde{\cap}G\ast B\tilde{=}\tilde{\varnothing}$, then $\left(\tilde{X},\tilde{\tau},E\right)$ is called as a ${P}_{2}$ space. **Theorem** **2.** Ref [40]. Let $\left(\tilde{X},\tilde{\tau},E\right)$ be a ${P}_{2}$ space, then it is also a ${P}_{1}$ space. **Theorem** **3.** Ref [40]. Any PAS topological space $\left(\tilde{X},\tilde{\tau},E\right)$ is a ${P}_{2}$ space. **Example** **3.** Ref [40]. Let ${e}_{1},{e}_{2}\in E,x\in X$ and $\left(X,{\tau}_{{e}_{1}}\right)$ and $\left(X,{\tau}_{{e}_{2}}\right)$ be indiscrete topological spaces over same universe $X$. Let $\left(\tilde{X},\tilde{\tau},E\right)$ be their PAS topology. So; $\tilde{\tau}=\left\{\tilde{\varnothing},\tilde{X},\left\{{X}_{\left\{{e}_{1}\right\}}\right\},\left\{{X}_{\left\{{e}_{2}\right\}}\right\}\right\}$ is an AS topological space over $\tilde{X}$. Therefore there exist AS open sets $\left\{{X}_{\left\{{e}_{1}\right\}}\right\}$ and $\left\{{X}_{\left\{{e}_{2}\right\}}\right\}$such that ${x}_{{e}_{1}}\tilde{\in}\left\{{X}_{\left\{{e}_{1}\right\}}\right\},$ ${x}_{{e}_{2}}\tilde{\in}\left\{{X}_{\left\{{e}_{2}\right\}}\right\}$and $\left\{{X}_{\left\{{e}_{1}\right\}}\right\}\tilde{\cap}\left\{{X}_{\left\{{e}_{2}\right\}}\right\}\tilde{=}\tilde{\varnothing}.$ So $\left(\tilde{X},\tilde{\tau},E\right)$ is a ${P}_{2}$ space.

**Definition** **22.** Ref [40]. Let $\left(\tilde{X},\tilde{\tau},E\right)$ be an AS topological space, $H\ast C$ be an AS closed set, $a\in E,x\in X$ such that ${x}_{a}\tilde{\notin}H\ast C$. If there exist $x\in X$ and AS open sets $F\ast A$and $G\ast B$ such that ${x}_{a}\tilde{\in}F\ast A,H\ast C\tilde{\subseteq}G\ast B$and $F\ast A\tilde{\cap}G\ast B\tilde{=}\tilde{\varnothing}$then $\left(\tilde{X},\tilde{\tau},E\right)$ is called as a Halime space. **Definition** **23.** Ref [40]. Let $\left(\tilde{X},\tilde{\tau},E\right)$be an AS topological space. Then it is said to be a ${P}_{3}$ space if it is both a Halime space and a ${P}_{1}$ space. **Example** **4.** Ref [40]. Let $X=\left\{x,y,z\right\}$ be a universal set, $E=\left\{{e}_{1},{e}_{2},{e}_{3}\right\}$ be a parameter set and let $\tilde{\tau}=\left\{\tilde{\varnothing},\tilde{X},\left\{{\left\{x,y\right\}}_{\left\{{e}_{1}\right\}},{\left\{x,z\right\}}_{\left\{{e}_{2}\right\}},{\left\{y,z\right\}}_{\left\{{e}_{3}\right\}}\right\},\left\{{\left\{z\right\}}_{\left\{{e}_{1}\right\}},{\left\{y\right\}}_{\left\{{e}_{2}\right\}},{\left\{x\right\}}_{\left\{{e}_{3}\right\}}\right\}\right\}$ such that every members be amply soft set $E\to P\left(X\right)$. Then $\left(\tilde{X},\tilde{\tau},E\right)$ is an AS topological space. Let us see if it is a ${P}_{3}$ space. Firstly, let us see if it is a Halime space.

For ${e}_{1}\in E$, choose ${x}_{{e}_{1}}$ and for an AS closed set $\left\{{\left\{z\right\}}_{\left\{{e}_{1}\right\}},{\left\{y\right\}}_{\left\{{e}_{2}\right\}},{\left\{x\right\}}_{\left\{{e}_{3}\right\}}\right\}$ such that ${x}_{{e}_{1}}\tilde{\notin}\left\{{\left\{z\right\}}_{\left\{{e}_{1}\right\}},{\left\{y\right\}}_{\left\{{e}_{2}\right\}},{\left\{x\right\}}_{\left\{{e}_{3}\right\}}\right\}\}$. There exists an AS open set $\left\{{\left\{z\right\}}_{\left\{{e}_{1}\right\}},{\left\{y\right\}}_{\left\{{e}_{2}\right\}},{\left\{x\right\}}_{\left\{{e}_{3}\right\}}\right\}\}$ such that $\left\{{\left\{z\right\}}_{\left\{{e}_{1}\right\}},{\left\{y\right\}}_{\left\{{e}_{2}\right\}},{\left\{x\right\}}_{\left\{{e}_{3}\right\}}\right\}\}\tilde{\subseteq}\left\{{\left\{z\right\}}_{\left\{{e}_{1}\right\}},{\left\{y\right\}}_{\left\{{e}_{2}\right\}},{\left\{x\right\}}_{\left\{{e}_{3}\right\}}\right\}\}$. There exists an AS open set $\left\{{\left\{x,y\right\}}_{\left\{{e}_{1}\right\}},{\left\{x,z\right\}}_{\left\{{e}_{2}\right\}},{\left\{y,z\right\}}_{\left\{{e}_{3}\right\}}\right\}$ such that ${x}_{{e}_{1}}\tilde{\in}\left\{{\left\{x,y\right\}}_{\left\{{e}_{1}\right\}},{\left\{x,z\right\}}_{\left\{{e}_{2}\right\}},{\left\{y,z\right\}}_{\left\{{e}_{3}\right\}}\right\}$ and $\left\{{\left\{x,y\right\}}_{\left\{{e}_{1}\right\}},{\left\{x,z\right\}}_{\left\{{e}_{2}\right\}},{\left\{y,z\right\}}_{\left\{{e}_{3}\right\}}\right\}\tilde{\cap}\left\{{\left\{z\right\}}_{\left\{{e}_{1}\right\}},{\left\{y\right\}}_{\left\{{e}_{2}\right\}},{\left\{x\right\}}_{\left\{{e}_{3}\right\}}\right\}\}\tilde{=}\tilde{\varnothing}$,

For ${e}_{2}\in E$, choose ${x}_{{e}_{2}}$ and for an AS closed set $\left\{{\left\{z\right\}}_{\left\{{e}_{1}\right\}},{\left\{y\right\}}_{\left\{{e}_{2}\right\}},{\left\{x\right\}}_{\left\{{e}_{3}\right\}}\right\}\}$ such that ${x}_{{e}_{2}}\tilde{\notin}\left\{{\left\{z\right\}}_{\left\{{e}_{1}\right\}},{\left\{y\right\}}_{\left\{{e}_{2}\right\}},{\left\{x\right\}}_{\left\{{e}_{3}\right\}}\right\}\}$. There exists an AS open set $\left\{{\left\{z\right\}}_{\left\{{e}_{1}\right\}},{\left\{y\right\}}_{\left\{{e}_{2}\right\}},{\left\{x\right\}}_{\left\{{e}_{3}\right\}}\right\}\}$ such that $\left\{{\left\{z\right\}}_{\left\{{e}_{1}\right\}},{\left\{y\right\}}_{\left\{{e}_{2}\right\}},{\left\{x\right\}}_{\left\{{e}_{3}\right\}}\right\}\}\tilde{\subseteq}\left\{{\left\{z\right\}}_{\left\{{e}_{1}\right\}},{\left\{y\right\}}_{\left\{{e}_{2}\right\}},{\left\{x\right\}}_{\left\{{e}_{3}\right\}}\right\}\}$. There exists an AS open set $\left\{{\left\{x,y\right\}}_{\left\{{e}_{1}\right\}},{\left\{x,z\right\}}_{\left\{{e}_{2}\right\}},{\left\{y,z\right\}}_{\left\{{e}_{3}\right\}}\right\}$ such that ${x}_{{e}_{2}}\tilde{\in}\left\{{\left\{x,y\right\}}_{\left\{{e}_{1}\right\}},{\left\{x,z\right\}}_{\left\{{e}_{2}\right\}},{\left\{y,z\right\}}_{\left\{{e}_{3}\right\}}\right\}$ and $\left\{{\left\{x,y\right\}}_{\left\{{e}_{1}\right\}},{\left\{x,z\right\}}_{\left\{{e}_{2}\right\}},{\left\{y,z\right\}}_{\left\{{e}_{3}\right\}}\right\}\tilde{\cap}\left\{{\left\{z\right\}}_{\left\{{e}_{1}\right\}},{\left\{y\right\}}_{\left\{{e}_{2}\right\}},{\left\{x\right\}}_{\left\{{e}_{3}\right\}}\right\}\}\tilde{=}\tilde{\varnothing},$ For ${e}_{3}\in E$, choose ${z}_{{e}_{3}}$ and for an AS closed set $\left\{{\left\{z\right\}}_{\left\{{e}_{1}\right\}},{\left\{y\right\}}_{\left\{{e}_{2}\right\}},{\left\{x\right\}}_{\left\{{e}_{3}\right\}}\right\}\}$ such that ${z}_{{e}_{3}}\tilde{\notin}\left\{{\left\{z\right\}}_{\left\{{e}_{1}\right\}},{\left\{y\right\}}_{\left\{{e}_{2}\right\}},{\left\{x\right\}}_{\left\{{e}_{3}\right\}}\right\}\}$ There exists an AS open set $\left\{{\left\{z\right\}}_{\left\{{e}_{1}\right\}},{\left\{y\right\}}_{\left\{{e}_{2}\right\}},{\left\{x\right\}}_{\left\{{e}_{3}\right\}}\right\}\}$
such that $\left\{{\left\{z\right\}}_{\left\{{e}_{1}\right\}},{\left\{y\right\}}_{\left\{{e}_{2}\right\}},{\left\{x\right\}}_{\left\{{e}_{3}\right\}}\right\}\left\}\tilde{\subseteq}\left\{{\left\{z\right\}}_{\left\{{e}_{1}\right\}},{\left\{y\right\}}_{\left\{{e}_{2}\right\}},{\left\{x\right\}}_{\left\{{e}_{3}\right\}}\right\}\right\}$. There exists an AS open set $\left\{{\left\{x,y\right\}}_{\left\{{e}_{1}\right\}},{\left\{x,z\right\}}_{\left\{{e}_{2}\right\}},{\left\{y,z\right\}}_{\left\{{e}_{3}\right\}}\right\}$ such that ${z}_{{e}_{3}}\tilde{\in}\left\{{\left\{x,y\right\}}_{\left\{{e}_{1}\right\}},{\left\{x,z\right\}}_{\left\{{e}_{2}\right\}},{\left\{y,z\right\}}_{\left\{{e}_{3}\right\}}\right\}$ and $\left\{{\left\{x,y\right\}}_{\left\{{e}_{1}\right\}},{\left\{x,z\right\}}_{\left\{{e}_{2}\right\}},{\left\{y,z\right\}}_{\left\{{e}_{3}\right\}}\right\}\tilde{\cap}\left\{{\left\{z\right\}}_{\left\{{e}_{1}\right\}},{\left\{y\right\}}_{\left\{{e}_{2}\right\}},{\left\{x\right\}}_{\left\{{e}_{3}\right\}}\right\}\}\tilde{=}\tilde{\varnothing},$

So $\left(\tilde{X},\tilde{\tau},E\right)$ is a Halime space from the Definition 22.

Finally, let us see if it is a ${P}_{1}$ space.

For ${e}_{1},{e}_{2}\in E$, there exist $y\in X$ and AS open sets $\left\{{\left\{x,y\right\}}_{\left\{{e}_{1}\right\}},{\left\{x,z\right\}}_{\left\{{e}_{2}\right\}},{\left\{y,z\right\}}_{\left\{{e}_{3}\right\}}\right\}$ and $\left\{{\left\{z\right\}}_{\left\{{e}_{1}\right\}},{\left\{y\right\}}_{\left\{{e}_{2}\right\}},{\left\{x\right\}}_{\left\{{e}_{3}\right\}}\right\}\}$ such that ${y}_{{e}_{1}}\tilde{\in}\left\{{\left\{x,y\right\}}_{\left\{{e}_{1}\right\}},{\left\{x,z\right\}}_{\left\{{e}_{2}\right\}},{\left\{y,z\right\}}_{\left\{{e}_{3}\right\}}\right\}$ and ${y}_{{e}_{2}}\tilde{\notin}\left\{{\left\{x,y\right\}}_{\left\{{e}_{1}\right\}},{\left\{x,z\right\}}_{\left\{{e}_{2}\right\}},{\left\{y,z\right\}}_{\left\{{e}_{3}\right\}}\right\}$; and ${y}_{{e}_{2}}\in \left\{{\left\{z\right\}}_{\left\{{e}_{1}\right\}},{\left\{y\right\}}_{\left\{{e}_{2}\right\}},{\left\{x\right\}}_{\left\{{e}_{3}\right\}}\right\}\}$ and ${y}_{{e}_{1}}\tilde{\notin}\left\{{\left\{z\right\}}_{\left\{{e}_{1}\right\}},{\left\{y\right\}}_{\left\{{e}_{2}\right\}},{\left\{x\right\}}_{\left\{{e}_{3}\right\}}\right\}\}.$

For ${e}_{1},{e}_{3}\in E$, there exist $x\in X$ and AS open sets $\left\{{\left\{x,y\right\}}_{\left\{{e}_{1}\right\}},{\left\{x,z\right\}}_{\left\{{e}_{2}\right\}},{\left\{y,z\right\}}_{\left\{{e}_{3}\right\}}\right\}$ and $\left\{{\left\{z\right\}}_{\left\{{e}_{1}\right\}},{\left\{y\right\}}_{\left\{{e}_{2}\right\}},{\left\{x\right\}}_{\left\{{e}_{3}\right\}}\right\}\}$ such that ${x}_{{e}_{1}}\tilde{\in}\left\{{\left\{x,y\right\}}_{\left\{{e}_{1}\right\}},{\left\{x,z\right\}}_{\left\{{e}_{2}\right\}},{\left\{y,z\right\}}_{\left\{{e}_{3}\right\}}\right\}$ and ${x}_{{e}_{3}}\tilde{\notin}\left\{{\left\{x,y\right\}}_{\left\{{e}_{1}\right\}},{\left\{x,z\right\}}_{\left\{{e}_{2}\right\}},{\left\{y,z\right\}}_{\left\{{e}_{3}\right\}}\right\}$; and ${x}_{{e}_{3}}\tilde{\in}\left\{{\left\{z\right\}}_{\left\{{e}_{1}\right\}},{\left\{y\right\}}_{\left\{{e}_{2}\right\}},{\left\{x\right\}}_{\left\{{e}_{3}\right\}}\right\}\}$ and ${x}_{{e}_{1}}\tilde{\notin}\left\{{\left\{z\right\}}_{\left\{{e}_{1}\right\}},{\left\{y\right\}}_{\left\{{e}_{2}\right\}},{\left\{x\right\}}_{\left\{{e}_{3}\right\}}\right\}\}$.

For ${e}_{2},{e}_{3}\in E$, there exist $y\in X$ and AS open sets $\left\{{\left\{x,y\right\}}_{\left\{{e}_{1}\right\}},{\left\{x,z\right\}}_{\left\{{e}_{2}\right\}},{\left\{y,z\right\}}_{\left\{{e}_{3}\right\}}\right\}$and $\left\{{\left\{z\right\}}_{\left\{{e}_{1}\right\}},{\left\{y\right\}}_{\left\{{e}_{2}\right\}},{\left\{x\right\}}_{\left\{{e}_{3}\right\}}\right\}\}$ such that ${y}_{{e}_{3}}\tilde{\in}\left\{{\left\{x,y\right\}}_{\left\{{e}_{1}\right\}},{\left\{x,z\right\}}_{\left\{{e}_{2}\right\}},{\left\{y,z\right\}}_{\left\{{e}_{3}\right\}}\right\}$and ${y}_{{e}_{2}}\tilde{\notin}\left\{{\left\{x,y\right\}}_{\left\{{e}_{1}\right\}},{\left\{x,z\right\}}_{\left\{{e}_{2}\right\}},{\left\{y,z\right\}}_{\left\{{e}_{3}\right\}}\right\}$; and ${y}_{{e}_{2}}\tilde{\in}\left\{{\left\{z\right\}}_{\left\{{e}_{1}\right\}},{\left\{y\right\}}_{\left\{{e}_{2}\right\}},{\left\{x\right\}}_{\left\{{e}_{3}\right\}}\right\}\}$ and ${y}_{{e}_{3}}\notin \left\{{\left\{z\right\}}_{\left\{{e}_{1}\right\}},{\left\{y\right\}}_{\left\{{e}_{2}\right\}},{\left\{x\right\}}_{\left\{{e}_{3}\right\}}\right\}\}$.

Therefore, $\left(\tilde{X},\tilde{\tau},E\right)$ is a ${P}_{1}$ space from the Definition 20. $\left(\tilde{X},\tilde{\tau},E\right)$ is a ${P}_{3}$ space because it is both a Halime space and a ${P}_{1}$ space.

**Theorem** **4.** Ref [40]. Let $\left(\tilde{X},\tilde{\tau},E\right)$ be a ${P}_{3}$ space, then it is also a ${P}_{2}$ space. **Definition** **24.** Ref [40]. Let $\left(\tilde{X},\tilde{\tau},E\right)$ be an AS topological space, $H\ast C$ and $K\ast D$ be AS closed sets such that $H\ast C\tilde{\cap}K\ast D=\tilde{\varnothing}$. If there exist AS open sets $F\ast A$ and $G\ast B$ such that $H\ast C\tilde{\subseteq}F\ast A,K\ast D\tilde{\subseteq}G\ast $B and $F\ast A\tilde{\cap}G\ast B\tilde{=}\tilde{\varnothing}$ then $\left(\tilde{X},\tilde{\tau},E\right)$ is called as an Orhan space. **Definition** **25.** Ref [40]. Let $\left(\tilde{X},\tilde{\tau},E\right)$ be an AS topological space. Then it is said to be a ${P}_{4}$ space if it is both an Orhan space and a ${P}_{1}$ space. **Example** **5.** Ref [40]. Let us consider the AS topology $\left(\tilde{X},\tilde{\tau},E\right)$ on the Example 4. Let us see if it is a ${P}_{4}$ space. $\left\{{\left\{x,y\right\}}_{\left\{{e}_{1}\right\}},{\left\{x,z\right\}}_{\left\{{e}_{2}\right\}},{\left\{y,z\right\}}_{\left\{{e}_{3}\right\}}\right\}$and $\left\{{\left\{z\right\}}_{\left\{{e}_{1}\right\}},{\left\{y\right\}}_{\left\{{e}_{2}\right\}},{\left\{x\right\}}_{\left\{{e}_{3}\right\}}\right\}\}$ are disjoint AS closed sets on it. We know that these are also disjoint AS open sets. So $\left(\tilde{X},\tilde{\tau},E\right)$ is an Orhan Space. Also $\left(\tilde{X},\tilde{\tau},E\right)$ is a ${P}_{1}$ space from the Example 4, so $\left(\tilde{X},\tilde{\tau},E\right)$ is a ${P}_{4}$ space from the Definition 25.

**Example** **6.** Ref [40]. Let us consider the PAS topology on the Example 3. Let us see if it is a ${P}_{4}$ space. First, let us see if it is an Orhan space. There exist AS closed sets $\left\{{X}_{\left\{{e}_{1}\right\}}\right\}$ and $\left\{{X}_{\left\{{e}_{2}\right\}}\right\}\}$ such that $\left\{{X}_{\left\{{e}_{1}\right\}}\right\}\tilde{\cap}\left\{{X}_{\left\{{e}_{2}\right\}}\right\}\}\tilde{=}\tilde{\varnothing}$. We know that $\left\{{X}_{\left\{{e}_{1}\right\}}\right\}$ and $\left\{{X}_{\left\{{e}_{2}\right\}}\right\}$ are also AS open sets such that $\left\{{X}_{\left\{{e}_{1}\right\}}\right\}\tilde{\subseteq}\left\{{X}_{\left\{{e}_{1}\right\}}\right\},\left\{{X}_{\left\{{e}_{2}\right\}}\right\}\}\tilde{\subseteq}\left\{{X}_{\left\{{e}_{2}\right\}}\right\}\}$ and $\left\{{X}_{\left\{{e}_{1}\right\}}\right\}\tilde{\cap}\left\{{X}_{\left\{{e}_{2}\right\}}\right\}\}\tilde{=}\tilde{\varnothing}$. So $\left(\tilde{X},\tilde{\tau},E\right)$ is an Orhan space.

Finally, let us see if it is a ${P}_{1}$ space.

There exist AS open sets $\left\{{X}_{\left\{{e}_{1}\right\}}\right\}$ and $\left\{{X}_{\left\{{e}_{2}\right\}}\right\}\}$ such that ${X}_{{e}_{1}}\tilde{\in}\left\{{X}_{\left\{{e}_{1}\right\}}\right\}$, ${X}_{{e}_{2}}\tilde{\in}\left\{{X}_{\left\{{e}_{1}\right\}}\right\}$ and ${x}_{{e}_{2}}\tilde{\in}\left\{{X}_{\left\{{e}_{2}\right\}}\right\},{x}_{{e}_{1}}\tilde{\notin}\left\{{X}_{\left\{{e}_{2}\right\}}\right\}$. So $\left(\tilde{X},\tilde{\tau},E\right)$ is a ${P}_{1}$ space. Because $\left(\tilde{X},\tilde{\tau},E\right)$ is both an Orhan space and a ${P}_{1}$ space, it is a ${P}_{4}$ space.

**Proposition** **4.** Ref [40]. Any PAS topological space $\left(\tilde{X},\tilde{\tau},E\right)$ may not be a ${P}_{4}$ space. **Example** **7.** Ref [40]. Let $\mathbb{R}$ be real numbers, $E=\left\{{e}_{1},{e}_{2}\right\}$ be a universal parameter set, $\left(\mathbb{R},{\tau}_{1}\right)$ be a discrete topological space and $\left(\mathbb{R},{\tau}_{2}\right)$ be a finite complement topological space and from the Definition 18, their PAS topology over$\tilde{\mathbb{R}}$ be $\left(\tilde{\mathbb{R}},\tilde{\tau},E\right).$ Now let us see if $\tilde{\mathbb{R}}$ is a ${\mathrm{P}}_{4}$ space.

Firstly, we can say that the PAS topology is a ${\mathrm{P}}_{1}$ space because of the Theorem 3. and the Theorem 2. Let us see it,

For ${e}_{1},{e}_{2};$ there exist $2\in \mathbb{R}$ and $\left\{{\left\{2\right\}}_{\left\{{e}_{1}\right\}},{\left(\mathbb{R}-\left\{2,3,5\right\}\right)}_{\left\{{e}_{2}\right\}}\right\},\left\{{\left\{3\right\}}_{\left\{{e}_{1}\right\}},{\left(\mathbb{R}-\left\{3,5\right\}\right)}_{\left\{{e}_{2}\right\}}\right\}\tilde{\in}\tilde{\tau}$ such that ${2}_{{e}_{1}}\tilde{\in}\left\{{\left\{2\right\}}_{\left\{{e}_{1}\right\}},{\left(\mathbb{R}-\left\{2,3,5\right\}\right)}_{\left\{{e}_{2}\right\}}\right\}$ and ${2}_{{e}_{2}}\tilde{\notin}\left\{{\left\{2\right\}}_{\left\{{e}_{1}\right\}},{\left(\mathbb{R}-\left\{2,3,5\right\}\right)}_{\left\{{e}_{2}\right\}}\right\}$; ${2}_{{e}_{2}}\tilde{\in}\left\{{\left\{3\right\}}_{\left\{{e}_{1}\right\}},{\left(\mathbb{R}-\left\{3,5\right\}\right)}_{\left\{{e}_{2}\right\}}\right\}$ and ${2}_{{e}_{1}}\tilde{\notin}\left\{{\left\{3\right\}}_{\left\{{e}_{1}\right\}},{\left(\mathbb{R}-\left\{3,5\right\}\right)}_{\left\{{e}_{2}\right\}}\right\}$. Therefore, it is clearly seen that $\left(\tilde{\mathbb{R}},\tilde{\tau},E\right)$ is a ${P}_{1}$ space.

Finally, let us see if $\tilde{\mathbb{R}}$ is an Orhan space. Let $U,V\tilde{\subseteq}\mathbb{R}$ be finite sets. Then we can choose $\left\{{U}_{\left\{{e}_{2}\right\}}\right\}$ and $\left\{{V}_{\left\{{e}_{2}\right\}}\right\}$ AS closed sets such that $\left\{{U}_{\left\{{e}_{2}\right\}}\right\}\tilde{\cap}\left\{{V}_{\left\{{e}_{2}\right\}}\right\}\tilde{=}\tilde{\varnothing}$ But for $A,B\subseteq E$, we cannot find any AS open sets $F\ast A$ and $G\ast B$ such that $\left\{{U}_{\left\{{e}_{2}\right\}}\right\}\tilde{\subseteq}F\ast A,\left\{{V}_{\left\{{e}_{2}\right\}}\right\}\tilde{\subseteq}G\ast B$ and $F\ast A\tilde{\cap}G\ast B\tilde{=}\tilde{\varnothing}.$ For this purpose, suppose that, there exist $\left\{{\left(R-U\right)}_{\left\{{e}_{2}\right\}}\right\},\left\{{\left(R-V\right)}_{\left\{{e}_{2}\right\}}\right\}$ be AS open sets such that $\left\{{V}_{\left\{{e}_{2}\right\}}\right\}\tilde{\subseteq}\left\{{\left(R-U\right)}_{\left\{{e}_{2}\right\}}\right\}$, $\left\{{U}_{\left\{{e}_{2}\right\}}\right\}\tilde{\subseteq}\left\{{\left(R-V\right)}_{\left\{{e}_{2}\right\}}\right\}$ and $\left\{{\left(R-U\right)}_{\left\{{e}_{2}\right\}}\right\}\tilde{\cap}\left\{{\left(R-V\right)}_{\left\{{e}_{2}\right\}}\right\}\tilde{=}\tilde{\varnothing}$. Therefore, we can obtain $U\cup V=\mathbb{R}$. This result contradicts the finite selection of $U$ and $V$. Hence, $\left(\tilde{\mathbb{R}},\tilde{\tau},E\right)$ is not an Orhan space and so it is not a ${P}_{4}$ space.

**Theorem** **5.** Ref [40]. Let $\left(\tilde{X},\tilde{\tau},E\right)$ be a ${P}_{4}$space, then it is also a ${P}_{3}$ space. **Conclusion** **1.** Ref [40]. Any AS topological space $\left(\tilde{X},\tilde{\tau},E\right)$ is a ${P}_{4}$ space $\u27f9{P}_{3}$ space$\u27f9{P}_{2}$ space$\u27f9{P}_{1}$ space$\u27f9{P}_{0}$space.