Two New Families of Supra-Soft Topological Spaces Defined by Separation Axioms

Abstract

This paper contributes to the field of supra-soft topology. We introduce and investigate supra $pp$-soft $T_j$ and supra $pt$-soft $T_j$-spaces $(j=0,1,2,3,4)$. These are defined in terms of different ordinary points; they rely on partial belong and partial non-belong relations in the first type, and partial belong and total non-belong relations in the second type. With the assistance of examples, we reveal the relationships among them as well as their relationships with classes of supra-soft topological spaces such as supra $tp$-soft $T_j$ and supra $tt$-soft $T_j$-spaces $(j=0,1,2,3,4)$. This work also investigates both the connections among these spaces and their relationships with the supra topological spaces that they induce. Some connections are shown with the aid of examples. In this regard, we prove that for $i=0,1$, possessing the $T_i$ property by a parametric supra-topological space implies possessing the $pp$-soft $T_i$ property by its supra-soft topological space. This relationship is invalid for the other types of soft spaces introduced in previous literature. We derive some results of $pp$-soft $T_i$-spaces from the cardinality numbers of the universal set and a set of parameters. We also demonstrate how these spaces behave as compared to their counterparts studied in soft topology and its generalizations (such as infra-soft topologies and weak soft topologies). Moreover, we investigated whether subspaces, finite product spaces, and soft $S^{🟉}$-continuous mappings preserve these axioms.

1. Introduction

This article is in continuation of earlier contributions in a fruitful area of research merging topology and soft set theory. The hybridization of the particular extension called supratopology with soft sets has produced an interesting structure called supra-soft topology, which is the exact focus for our research. Firstly, the state of the art in this field will be briefly summarized.

In order to characterize situations containing uncertainties, the structure called `soft set’ [1] was launched in 1999. Soon afterwards, some operations on soft sets (such as their union and intersection) and operators (like complements of soft sets) were formulated in [2]. This paper defined the null and absolute soft sets, which act as a soft counterpart of the universal and empty crisp sets, respectively. As to their applicability, Maji et al. [3], later improved by [4], succeeded in using soft sets for decision making. Despite the shortcomings of some results and concepts in these seminal works, they form the foundational basis of soft set theory, together with a remarkable recent addition: their semantical interpretation [5]. Actually, the literature has produced many types of operations between soft sets. More opportunities to make full use of soft set theory have been provided in both theoretical [6,7,8] and applied studies (in computer science [9], medical science [10,11], computational biology [12], etc.).

In addition, powerful mathematical structures have been exported to the soft set field [13], thus emphasizing its importance for the analysis of abstract theories. Particularly, topological notions have been hybridized with soft set theory since 2011. This year, Shabir and Naz [14] proposed the concept of soft topology. Zorlutuna et al. [15] set forth the idea of soft point. This is a good tool for the investigation of properties of soft interior points and soft neighborhood systems, and in fact, it was independently reformulated by both [16] and [17]. While [16] used the new phrasing of soft point to investigate soft metric spaces, Ref. [17] employed it to study soft neighborhood systems and uncover some relations of the soft limit points of a soft set. The comparative performance of soft topologies and standard topologies has been the subject of studies such as [18,19]. Recently, abstract soft topological concepts such as compactness, separation axioms, and generalized open sets were employed to address practical problems in the areas of economic [20] and nutrition systems [21].

In 1983, Mashhour et al. [22] proposed the concept of supratopological spaces. In their model, supratopologies are collections of subsets generalizing the axiomatics for a topological space by dispensing with the postulate that the collection is closed under finite intersections. Then, El-Sheikh and Abd El-Latif [23] exported that model to soft set theory in 2014, and they conceived of the concept of supra-soft topology. Quite naturally, supra-soft topological spaces contain soft topological spaces. Several researchers studied essential notions related to the new structure, inclusive of supra-soft continuity [23], new generalizations of supra-soft open sets, and various classes of supra-soft separation axioms [24,25]. By utilizing soft stacks, Ref. [26] conceived a supra-soft topology from a soft topology. However, we note that the research of the fundamental concepts and notions of supra-soft topological spaces has not yet received the required consideration. As a result, a number of interesting notions in this structure still need to be formulated and properly discussed. We contribute to this area with an original inspection of the class of `separation axioms’ that are meaningful for the investigation of supra-soft topology.

Several arguments justify the study of topological concepts within the frame of supra-topology. First, this setting suffices to preserve some topological characteristics and properties under conditions that do not require a topology; for example, the supra-interior and supra-closure points of a set are, respectively, still supra-open and supra-closed sets; a supra-closed subset of a supra-compact space is supra-compact, etc. Second, studying topological ideas via supra-topology produces a richer variety of concepts, especially over a finite set; for instance, the only $T_1$-topology defined on a finite set is the discrete topology (which is a trivial case, and hence meaningless in application areas), whereas there are several sorts of supra-topologies that produce $T_1$-spaces. Furthermore, the supra-topological frames show the easiness and diversity of building examples that satisfy supra $T_i$-spaces compared with their counterparts on classical topology—especially those related to strong types of separation axioms. Third, supra-topology provides an appropriate environment to describe many real-life problems, which can be noted via rough set approximation operators produced by topological approaches [27]. To illustrate this point, notice that most generalizations of open sets form a supra topology but not a topology (for they are only closed under arbitrary union), which affects the performance and properties of lower and upper approximation operators in rough set theory. It is worth noting that the abstract and theoretical extension of topology called “infra-topology” was also applied in the analysis of information systems by the theory of rough sets [27].

Soft separation axioms stand out among the most relevant characteristics of soft topology. This is also the case in point-set topology. With the help of these properties, one can establish the structured categories of well-behaved soft topological or just topological spaces. A large variety of soft separation axioms have been proposed (and some facts have been corrected [28]), which can be ascribed to the factors of two kinds. Firstly, the objects that we want to separate: one can opt for either ordinary points or soft points. Secondly, the belongingness and non-belongingness relations used in the definitions: here, one can opt for either partial or total versions. In this regard, we emphasize the fact that El-shafei et al. [28] introduced new types of relations between ordinary points and soft sets, namely partial belong and total non-belong relations; they add to the relations given in the foundational [14]. Further information on this issue can be found in [14,28,29,30,31,32]. Likewise, in this paper, we investigate separation axioms in the context of supra-soft topology beyond the pioneering analysis in Al-shami and El-Shafei [33]. The two types of supra-soft separation axioms defined herein represent new classes of supra-soft topological spaces wider than those given in [33]. The relationships among these classes of soft axioms are elucidated (cf., Propositions 5, 6 and 16 plus some interesting counterexamples). We note that using a partial belong relation in the definitions of new two types of supra-soft separation axioms makes initiating examples that more easily prove certain relations among topological concepts. One of the divergences between them is the sufficient conditions that guarantee the existence of some soft spaces; for example, supra $tt$-soft $T_1$-space implies that every soft point is supra-soft closed, as proven in [33], whereas this characteristic does not hold for spaces of supra $pp$-soft $T_1$ and supra $pt$-soft $T_1$ introduced herein (cf., Proposition 17 and Example 12).

This article is organized as follows. We mention some definitions and properties that should help the reader understand this research in Section 2. Then, Section 3 defines the concepts of supra $pp$-soft $T_j$-spaces $(j=0,1,2,3,4)$, and investigates their main properties. In Section 4, we formulate the concepts of supra $pt$-soft $T_j$-spaces $(j=0,\dots ,4)$. Then, we disclose some relationships among them, as well as with respect to supra $pt$-soft $T_j$-spaces $(j=0,\dots ,4)$. To end this paper, Section 5 contains some conclusions and hints at some directions for future research.

2. Preliminary Concepts

This section recalls some notions that we need in this paper.

The notation $2^{\mathcal{O}}$ captures the set of all subsets of $\mathcal{O}$.

2.1. Elements of Soft-Set Theory

Definition 1

([1]). The pair $(G,\mathcal{M})$ is a soft set over $\mathcal{O}$, a non-empty set, when G is a mapping from the set of parameters $\mathcal{M}$ into $2^{\mathcal{O}}$.

To avoid cumbersome notation, henceforth, a soft set is symbolized by $G_{\mathcal{M}}$ instead of $(G,\mathcal{M})$. For convenience, it is abbreviated by the notation $G_{\mathcal{M}}=\{(m,G(m)):m\in \mathcal{M}$ and $G(m)\in 2^{\mathcal{O}}\}$.

The collection of all soft sets over $\mathcal{O}$ when the set of parameters is $\mathcal{M}$ will be expressed by $S({\mathcal{O}}_{\mathcal{M}})$.

The standard operations on sets were adapted to soft-set theory. For example, complementarity is defined as follows:

Definition 2

([7]). The complement of the soft set $G_{\mathcal{M}}$, symbolized by $G_{\mathcal{M}}^c$, is the soft set $G_{\mathcal{M}}^c={(G^c)}_{\mathcal{M}}$ with $G^c:\mathcal{M}\to 2^{\mathcal{O}}$ defined as $G^c(m)=\mathcal{O}\setminus G(m)$ for every $m\in \mathcal{M}$.

Extreme examples of soft sets include the following instances:

Definition 3

([2]). The null soft set over $\mathcal{O}$, symbolized by $\tilde{\varphi }$, is the soft set that satisfies $\varphi (m)=\varnothing$ for every $m\in \mathcal{M}$. Its complement is called the absolute soft set, and it will be denoted as $\tilde{\mathcal{O}}$.

Definition 4

([7]). Let $G_{\mathcal{M}}$ and $F_{\mathcal{N}}$ be two soft sets over $\mathcal{O}$. Their intersection, symbolized by $G_{\mathcal{M}}\tilde{\cap }{F}_{\mathcal{N}}$, is a soft set $H_{\mathcal{A}}$, where $\mathcal{A}=\mathcal{M}\cap \mathcal{N}\ne \varnothing$, and the mapping $H:\mathcal{A}\to 2^{\mathcal{O}}$ is given by $H(a)=G(a)\cap F(a)$ for each $a\in \mathcal{A}$.

Definition 5

([2]). Let $G_{\mathcal{M}}$ and $F_{\mathcal{N}}$ be two soft sets over $\mathcal{O}$. Their union of two soft sets $G_{\mathcal{M}}$ and $F_{\mathcal{N}}$ over $\mathcal{O}$, symbolized by $G_{\mathcal{M}}\tilde{\cup }{F}_{\mathcal{N}}$, is a soft set $H_{\mathcal{A}}$, where $\mathcal{A}=\mathcal{M}\cup \mathcal{N}$ and a mapping $H:\mathcal{A}\to 2^{\mathcal{O}}$ is given as follows:

$$H(a)=\left\{\begin{array}{ccc}\hfill G(a)& :& \hfill a\in \mathcal{M}\setminus \mathcal{N}\\ \hfill F(a)& :& \hfill a\in \mathcal{N}\setminus \mathcal{M}\\ \hfill G(a)\cup F(a)& :& \hfill a\in \mathcal{M}\cap \mathcal{N}\end{array}\right.$$

Next, we recall the formal meaning of soft subsethood and soft equality:

Definition 6

([34]). Let $F_{\mathcal{N}}$ and $G_{\mathcal{M}}$ be soft sets. Then, $G_{\mathcal{M}}$ is a soft subset of $F_{\mathcal{N}}$, represented as $G_{\mathcal{M}}\tilde{\subseteq }{F}_{\mathcal{N}}$, when we have both $\mathcal{M}\subseteq \mathcal{N}$ and $G(m)\subseteq F(m)$ for each $m\in \mathcal{M}$.

We say that $G_{\mathcal{M}}$ and $F_{\mathcal{N}}$ are soft equal when both $G_{\mathcal{M}}\tilde{\subseteq }{F}_{\mathcal{N}}$ and $F_{\mathcal{N}}\tilde{\subseteq }{G}_{\mathcal{M}}$ hold true.

Soft points are an important element for the discussion of many other concepts:

Definition 7

([16,17]). A soft point $B_{\mathcal{M}}$ over $\mathcal{O}$ is a soft set for which there is $m\in \mathcal{M}$ with the property that $B(m)$ is a singleton $\left\{k\right\}$, and $B(m^{\prime })$ is the empty set for each $m^{\prime }\ne m$. This soft point will be briefly symbolized by $B_m^k$.

In addition to the null/absolute soft sets and to soft points, other basic types of soft sets are given by the next two related definitions:

Definition 8

([14]). The soft set over $\mathcal{O}$ denoted as $k_{\mathcal{M}}$ is such that $k(m)=\left\{k\right\}$ for each $m\in \mathcal{M}$.

Definition 9

([28]). A stable soft set over $\mathcal{O}$ is a soft set such that all their components are equal.

Notice that, by definition, each $k_{\mathcal{M}}$ is a stable soft set, its components being equal to the singleton $\left\{k\right\}$.

Definition 10

([29]). The product of $G_{\mathcal{M}}$ and $H_{\mathcal{N}}$, denoted by ${(G\times H)}_{\mathcal{M}\times \mathcal{N}}$, is the soft set such that $(G\times H)(m,n)=G(m)\times H(n)$ when $(m,n)\in \mathcal{M}\times \mathcal{N}$.

Soft mappings and some of their special features are recalled in the next two definitions:

Definition 11

([17]). A soft mapping from $S({\mathcal{O}}_{\mathcal{M}})$ to $S(Y_{\mathcal{N}})$ is a pair of mappings $(f,\varphi )$, alternatively represented as $f_{\varphi }$, with $f:\mathcal{O}\to Y$, $\varphi :\mathcal{M}\to \mathcal{N}$. Fix $G_A$ and $H_C$, respective subsets of $S({\mathcal{O}}_{\mathcal{M}})$ and $S(Y_{\mathcal{N}})$. The image of $G_A$ and the pre-image of $H_C$ are

(i) 

$f_{\varphi }(G_A)=(f(G),\mathcal{N})$, the soft set in $S(Y_{\mathcal{N}})$ that satisfies

$$f(G)(n)=\left\{\begin{array}{cc}\hfill {\tilde{\cup }}_{m\in {\varphi }^{-1}(n)\cap A}f(G(m)):& \hfill {\varphi }^{-1}(n)\ne \varnothing \\ \hfill \varnothing :& \hfill {\varphi }^{-1}(n)=\varnothing \end{array}\right.$$

for each $n\in \mathcal{N}$.

(ii) 

$f_{\varphi }^{-1}(H_N)=(f^{-1}(H),\mathcal{M})$, the soft set in $S({\mathcal{O}}_{\mathcal{M}})$ that satisfies

$$f^{-1}(H)(m)=\left\{\begin{array}{cc}\hfill f^{-1}(H(\varphi (m))):& \hfill \varphi (m)\in C\\ \hfill \varnothing :& \hfill \varphi (m)\notin C\end{array}\right.$$

for each $m\in \mathcal{M}$.

Definition 12

([17]). We say that the soft mapping $f_{\varphi }:S({\mathcal{O}}_{\mathcal{M}})\to S(Y_{\mathcal{N}})$ is injective (respectively, surjective or bijective) when both mappings f and ϕ are injective (respectively, surjective or bijective).

Then, we recall some belongingness relations between points in the reference set $\mathcal{O}$ and soft sets over $\mathcal{O}$:

Definition 13

([14,28]). Fix $k\in \mathcal{O}$ and a soft set $G_{\mathcal{M}}$ over $\mathcal{O}$. Then, we represent:

(i) 

$k\in G_{\mathcal{M}}$ (it reads as k totally belongs to $G_{\mathcal{M}}$) if $k\in G(m)$ for all $m\in \mathcal{M}$.

(ii) 

$k\notin G_{\mathcal{M}}$ (it reads as k does not partially belong to $G_{\mathcal{M}}$) when there is $m\in \mathcal{M}$ with $k\notin G(m)$.

(iii) 

$k⋐G_{\mathcal{M}}$ (it reads as k partially belongs to $G_{\mathcal{M}}$) when there is $m\in \mathcal{M}$ with $k\in G(m)$.

(iv) 

$k\cancel{⋐}{G}_{\mathcal{M}}$ (it reads as k, does not totally belong to $G_{\mathcal{M}}$) when $k\notin G(m)$ for all $m\in \mathcal{M}$.

The behavior of these relations under some types of soft mappings is described in the next proposition.

Proposition 1

([28]). Consider a soft mapping $g_{\varphi }:S({\mathcal{O}}_{\mathcal{M}})\to S(Y_{\mathcal{N}})$ and let $G_{\mathcal{M}}$ and $H_{\mathcal{N}}$ be respective soft sets in $S({\mathcal{O}}_{\mathcal{M}})$ and $S(Y_{\mathcal{N}})$. The following results are true:

(i) 

If $k⋐G_{\mathcal{M}}$, then $g(p)⋐g_{\varphi }(G_{\mathcal{M}})$.

(ii) 

If $k\cancel{⋐}{G}_{\mathcal{M}}$ and g is injective, then $g(p)\cancel{⋐}{g}_{\varphi }(G_{\mathcal{M}})$.

(iii) 

If $k\notin G_{\mathcal{M}}$ and $g_{\varphi }$ is injective, then $g(p)\notin g_{\varphi }(G_{\mathcal{M}})$.

(iv) 

If $q⋐H_{\mathcal{N}}$ and ϕ is surjective, then $k\in g^{-1}(q)$ implies $k⋐g_{\varphi }^{-1}(H_{\mathcal{N}})$.

(v) 

If $q\cancel{⋐}{H}_{\mathcal{N}}$, then $k\in g^{-1}(y)$ implies $k\cancel{⋐}{g}_{\varphi }^{-1}(H_{\mathcal{N}})$.

(vi) 

If $q\notin H_{\mathcal{N}}$ and ϕ is surjective, then $k\in g^{-1}(q)$ implies $k\notin g_{\varphi }^{-1}(H_{\mathcal{N}})$.

2.2. Elements of Supra-Soft Topology

Now, we proceed to give some introduction to the main notions in supra-soft topological spaces. We begin with the fundamental concept in this field that owes to the idea of a supratopological space:

Definition 14

([22,35]). A supratopology on $\mathcal{O}$ is a collection μ of subsets of $\mathcal{O}$ that is closed under arbitrary unions and satisfies $\mathcal{O}\in \mu$.

When exported to a soft topological context, we obtain:

Definition 15

([23,24]). Let us fix a set of parameters $\mathcal{M}$. A collection δ of soft sets over $\mathcal{O}$ under $\mathcal{M}$ is a supra-soft topology on $\mathcal{O}$ when $\tilde{\mathcal{O}}$ is a member of δ and the collection is closed under arbitrary soft unions.

We say in this case that the triple $(\mathcal{O},\delta ,\mathcal{M})$ is a supra-soft topological space (briefly, supra-soft${}_{TS}$).

We call the members of δ supra-soft open sets; and their complements are called supra-soft closed sets.

The concepts above are related by the following property:

Proposition 2.

Let $(\mathcal{O},\delta ,\mathcal{M})$ be a supra-soft${}_{TS}$. Then, for each $m\in \mathcal{M}$, the collection ${\delta }_m=\{G(m):G_{\mathcal{M}}\in \delta \}$ is a supra topology on $\mathcal{O}$.

Interiors and closures are naturally defined in the case of supra-soft${}_{TSs}$. Additionally, their basic features are easy to prove.

Definition 16

([36]). Fix a soft subset $H_{\mathcal{M}}$ of $(\mathcal{O},\delta ,\mathcal{M})$. Then, $In{t}_{\delta }(H_{\mathcal{M}})$ is the soft union of all the supra-soft open sets that are soft subsets of $H_{\mathcal{M}}$, and $C{l}_{\delta }(H_{\mathcal{M}})$ is the soft intersection of all the supra-soft closed sets such that $H_{\mathcal{M}}$ is a soft subset of them.

Theorem 1

([36]). Suppose that $H_{\mathcal{M}}$ and $F_{\mathcal{M}}$ are two soft subsets of $(\mathcal{O},\delta ,\mathcal{M})$. The following properties hold true:

(i) 

If $H_{\mathcal{M}}\tilde{\subseteq }{F}_{\mathcal{M}}$, then $C{l}_{\delta }(H_{\mathcal{M}})\tilde{\subseteq }C{l}_{\delta }(F_{\mathcal{M}})$.

(ii) 

$B_m^k\in C{l}_{\delta }(H_{\mathcal{M}})$ if and only if $G_{\mathcal{M}}\tilde{\cap }{H}_{\mathcal{M}}\ne \tilde{\varphi }$ for every supra-soft open set $G_{\mathcal{M}}$ containing $B_m^k$.

Relative topologies are defined in the following terms:

Definition 17

([33]). Suppose that Y is a non-empty subset of $\mathcal{O}$. Let $(\mathcal{O},\delta ,\mathcal{M})$ be a supra-soft${}_{TS}$. The collection ${\delta }_Y=\{\tilde{Y}\tilde{\cap }{G}_{\mathcal{M}}:G_{\mathcal{M}}\in \delta \}$ is the supra-soft relative topology on Y. We also say that $(Y,{\delta }_Y,\mathcal{M})$ is a supra-soft subspace of $(\mathcal{O},\delta ,\mathcal{M})$.

A suitably defined idea of `homeomorphism’ allows one to conceive of properties that are supra-soft topological:

Definition 18

([33]). A soft mapping $f_{\varphi }:(\mathcal{O},\delta ,\mathcal{M})\to (Y,\upsilon ,\mathcal{N})$ is:

(i) 

Soft $S^{🟉}$-continuous when $f_{\varphi }^{-1}(G_{\mathcal{N}})$ is a supra-soft open set for each $G_{\mathcal{N}}\in \upsilon$.

(ii) 

Soft $S^{🟉}$-open (respectively, soft $S^{🟉}$-closed) when $f_{\varphi }(F_{\mathcal{M}})$ is a supra-soft open set (respectively, supra-soft closed set) for each $F_{\mathcal{M}}\in \delta$ (respectively, $F_{\mathcal{M}}^c\in \delta$).

A bijective soft mapping is called a soft $S^{🟉}$-homeomorphism provided it is soft $S^{🟉}$-continuous and soft $S^{🟉}$-open. A property that is preserved by a soft $S^{🟉}$-homeomorphisms is called a supra-soft topological property [33].

Finally, in this section, we recall some separation axioms pertaining to this context:

Definition 19

([33]). The supra-soft${}_{TS}$$(\mathcal{O},\delta ,\mathcal{M})$ is supra-soft normal provided that any two disjoint supra-soft closed sets can be separated by disjoint supra-soft open sets.

Definition 20

([33]). The supra-soft${}_{TS}$$(\mathcal{O},\delta ,\mathcal{M})$ is called:

(i) 

Supra $tp$-soft $T_0$ if there exists a member $G_{\mathcal{M}}\in \delta$ for every $k\ne q\in \mathcal{O}$ that satisfies $k\in G_{\mathcal{M}},q\notin G_{\mathcal{M}}$, or $q\in G_{\mathcal{M}},k\notin G_{\mathcal{M}}$.

(ii) 

Supra $tp$-soft $T_1$ if there exist two members $G_{\mathcal{M}},F_{\mathcal{M}}\in \delta$ for every $k\ne q\in \mathcal{O}$ that satisfy $k\in G_{\mathcal{M}},q\notin G_{\mathcal{M}}$, and $q\in F_{\mathcal{M}},k\notin F_{\mathcal{M}}$.

(iii) 

Supra $tp$-soft $T_2$ if there exist disjoint members $G_{\mathcal{M}},F_{\mathcal{M}}\in \delta$ for every $k\ne q\in \mathcal{O}$ that satisfy $k\in G_{\mathcal{M}},q\notin G_{\mathcal{M}}$, and $q\in F_{\mathcal{M}},k\notin F_{\mathcal{M}}$.

(iv) 

Supra $tp$-soft regular when, for each supra-soft closed set $H_{\mathcal{M}}$ such that $k\notin H_{\mathcal{M}}$, disjoint supra-soft open sets $G_{\mathcal{M}}$ and $F_{\mathcal{M}}$ exist with the properties $H_{\mathcal{M}}\tilde{\subseteq }{G}_{\mathcal{M}}$ and $a\in F_{\mathcal{M}}$.

(v) 

Supra $tp$-soft $T_3$ (respectively, a supra $tp$-soft $T_4$) when it is both supra $tp$-soft regular (respectively, supra-soft normal) and supra $tp$-soft $T_1$.

Whilst the concepts of supra $tp$-soft $T_j$-spaces and supra $tt$-soft $T_j$-spaces were studied in [33] under the name of “supra-soft $T_i$-spaces” and “supra p-soft $T_i$-spaces”, respectively, we believe that the adapted terminology that we employ here, namely supra $tt$-soft regular and supra $tt$-soft $T_j$-spaces, is better, because the previous two types of soft separation axioms and the two types introduced in this study are distinguished in terms of the types of belongingness and non-belongingness relations between ordinary points and supra-soft open and supra-soft closed sets. Furthermore, supra $tt$-soft regular and supra $tt$-soft $T_j$-spaces were defined in [33] as an alternative to the corresponding $tt$-version in the above definition. They are, respectively, stated by a routine replacement of ∉ in Definition 20 by the relation $\cancel{⋐}$.

Proposition 3

([33]). The following statements hold true:

(i) 

Supra $tp$-soft $T_j$-spaces are supra $tp$-soft $T_{j-1}$ for $j=1,2,3$.

(ii) 

Supra $tt$-soft $T_j$-spaces are supra $tt$-soft $T_{j-1}$ for $j=1,2,3,4$.

(iii) 

Supra $tt$-soft $T_j$-spaces are supra $tp$-soft $T_j$ for $j=0,1,2,4$; and supra $tp$-soft $T_3$-spaces are supra $tt$-soft $T_3$.

Remark 1.

The spaces of supra $tp$-soft $T_3$ and supra $tp$-soft $T_4$ are independent of each other [33].

Proposition 4

([33]). Supra $tp$-soft regular spaces are stable.

An equivalence between the concepts of extended (cf., Definition 21 below) and enriched soft topology was proven in [18], which also made use of this technique to establish many results that associated a soft topology with its parametric topologies. We can now define an extended supra-soft topology in a similar way:

Definition 21.

A supra-soft topology δ on $\mathcal{O}$ under a set of parameters $\mathcal{M}$ is called an extended supra-soft topology if $\delta =\{G_{\mathcal{M}}:G(m)\in {\delta }_m$ for each $m\in \mathcal{M}\}$, where ${\delta }_m$ is a parametric supra topology on $\mathcal{O}$.

3. Supra $\mathit{pp}$-Soft ${\mathit{T}}_{\mathit{j}}$-Spaces $(\mathit{j}=\mathbf{0},\mathbf{1},\mathbf{2},\mathbf{3},\mathbf{4})$

In this section, we introduce the novel separation axioms that jointly form the family of supra $pp$-soft $T_j$-spaces. We do this in our next definition:

Definition 22.

The supra-soft${}_{TS}$$(\mathcal{O},\delta ,\mathcal{M})$ is called:

(i) 

Supra $pp$-soft $T_0$ if there exists a member $G_{\mathcal{M}}\in \delta$ for every $k\ne q\in \mathcal{O}$ that satisfies $k⋐G_{\mathcal{M}},q\notin G_{\mathcal{M}}$, or $q⋐G_{\mathcal{M}},k\notin G_{\mathcal{M}}$.

(ii) 

Supra $pp$-soft $T_1$ if there exist two members $G_{\mathcal{M}},F_{\mathcal{M}}\in \delta$ for every $k\ne q\in \mathcal{O}$ that satisfy $k⋐G_{\mathcal{M}},q\notin G_{\mathcal{M}}$, and $q⋐F_{\mathcal{M}},k\notin F_{\mathcal{M}}$.

(iii) 

Supra $pp$-soft $T_2$ if there exist disjoint members $G_{\mathcal{M}},F_{\mathcal{M}}\in \delta$ for every $k\ne q\in \mathcal{O}$ that satisfy $k⋐G_{\mathcal{M}},q\notin G_{\mathcal{M}}$, and $q⋐F_{\mathcal{M}},k\notin F_{\mathcal{M}}$.

(iv) 

Supra $pp$-soft regular when for each supra-soft closed set $H_{\mathcal{M}}$ such that $k\notin H_{\mathcal{M}}$, disjoint supra-soft open sets $G_{\mathcal{M}}$ and $F_{\mathcal{M}}$ exist with the properties $H_{\mathcal{M}}\tilde{\subseteq }{G}_{\mathcal{M}}$ and $k⋐F_{\mathcal{M}}$.

(v) 

Supra $pp$-soft $T_3$ (respectively, supra $pp$-soft $T_4$) if it is supra $pp$-soft regular (respectively, supra-soft normal) and supra $pp$-soft $T_1$.

Firstly, in this section, we reveal the relationship between these separation axioms as well as their relationships with supra $tt$-soft $T_j$ and supra $tp$-soft $T_j$-spaces.

Proposition 5.

(i) 

Every supra $pp$-soft $T_j$-space $(\mathcal{O},\delta ,\mathcal{M})$ is supra $pp$-soft $T_{j-1}$ for $j=1,2,3$.

(ii) 

Every supra $tp$-soft $T_j$-space $(\mathcal{O},\delta ,\mathcal{M})$ is supra $pp$-soft $T_j$ for $j=0,1,2,3,4$.

Proof. 

We first prove (i). The proofs of the two cases $j=1$ and $j=2$ come directly from the definition above. To prove the case of $j=3$, let $k\ne q\in \mathcal{O}$. Since $(\mathcal{O},\delta ,\mathcal{M})$ is supra $pp$-soft $T_1$, then we have two members $U_{\mathcal{M}}$ and $V_{\mathcal{M}}$ of $\delta$ that satisfy $k⋐U_{\mathcal{M}},q\notin U_{\mathcal{M}}$, and $q⋐V_{\mathcal{M}}$, $k\notin V_{\mathcal{M}}$. Now, $k\notin U_{\mathcal{M}}^c$. Since $(\mathcal{O},\delta ,\mathcal{M})$ is supra $pp$-soft regular, then we have disjoint supra-soft open sets $G_{\mathcal{M}}$ and $H_{\mathcal{M}}$ that satisfy $U_{\mathcal{M}}^c\tilde{\subseteq }{G}_{\mathcal{M}}$, and $k⋐H_{\mathcal{M}}$. It is clear that $q⋐G_{\mathcal{M}}$. Since $G_{\mathcal{M}}$ and $H_{\mathcal{M}}$ are disjoint, then $q\notin H_{\mathcal{M}}$ and $k\notin G_{\mathcal{M}}$; therefore, $(\mathcal{O},\delta ,\mathcal{M})$ is supra $pp$-soft $T_2$.

Since $k\in G_{\mathcal{M}}$ implies $k⋐G_{\mathcal{M}}$, we obtain the proof of (ii). □

The three following examples point out that the converses of the statements in Proposition 5 fail to hold true.

Example 1.

Let $\mathcal{M}=\{m_1,m_2,m_3\}$ denote a set of parameters. Consider a universe $\mathcal{O}=\{p,q,r\}$ and the following soft sets over $\mathcal{O}$:

$F_{\mathcal{M}}=\{(m_1,\left\{p\right\}),(m_2,\mathcal{O}),(m_3,\mathcal{O})\}$,

$G_{\mathcal{M}}=\{(m_1,\mathcal{O}),(m_2,\left\{q\right\}),(m_3,\mathcal{O})\}$, and

$H_{\mathcal{M}}=\{(m_1,\mathcal{O}),(m_2,\mathcal{O}),(m_3,\left\{r\right\})\}$.

Now, notice that the family $\delta =\{\tilde{\varphi },\tilde{\mathcal{O}},F_{\mathcal{M}},G_{\mathcal{M}},H_{\mathcal{M}}\}$ defines a supra-soft topology on $\mathcal{O}$. Upon inspection, one can check that $(\mathcal{O},\delta ,\mathcal{M})$ is supra $pp$-soft $T_1$. However, disjoint supra-soft open sets do not exist. Therefore, $(\mathcal{O},\delta ,\mathcal{M})$ is not supra $pp$-soft $T_2$.

Example 2.

For $\mathcal{M}=\{m_1,m_2\}$, let $H_{\mathcal{M}}=\{(m_1,\left\{p\right\}),(m_2,\varnothing )\}$ be a soft set on $\mathcal{O}=\{p,q\}$. Then, $\delta =\{\tilde{\varphi },\tilde{\mathcal{O}},H_{\mathcal{M}}\}$ is a supra-soft topology on $\mathcal{O}$. It can be easily checked that $(\mathcal{O},\delta ,\mathcal{M})$ is supra $pp$-soft $T_0$; however, it is not a supra $pp$-soft $T_1$.

Example 3.

Over the universal set $\mathcal{O}=\{p,q\}$ with a parameters set $\mathcal{M}=\{m_1,m_2\}$, the next soft sets are defined.

$F_{\mathcal{M}}=\{(m_1,\left\{p\right\}),(m_2,\left\{q\right\})\}$ and

$G_{\mathcal{M}}=\{(m_1,\left\{q\right\}),(m_2,\left\{p\right\})\}$.

The collection $\delta =\{\tilde{\varphi },\tilde{\mathcal{O}},F_{\mathcal{M}},G_{\mathcal{M}}\}$ defines a supra-soft topology on $\mathcal{O}$. As a matter of fact, $(\mathcal{O},\delta ,\mathcal{M})$ is both supra $pp$-soft $T_3$ and supra $pp$-soft $T_4$. In contrast, there exist no supra open sets totally containing either a or q. Therefore, $(\mathcal{O},\delta ,\mathcal{M})$ is not supra $tp$-soft $T_3$; also, it is not supra $tp$-soft $T_4$.

The next two examples explain that supra $pp$-soft $T_3$ and supra $pp$-soft $T_4$-spaces are independent of each other.

Example 4.

Let the next soft sets be defined on the universal set $\mathcal{O}=\{p,q\}$ and a set of parameters $\mathcal{M}=\{m_1,m_2\}$:

$F_{\mathcal{M}}=\{(m_1,\left\{p\right\}),(m_2,\varnothing )\}$;

$G_{\mathcal{M}}=\{(m_1,\varnothing ),(m_2,\left\{q\right\})\}$;

$H_{\mathcal{M}}=\{(m_1,\left\{p\right\}),(m_2,\left\{q\right\})\}$.

The collection $\delta =\{\tilde{\varphi },\tilde{\mathcal{O}},F_{\mathcal{M}},G_{\mathcal{M}},H_{\mathcal{M}}\}$ defines a supra-soft topology on $\mathcal{O}$. Although $(\mathcal{O},\delta ,\mathcal{M})$ is $pp$-soft $T_4$, it is not supra $pp$-soft $T_3$.

Example 5.

A supra-soft topology can be identified with a supra topology when $\mathcal{M}$ is a singleton, and then supra $pp$-soft $T_i$-spaces coincide with supra $T_i$-spaces. Therefore, Example 5.12 of [35] provides an example of a supra $pp$-soft $T_3$-space that fails to be supra $pp$-soft $T_4$.

As a direct consequence of Propositions 3 and 5, one obtains the next proposition:

Proposition 6.

Every supra $tt$-soft $T_j$-space is a supra $pp$-soft $T_{j-1}$-space for $j=0,1,2,4$.

The two following examples guarantee that supra $tt$-soft $T_3$ and supra $pp$-soft $T_3$-spaces are independent concepts.

Example 6.

Reconsider $(\mathcal{O},\delta ,\mathcal{M})$ studied in Example 3. Although $(\mathcal{O},\delta ,\mathcal{M})$ is $pp$-soft $T_3$, it is not supra $tt$-soft $T_3$.

Example 7.

Let the next soft sets be defined on the universal set $\mathcal{O}=\{p,q\}$ and a set of parameters $\mathcal{M}=\{m_1,m_2\}$:

$U_{1\mathcal{M}}=\{(m_1,\left\{p\right\}),(m_2,\left\{p\right\})\}$;

$U_{2\mathcal{M}}=\{(m_1,\left\{q\right\}),(m_2,\left\{q\right\})\}$;

$U_{3\mathcal{M}}=\{(m_1,\left\{p\right\}),(m_2,\left\{q\right\})\}$;

$U_{4\mathcal{M}}=\{(m_1,\mathcal{O}),(m_2,\left\{q\right\})\}$;

$U_{5\mathcal{M}}=\{(m_1,\left\{p\right\}),(m_2,\mathcal{O})\}$;

$U_{6\mathcal{M}}=\{(m_1,\mathcal{O}),(m_2,\left\{p\right\})\}$.

Then, $\delta =\{\tilde{\varphi },\tilde{\mathcal{O}},U_{i\mathcal{M}}:i=1,2,...,6\}$ defines a supra-soft topology on $\mathcal{O}$. It can be noted that $(\mathcal{O},\delta ,\mathcal{M})$ is supra $tt$-soft $T_3$. In contrast, $U_{3\mathcal{M}}^c$ is a supra-soft closed set such that $k\notin U_{3\mathcal{M}}^c$. Because there do not exist two disjoint supra-soft open sets that satisfy the condition of a supra $pp$-soft regular space, we conclude that $(\mathcal{O},\delta ,\mathcal{M})$ is not supra $pp$-soft $T_3$.

The next conclusion can be easily proven. Its proof is omitted.

Proposition 7.

When $(\mathcal{O},\delta ,\mathcal{M})$ is a supra- soft${}_{TS}$ such that $\mid \mathcal{O}\mid \mid \mathcal{M}\mid \ge 4$, the following properties hold true:

(i) 

If $(\mathcal{O},\delta ,\mathcal{M})$ is a supra $pp$-soft $T_1$-space (supra $pp$-soft $T_4$-space), then it includes one proper supra-soft open set at least.

(ii) 

If $(\mathcal{O},\delta ,\mathcal{M})$ is a supra $pp$-soft $T_2$-space (supra $pp$-soft $T_3$-space), then it includes two proper supra-soft open sets at least.

Proposition 8.

If $U_{\mathcal{M}}$ is a supra-soft open set in $(\mathcal{O},\delta ,\mathcal{M})$ such that $U_{\mathcal{M}}$ and its complement are full soft sets, then $(\mathcal{O},\delta ,\mathcal{M})$ is supra $pp$-soft $T_1$.

Proof. 

Let $k\ne q\in \mathcal{O}$. Since $U_{\mathcal{M}}$ is a full soft set, then $k⋐U_{\mathcal{M}}$ and $q⋐U_{\mathcal{M}}$; and since $U_{\mathcal{M}}^c$ is also full, then $k\notin U_{\mathcal{M}}$ and $q\notin U_{\mathcal{M}}$. By hypothesis, $U_{\mathcal{M}}$ is a supra-soft open set; hence, $(\mathcal{O},\delta ,\mathcal{M})$ is a supra $pp$-soft $T_1$-space. □

Corollary 1.

If $U_{\mathcal{M}}$ is a supra-soft open subset of $(\mathcal{O},\delta ,\mathcal{M})$ such that $U_{\mathcal{M}}$ is also a partition soft set, then $(\mathcal{O},\delta ,\mathcal{M})$ is supra $pp$-soft $T_1$.

Proposition 9.

If $U_{\mathcal{M}}$ and $U_{\mathcal{M}}^c$ are full supra-soft clopen sets in $(\mathcal{O},\delta ,\mathcal{M})$, then $(\mathcal{O},\delta ,\mathcal{M})$ is supra $pp$-soft $T_2$.

Proof. 

Let $k\ne q\in \mathcal{O}$. By hypothesis, since $U_{\mathcal{M}}$ and $U_{\mathcal{M}}^c$ are full soft sets, then $k⋐U_{\mathcal{M}}$ and $q\notin U_{\mathcal{M}}$; and $q⋐U_{\mathcal{M}}^c$ and $k\notin U_{\mathcal{M}}^c$. By hypothesis, $U_{\mathcal{M}}$ is a supra-soft clopen set; hence, $(\mathcal{O},\delta ,\mathcal{M})$ is a supra $pp$-soft $T_2$-space. □

Corollary 2.

If $U_{\mathcal{M}}$ is a supra-soft clopen subset of $(\mathcal{O},\delta ,\mathcal{M})$ such that $U_{\mathcal{M}}$ is partition soft set, then $(\mathcal{O},\delta ,\mathcal{M})$ is supra $pp$-soft $T_2$.

In what follows, we establish some results that associate supra $pp$-soft $T_j$-spaces with supra $T_j$-spaces in the parametric supra topological spaces.

Theorem 2.

If $(\mathcal{O},{\delta }_m)$ is supra $T_j$, then $(\mathcal{O},\delta ,\mathcal{M})$ is a supra $pp$-soft $T_j$-space for $j=0,1$.

Proof. 

When $j=1$, let $k\ne q\in \mathcal{O}$. Since $(\mathcal{O},{\delta }_m)$ is supra $T_1$. Then, there are two members $U,V$ in ${\delta }_m$ such that $k\in U$, $q\notin U$ and $q\in V$, $k\notin V$. This necessitates the existence of two members $G_{\mathcal{M}},H_{\mathcal{M}}$ in $\delta$ satisfy $G(m)=U$ and $H(m)=V$. Note that $k⋐G_{\mathcal{M}}$, $q\notin G_{\mathcal{M}}$, and $q⋐H_{\mathcal{M}}$, $k\notin H_{\mathcal{M}}$. Hence, $(\mathcal{O},\delta ,\mathcal{M})$ is supra $pp$-soft $T_1$.

A similar technique is followed to prove $j=0$. □

Remark 2.

From Example 1, we see that $(\mathcal{O},\delta ,\mathcal{M})$ is supra $pp$-soft $T_1$, but $(\mathcal{O},{\delta }_{m_1})$, $(\mathcal{O},{\delta }_{m_2})$ and $(\mathcal{O},{\delta }_{m_3})$, which represent parametric supra topological spaces, are not supra $T_0$.

The spaces of supra $pp$-soft $T_j$ and supra $T_j$ (in the parametric supra topological spaces) are not navigated in the cases of $j=2,3,4$. The following example explains this fact.

Example 8.

Let there be the two next soft sets over $\mathcal{O}=\{p,q\}$ with a parameter set $\mathcal{M}=\{m_1,m_2\}$ as follows:

$G_{\mathcal{M}}=\{(m_1,\mathcal{O}),(m_2,\varnothing )\}$;

$H_{\mathcal{M}}=\{(m_1,\varnothing ),(m_2,\mathcal{O})\}$.

Then, $\delta =\{\tilde{\varphi },\tilde{\mathcal{O}},G_{\mathcal{M}},H_{\mathcal{M}}\}$ is a supra-soft topology on $\mathcal{O}$. Now, $(\mathcal{O},\delta ,\mathcal{M})$ is supra $pp$-soft $T_4$ and supra $pp$-soft $T_3$. However, $(\mathcal{O},{\delta }_{m_1})$ and $(\mathcal{O},{\delta }_{m_2})$ are not supra $T_0$.

Example 9.

Assume that $(\mathcal{O},\delta ,\mathcal{M})$ is the supra-soft${}_{TS}$ of Example 7. It was demonstrated that $(\mathcal{O},\delta ,\mathcal{M})$ is not supra $pp$-soft $T_3$. Furthermore, it is not a supra $pp$-soft $T_4$-space because disjoint supra-soft open sets containing the disjoint supra-soft closed sets $U_{{\mathcal{M}}_3}^c$ and $U_{{\mathcal{M}}_6}^c$ do not exist. On the other hand, $(\mathcal{O},{\delta }_{m_1})$ and $(\mathcal{O},{\delta }_{m_2})$ are supra $T_4$.

Example 10.

Consider the next four soft sets over $\mathcal{O}=\{p,q\}$ with a parameter set $\mathcal{M}=\{m_1,m_2\}$ as follows:

$$U_{{\mathcal{M}}_1}=\{(m_1,\left\{p\right\}),(m_2,\mathcal{O})\};$$

$$U_{{\mathcal{M}}_2}=\{(m_1,\left\{q\right\}),(m_2,\mathcal{O})\};$$

$$U_{{\mathcal{M}}_3}=\{(m_1,\mathcal{O}),(m_2,\left\{p\right\})\}$$

$$U_{{\mathcal{M}}_4}=\{(m_1,\mathcal{O}),(m_2,\left\{q\right\})\}.$$

Then, $\delta =\{\tilde{\varphi },\tilde{\mathcal{O}},U_{{\mathcal{M}}_i}:i=1,2,3,4\}$ is a supra-soft topology on $\mathcal{O}$. It is clear that $(\mathcal{O},\delta ,\mathcal{M})$ is not supra $pp$-soft $T_2$; however, $(\mathcal{O},{\delta }_{m_1})$ and $(\mathcal{O},{\delta }_{m_2})$ are supra $T_2$.

Theorem 3.

Let $(\mathcal{O},\delta ,\mathcal{M})$ be extended. If all $(\mathcal{O},{\delta }_m)$ are supra $T_j$, then $(\mathcal{O},\delta ,\mathcal{M})$ is supra $pp$-soft $T_j$ for $j=0,1,3,4$.

Proof. 

The theorem is demonstrated in Theorem 2 when $j=0,1$. For the proof of the other cases, it suffices to prove the properties of supra $pp$-soft regular and supra-soft normal.

First, assume that $H_{\mathcal{M}}$ is a supra-soft closed set and let $k\notin H_{\mathcal{M}}$. Then, $k\notin H(m)$ for a parameter $m\in \mathcal{M}$. Since $H(m)$ is a supra closed set, and $(\mathcal{O},{\delta }_m)$ is supra regular, we find two disjoint members $U,V\in \delta$ such that $H(m)\subseteq U$ and $k\in V$. By the extension of $(\mathcal{O},\delta ,\mathcal{M})$, there exist two members $G_{\mathcal{M}}$ and $F_{\mathcal{M}}$ of $\delta$ defined as follows

$G(m)=U$ and $G(m^{\prime })=\mathcal{O}$ for all the others $m^{\prime }\ne m$

$F(m)=V$ and $F(m)=\varnothing$ for all the others $m^{\prime }\ne m$

This directly leads to that $G_{\mathcal{M}}$ and $F_{\mathcal{M}}$ being disjoint such that $H_{\mathcal{M}}\tilde{\subseteq }{G}_{\mathcal{M}}$ and $k⋐F_{\mathcal{M}}$, which ends the proof that $(\mathcal{O},\delta ,\mathcal{M})$ is supra $pp$-soft regular.

Second, we show that $(\mathcal{O},\delta ,\mathcal{M})$ is supra-soft normal. To do this, let $H_{\mathcal{M}},L_{\mathcal{M}}$ be disjoint supra-soft closed sets. This leads to obtain $H(m)$ and $L(m)$ as disjoint supra closed sets for every $m\in \mathcal{M}$. Since $(\mathcal{O},{\delta }_m)$ is supra normal, then there exist two disjoint supra open sets U and V such that $H(m)\subseteq U$ and $L(m)\subseteq V$. Again, by the extension of $(\mathcal{O},\delta ,\mathcal{M})$, there exist two members $G_{\mathcal{M}},F_{\mathcal{M}}$ of $(\mathcal{O},\delta ,\mathcal{M})$ defined as follows

$G(m)=U$ and $G(m^{\prime })=\varnothing$ for all the others $m^{\prime }\ne m$

$F(m)=V$ and $G(m^{\prime })=\varnothing$ for all the others $m^{\prime }\ne m$

Now, ${\tilde{\cup }}_{\forall m\in \mathcal{M}}{G}_{\mathcal{M}}$ and ${\tilde{\cup }}_{\forall m\in \mathcal{M}}{F}_{\mathcal{M}}$ are disjoint supra-soft open sets such that $H_{\mathcal{M}}\tilde{\subseteq }{\tilde{\cup }}_{\forall m\in \mathcal{M}}{G}_{\mathcal{M}}$ and $L_{\mathcal{M}}\tilde{\subseteq }{\tilde{\cup }}_{\forall m\in \mathcal{M}}{F}_{\mathcal{M}}$. Thus, $(\mathcal{O},\delta ,\mathcal{M})$ is supra-soft normal. Hence, it is supra $pp$-soft $T_4$. □

The converse of the above theorem does need to be not true as illustrated in Example 8.

Proposition 10.

An extended supra-soft${}_{TS}$$(\mathcal{O},\delta ,\mathcal{M})$ is supra $pp$-soft $T_2$.

Proof. 

Let $k\ne q\in \mathcal{O}$ and let $F_{\mathcal{M}}$ and $G_{\mathcal{M}}$ be given by the following formulas

$F(m_i)=G(m_j)=\mathcal{O}$, when $i\ne j$

$F(m_{i^{\prime }})=G(m_{j^{\prime }})=\varnothing$ for all $i^{\prime }\ne i$ and $j^{\prime }\ne j$.

By the extension of $\delta$, $F_{\mathcal{M}}$ and $G_{\mathcal{M}}$ are disjoint supra-soft open sets such that $k⋐G_{\mathcal{M}},q\notin G_{\mathcal{M}}$, and $q⋐F_{\mathcal{M}},k\notin F_{\mathcal{M}}$. Hence, we prove the desired result. □

Example 3 demonstrates that the converse of the above proposition fails.

Theorem 4.

Let $(\mathcal{O},\delta ,\mathcal{M})$ be stable. Then, $(\mathcal{O},{\delta }_m)$ is a supra $T_j$-space if and only if $(\mathcal{O},\delta ,\mathcal{M})$ is a supra $pp$-soft $T_j$-space for all j.

Proof. 

By the stability of $(\mathcal{O},\delta ,\mathcal{M})$, we obtain U as a member of $(\mathcal{O},{\delta }_m)$ if and only if $\{(m,U):m\in \mathcal{M}\}$ is a member of $(\mathcal{O},\delta ,\mathcal{M})$. Hence, the proof is complete. □

Corollary 3.

Let $(\mathcal{O},\delta ,\mathcal{M})$ be supra $tp$-soft regular. Then, $(\mathcal{O},{\delta }_m)$ is supra $T_j$ iff $(\mathcal{O},\delta ,\mathcal{M})$ is supra $pp$-soft $T_j$ for all j.

Proposition 11.

$(\mathcal{O},\delta ,\mathcal{M})$ is a supra $pp$-soft regular space if and only if, when $k\in \mathcal{O}$ and $F_{\mathcal{M}}$ is a supra-soft open set partially containing a, a supra-soft open set $G_{\mathcal{M}}$ exists such that $k⋐G_{\mathcal{M}}\tilde{\subseteq }C{l}_{\delta }(G_{\mathcal{M}})\tilde{\subseteq }{F}_{\mathcal{M}}$.

Proof. 

Necessity: Let $k\in \mathcal{O}$ and $F_{\mathcal{M}}$ be a supra-soft open set that partially contains a. Then, $F_{\mathcal{M}}^c$ is a supra-soft closed set and $k\notin F_{\mathcal{M}}^c$. By hypothesis, there are disjoint supra-soft open sets $U_{\mathcal{M}}$ and $V_{\mathcal{M}}$ such that $F_{\mathcal{M}}^c\tilde{\subseteq }{U}_{\mathcal{M}}$ and $k⋐V_{\mathcal{M}}$. Obviously, $V_{\mathcal{M}}\tilde{\subseteq }{U}_{\mathcal{M}}^c\tilde{\subseteq }{F}_{\mathcal{M}}$. Thus, $C{l}_{\delta }(U_{\mathcal{M}})\tilde{\subseteq }{V}_{\mathcal{M}}^c\tilde{\subseteq }{F}_{\mathcal{M}}$.

Sufficiency: Let $H_{\mathcal{M}}$ be a supra-soft closed set. Suppose that $k\in \mathcal{O}$ satisfies $k\notin H_{\mathcal{M}}$. Then, $k⋐H_{\mathcal{M}}^c$. By hypothesis, there exists a supra-soft open set $G_{\mathcal{M}}$ such that $k⋐G_{\mathcal{M}}\tilde{\subseteq }C{l}_{\delta }(G_{\mathcal{M}})\tilde{\subseteq }{H}_{\mathcal{M}}^c$. Obviously, $H_{\mathcal{M}}\tilde{\subseteq }{\left[C{l}_{\delta }(G_{\mathcal{M}})\right]}^c$. By the disjointness of $G_{\mathcal{M}}$ and ${\left[C{l}_{\delta }(G_{\mathcal{M}})\right]}^c$, the desired finding is obtained. □

Theorem 5.

When a supra-soft${}_{TS}$$(\mathcal{O},\delta ,\mathcal{M})$ is supra $pp$-soft regular, the following spaces are identical:

(i) 

$(\mathcal{O},\delta ,\mathcal{M})$ is supra $pp$-soft $T_2$.

(ii) 

$(\mathcal{O},\delta ,\mathcal{M})$ is supra $pp$-soft $T_1$.

(iii) 

$(\mathcal{O},\delta ,\mathcal{M})$ is supra $pp$-soft $T_0$.

Proof. 

By Proposition 5, the directions $(i)\to (ii)\to (iii)$ are proven.

To prove $(iii)\to (i)$, let $k\ne q\in \mathcal{O}$. Since $(\mathcal{O},\delta ,\mathcal{M})$ is supra $pp$-soft $T_0$, then we have a supra-soft open set $G_{\mathcal{M}}$ such that $k⋐G_{\mathcal{M}}$ and $q\notin G_{\mathcal{M}}$, or $q⋐G_{\mathcal{M}}$ and $k\notin G_{\mathcal{M}}$. Say, $k⋐G_{\mathcal{M}}$ and $q\notin G_{\mathcal{M}}$. It is clear that $k\notin G_{\mathcal{M}}^c$ and $q⋐G_{\mathcal{M}}^c$. Since $(\mathcal{O},\delta ,\mathcal{M})$ is supra $pp$-soft regular, we can assure the existence of two disjoint supra-soft open sets $U_{\mathcal{M}}$ and $V_{\mathcal{M}}$ that satisfy both $k⋐U_{\mathcal{M}}$ and $q⋐G_{\mathcal{M}}^c\tilde{\subseteq }{V}_{\mathcal{M}}$. Since $U_{\mathcal{M}}$ and $V_{\mathcal{M}}$ are disjoint, then $q\notin U_{\mathcal{M}}$ and $k\notin V_{\mathcal{M}}$. This ends the proof that $(\mathcal{O},\delta ,\mathcal{M})$ is supra $pp$-soft $T_2$. □

Theorem 6.

For each $j=0,1,2,3$, supra-soft $T_j$ is a supra-soft hereditary property.

Proof. 

We prove the claim when $j=3$. The other cases follow similar lines.

Let $(Y,{\delta }_Y,\mathcal{M})$ be a subspace of $(\mathcal{O},\delta ,\mathcal{M})$ which is supra $pp$-soft $T_1$. First, we show that $(Y,{\delta }_Y,\mathcal{M})$ is supra $pp$-soft $T_1$. Let $k\ne q\in Y$. Then, $\delta$ contains two members $G_{\mathcal{M}}$ and $F_{\mathcal{M}}$ such that $k⋐G_{\mathcal{M}},q\notin G_{\mathcal{M}}$ and $q⋐F_{\mathcal{M}},k\notin F_{\mathcal{M}}$. According to the definition of a subspace, we have $k⋐U_{\mathcal{M}}=\tilde{Y}\tilde{\cap }{G}_{\mathcal{M}}$ and $q⋐V_{\mathcal{M}}=\tilde{Y}\tilde{\cap }{F}_{\mathcal{M}}$. Furthermore, $q\notin U_{\mathcal{M}}$ and $k\notin V_{\mathcal{M}}$. Thus, $(Y,{\delta }_Y,\mathcal{M})$ is supra $pp$-soft $T_1$.

To prove the supra $pp$-soft regularity of $(Y,{\delta }_Y,\mathcal{M})$, let $k\in Y$ and $L_{\mathcal{M}}$ be a supra-soft closed subset of $(Y,{\delta }_Y,\mathcal{M})$ such that $k\notin L_{\mathcal{M}}$. Then, there exists a supra-soft closed subset $H_{\mathcal{M}}$ of $(\mathcal{O},\delta ,\mathcal{M})$ such that $L_{\mathcal{M}}=\tilde{Y}\tilde{\cap }{H}_{\mathcal{M}}$. Since $k\notin H_{\mathcal{M}}$, there exist disjoint members $G_{\mathcal{M}}$ and $F_{\mathcal{M}}$ of $\delta$ such that $H_{\mathcal{M}}\tilde{\subseteq }{G}_{\mathcal{M}}$ and $k⋐F_{\mathcal{M}}$. Now, we find that $L_{\mathcal{M}}\tilde{\subseteq }\tilde{Y}\tilde{\cap }{G}_{\mathcal{M}}$ and $k⋐\tilde{Y}\tilde{\cap }{F}_{\mathcal{M}}$ and $(\tilde{Y}\tilde{\cap }{G}_{\mathcal{M}})\tilde{\cap }(\tilde{Y}\tilde{\cap }{F}_{\mathcal{M}})=\tilde{\varphi }$. Thus, $(Y,{\delta }_Y,\mathcal{M})$ is supra $pp$-soft regular. Hence, $(Y,{\delta }_Y,\mathcal{M})$ is supra $pp$-soft $T_3$. □

Theorem 7.

The property of being a supra-soft $T_4$-space is a supra-soft closed hereditary property.

Proof. 

Obvious. □

Theorem 8.

The finite product of supra $pp$-soft $T_j$-spaces is supra $pp$-soft $T_j$ for $j=0,1,2$.

Proof. 

For $j=2$. Without loss of generality, let $(\mathcal{O},\delta ,\mathcal{M})$ and $(Y,\upsilon ,\mathcal{N})$ be supra $pp$-soft $T_2$-spaces. Let us assume that $(k_1,q_1)\ne (k_2,q_2)$ in $\mathcal{O}\times Y$. Then, $k_1\ne k_2$ or $q_1\ne q_2$. Say, $k_1\ne k_2$. Then, there are two disjoint members $U_{\mathcal{M}},V_{\mathcal{N}}$ of $\delta$ such that $k_1⋐U_{\mathcal{M}}$ and $k_2\notin U_{\mathcal{M}}$; and $k_2⋐V_{\mathcal{N}}$ and $k_1\notin V_{\mathcal{N}}$. Now, $U_{\mathcal{M}}\times \tilde{\mathcal{O}}$ and $V_{\mathcal{N}}\times \tilde{Y}$ are supra-soft open sets such that $(k_1,q_1)⋐U_{\mathcal{M}}\times \tilde{\mathcal{O}}$ and $(k_2,q_2)\notin U_{\mathcal{M}}\times \tilde{\mathcal{O}}$, and $(k_2,q_2)⋐V_{\mathcal{N}}\times \tilde{Y}$ and $(k_1,q_1)\notin V_{\mathcal{N}}\times \tilde{Y}$. Since $[U_{\mathcal{M}}\times \tilde{\mathcal{O}}]\tilde{\cap }[V_{\mathcal{N}}\times \tilde{Y}]=\widetilde{{\varnothing }_{\mathcal{M}\times \mathcal{N}}}$, then $\mathcal{O}\times Y$ is supra $pp$-soft $T_2$. □

Proposition 12.

Let $f_{\varphi }:(\mathcal{O},\delta ,\mathcal{M})\to (Y,\upsilon ,\mathcal{N})$ be a soft $S^{🟉}$-continuous mapping such that f and ϕ are, respectively, injective and surjective. If $(Y,\upsilon ,\mathcal{N})$ is supra $pp$-soft $T_j$, then $(\mathcal{O},\delta ,\mathcal{M})$ is supra $pp$-soft $T_j$ for $j=2,1,0$.

Proof. 

When $j=2$. Let $k\ne q\in \mathcal{O}$. Then, there are only two points $x\ne y\in Y$ with $f(k)=x$ and $f(q)=y$ because f is injective. Since $(Y,\upsilon ,\mathcal{N})$ is supra $pp$-soft $T_2$, there are two disjoint members $G_{\mathcal{N}}$ and $F_{\mathcal{N}}$ of $\delta$ such that $x⋐G_{\mathcal{N}}$, $y\notin G_{\mathcal{N}}$ and $y⋐F_{\mathcal{N}}$, $x\notin F_{\mathcal{N}}$. Since $\varphi$ is surjective, Proposition 1 provides us with $k⋐f_{\varphi }^{-1}(G_{\mathcal{N}})$, $q\notin f_{\varphi }^{-1}(G_{\mathcal{N}})$ and $q⋐f_{\varphi }^{-1}(F_{\mathcal{N}})$, $k\notin f_{\varphi }^{-1}(F_{\mathcal{N}})$. By the soft $S^{🟉}$-continuity of $f_{\varphi }$, we obtained $f_{\varphi }^{-1}(G_{\mathcal{N}})$ and $f_{\varphi }^{-1}(F_{\mathcal{N}})$ as supra-soft open sets. Obviously, they are disjoint as well. Hence, $(\mathcal{O},\delta ,\mathcal{M})$ is supra $pp$-soft $T_2$. □

For all j, it can be proven that the next results follow a similar argument; hence, we omit their proofs.

Proposition 13.

Let $f_{\varphi }:(\mathcal{O},\delta ,\mathcal{M})\to (Y,\upsilon ,\mathcal{N})$ be a bijective soft $S^{🟉}$-continuous. If $(Y,\upsilon ,\mathcal{N})$ is supra $pp$-soft $T_j$, then $(\mathcal{O},\delta ,\mathcal{M})$ is supra $pp$-soft $T_j$.

Proposition 14.

Let $f_{\varphi }:(\mathcal{O},\delta ,\mathcal{M})\to (Y,\upsilon ,\mathcal{N})$ be a bijective soft $S^{🟉}$-open. If $(\mathcal{O},\delta ,\mathcal{M})$ is supra $pp$-soft $T_j$, then $(Y,\upsilon ,\mathcal{N})$ is supra $pp$-soft $T_j$.

Proposition 15.

The property of being a supra $pp$-soft $T_j$-space is preserved under an $S^{🟉}$-homeomorphism map.

4. Supra $\mathit{pt}$-Soft ${\mathit{T}}_{\mathit{j}}$-Spaces $(\mathit{j}=\mathbf{0},\mathbf{1},\mathbf{2},\mathbf{3},\mathbf{4})$

This section introduces the novel separation axioms that jointly form the family of supra $pt$-soft $T_j$-spaces. They are presented in Definition 23 below:

Definition 23.

A supra-soft${}_{TS}$$(\mathcal{O},\delta ,\mathcal{M})$ is said to be:

(i) 

Supra $pt$-soft $T_0$ if there exists a member $G_{\mathcal{M}}\in \delta$ for every $k\ne q\in \mathcal{O}$ satisfies $k⋐G_{\mathcal{M}},q\cancel{⋐}{G}_{\mathcal{M}}$, or $q⋐G_{\mathcal{M}},k\cancel{⋐}{G}_{\mathcal{M}}$.

(ii) 

Supra $pt$-soft $T_1$ if there exist two members $G_{\mathcal{M}},F_{\mathcal{M}}\in \delta$ for every $k\ne q\in \mathcal{O}$ that satisfy $k⋐G_{\mathcal{M}},q\cancel{⋐}{G}_{\mathcal{M}}$, and $q⋐F_{\mathcal{M}},k\cancel{⋐}{F}_{\mathcal{M}}$.

(iii) 

Supra $pt$-soft $T_2$ if there exist disjoint members $G_{\mathcal{M}},F_{\mathcal{M}}\in \delta$ for every $k\ne q\in \mathcal{O}$ that satisfy $k⋐G_{\mathcal{M}},q\cancel{⋐}{G}_{\mathcal{M}}$, and $q⋐F_{\mathcal{M}},k\cancel{⋐}{F}_{\mathcal{M}}$.

(iv) 

Supra $pt$-soft regular if for every supra-soft closed set $H_{\mathcal{M}}$ such that $k\cancel{⋐}{H}_{\mathcal{M}}$, there exist disjoint members $G_{\mathcal{M}},F_{\mathcal{M}}\in \delta$ such that $H_{\mathcal{M}}\tilde{\subseteq }{G}_{\mathcal{M}}$ and $k⋐F_{\mathcal{M}}$.

(v) 

Supra $pt$-soft $T_3$ (resp. supra $pt$-soft $T_4$) if it is supra $pt$-soft regular (resp. supra-soft normal) and supra $pt$-soft $T_1$.

Proposition 16.

The next properties are satisfied:

(i) 

Supra $pt$-soft $T_j$-spaces are supra $pt$-soft $T_{j-1}$ for $j=1,2$.

(ii) 

Supra $pt$-soft $T_j$-spaces are supra $pp$-soft $T_j$ for $j=0,1,2,4$.

(iii) 

Supra $tt$-soft $T_j$-spaces are supra $pt$-soft $T_j$ for all j.

Proof. 

(i) is a direct consequence of the above definition.

Since $k\cancel{⋐}{G}_{\mathcal{M}}$ implies $k\notin G_{\mathcal{M}}$, we obtain the proofs of (ii) and (iii). □

We provide the following examples to illustrate that the converse of the above proposition is not always true.

Example 11.

It can be easily checked that $(\mathcal{O},\delta ,\mathcal{M})$ displayed in Example 2 is a supra $pt$-soft $T_0$-space; however, it is not supra $pt$-soft $T_1$.

Example 12.

Let the next soft sets be defined over $\mathcal{O}$ and $\mathcal{M}$, where $\mathcal{O}=\{p,q,r\}$ and $\mathcal{M}=\{m_1,m_2\}$

$F_{1\mathcal{M}}=\{(m_1,\left\{q\right\}),(m_2,\left\{r\right\})\}$;

$F_{2\mathcal{M}}=\{(m_1,\left\{p\right\}),(m_2,\{p,q\})\}$;

$F_{3\mathcal{M}}=\{(m_1,\left\{p\right\}),(m_2,\{p,r\})\}$;

$F_{4\mathcal{M}}=\{(m_1,\left\{p\right\}),(m_2,\mathcal{O})\}$;

$F_{5\mathcal{M}}=\{(m_1,\{p,q\}),(m_2,\{p,r\})\}$ and

$F_{6\mathcal{M}}=\{(m_1,\{p,q\}),(m_2,\mathcal{O})\}$;

Now, the above six soft sets plus the absolute and null soft sets form a supra-soft topology δ on $\mathcal{O}$. Note that $(\mathcal{O},\delta ,\mathcal{M})$ is a supra $pt$-soft $T_1$. In contrast, there are no disjoint members of δ satisfying a condition of supra $pt$-soft $T_2$ for $p\ne r$, which means that $(\mathcal{O},\delta ,\mathcal{M})$ is not supra $pt$-soft $T_2$.

Example 13.

Example 8 shows that $(\mathcal{O},\delta ,\mathcal{M})$ is both supra $pp$-soft $T_4$ and supra $pp$-soft $T_3$. On the other hand, there is no member in δ (except for the null soft set) that does not totally include k or q, which implies that $(\mathcal{O},\delta ,\mathcal{M})$ is not supra $pt$-soft $T_0$.

Example 14.

Let $(\mathcal{O},\delta ,\mathcal{M})$ be the same as in Example 4. It can be shown that $(\mathcal{O},\delta ,\mathcal{M})$ is supra $pt$-soft $T_j$ for $j=2,3,4$. However, there is no member in δ (except for the absolute soft set) that totally contains k or q; therefore, $(\mathcal{O},\delta ,\mathcal{M})$ is not supra $tt$-soft $T_0$.

In the next example, we clarify that the systematic relation $T_3$ implies that $T_2$ does not hold true for their counterparts: supra $pt$-soft $T_2$ and supra $pt$-soft $T_3$.

Example 15.

Define the soft sets over $\mathcal{O}$ and $\mathcal{M}$, where $\mathcal{O}=\{p,q,r\}$ and $\mathcal{M}=\{m_1,m_2\}$,

$F_{1\mathcal{M}}=\{(m_1,\{p,q\}),(m_2,\varnothing )\}$;

$F_{2\mathcal{M}}=\{(m_1,\{p,r\}),(m_2,\varnothing )\}$;

$F_{3\mathcal{M}}=\{(m_1,\{q,r\}),(m_2,\varnothing )\}$;

$F_{4\mathcal{M}}=\{(m_1,\mathcal{O}),(m_2,\varnothing )\}$;

Now, the above four soft sets plus the absolute and null soft sets form a supra-soft topology δ on $\mathcal{O}$. It can be noted that $(\mathcal{O},\delta ,\mathcal{M})$ is supra $pt$-soft $T_3$. In contrast, there are no disjoint members of δ satisfying a condition of supra $pt$-soft $T_2$ for any two distinct points.

The behaviors and features of supra $pt$-soft $T_j$-spaces under different circumstances are investigated in the following.

Proposition 17.

$(\mathcal{O},\delta ,\mathcal{M})$ is a supra $pt$-soft $T_1$-space if $k_{\mathcal{M}}$ is a supra-soft closed set for all $k\in \mathcal{O}$.

Proof. 

Let $k\ne q$. By hypothesis, $k_{\mathcal{M}}$ and $q_{\mathcal{M}}$ are supra-soft closed sets. Then, $q\in {(a_{\mathcal{M}})}^c={(\mathcal{O}\setminus \left\{p\right\})}_{\mathcal{M}}$ and $k\in {(b_{\mathcal{M}})}^c={(\mathcal{O}\setminus \left\{q\right\})}_{\mathcal{M}}$. Since $k\cancel{⋐}{(\mathcal{O}\setminus \left\{p\right\})}_{\mathcal{M}}$ and $q\cancel{⋐}{(\mathcal{O}\setminus \left\{q\right\})}_{\mathcal{M}}$, then $(\mathcal{O},\delta ,\mathcal{M})$ is $pt$-soft $T_1$. □

The converse of this property fails to hold true, as shown in Example 12.

Theorem 9.

If $(\mathcal{O},\delta ,\mathcal{M})$ has a basis of supra-soft clopen sets, then $(\mathcal{O},\delta ,\mathcal{M})$ is supra $pt$-soft regular.

Proof. 

Consider that $H_{\mathcal{M}}$ is a supra-soft closed set and let $k\notin H_{\mathcal{M}}$. Then, $k⋐H_{\mathcal{M}}^c$ which is a supra-soft open set. By hypothesis, the basis contains a supra-soft clopen set $F_{\mathcal{M}}$ such that $k⋐F_{\mathcal{M}}\tilde{\subseteq }{H}_{\mathcal{M}}^c$. Now, $H_{\mathcal{M}}\tilde{\subseteq }{F}_{\mathcal{M}}^c$. Obviously, $F_{\mathcal{M}}$ and $F_{\mathcal{M}}^c$ are disjoint supra-soft open sets; hence, $(\mathcal{O},\delta ,\mathcal{M})$ is supra $pt$-soft regular. □

Theorem 10.

Let $(\mathcal{O},\delta ,\mathcal{M})$ be a $pp$-soft regular space. Then, every supra $pt$-soft $T_j$-space $(\mathcal{O},\delta ,\mathcal{M})$ is supra $tp$-soft $T_j$ for $j=0,1$.

Proof. 

When $j=0$. Let $k\ne q$. Then, $\delta$ contains a supra-soft open set $G_{\mathcal{M}}$ such that $k⋐G_{\mathcal{M}}$ and $q\cancel{⋐}{G}_{\mathcal{M}}$, or $q⋐G_{\mathcal{M}}$ and $k\cancel{⋐}{G}_{\mathcal{M}}$. Say, $k⋐G_{\mathcal{M}}$ and $q\cancel{⋐}{G}_{\mathcal{M}}$. Since $k\notin G_{\mathcal{M}}^c$, it follows by supra $pp$-soft regularity that $\delta$ contains two disjoint supra-soft open sets $V_{\mathcal{M}}$ and $W_{\mathcal{M}}$ such that $G_{\mathcal{M}}^c\tilde{\subseteq }{V}_{\mathcal{M}}$ and $k⋐W_{\mathcal{M}}$. Thus, $q\in V_{\mathcal{M}}$ and $k\notin V_{\mathcal{M}}$ which completes the proof.

In a similar way, $j=1$ is proven. □

The spaces of supra $pp$-soft regular (supra $pt$-soft regular, supra-soft normal) need not be $pt$-soft $T_j$, $tp$-soft $T_j$ and $tt$-soft $T_j$ for all j. The example below points out this fact.

Example 16.

Consider $(\mathcal{O},\delta ,\mathcal{M})$ to be defined by Example 3. One can observe that $(\mathcal{O},\delta ,\mathcal{M})$ is supra $pp$-soft regular and supra-soft normal. Note that δ does not include a supra-soft open set (excepting the absolute soft set) totally including k or q. Furthermore, δ does not include a supra-soft open set (except for the null soft set) does not totally including k or q. This means that $(\mathcal{O},\delta ,\mathcal{M})$ is not $pt$-soft $T_j$, or $tp$-soft $T_j$, or $tt$-soft $T_j$ for all j.

Theorem 11.

Suppose that $(\mathcal{O},\delta ,\mathcal{M})$ is extended. When $j=0$ or $j=1$, $(\mathcal{O},\delta ,\mathcal{M})$ is supra $tp$-soft $T_j$ iff it is supra $pt$-soft $T_j$.

Proof. 

We proceed with the argument for $j=0$, the case $j=1$ is analogous.

Necessity: Let $k\ne q$. There must be a member $G_{\mathcal{M}}$ of $\delta$ such that $k\in G_{\mathcal{M}}$ and $q\notin G_{\mathcal{M}}$, or $q\in G_{\mathcal{M}}$ and $k\notin G_{\mathcal{M}}$. Say, $k\in G_{\mathcal{M}}$ and $q\notin G_{\mathcal{M}}$. If $q\notin G(m)$ for all $m\in \mathcal{M}$, then the proof finishes. Otherwise, we do not lose generality if we consider that there is $m\in \mathcal{M}$ such that $q\notin G(m)$ and $q\in G(m^{\prime })$ for each $m^{\prime }\ne m$. Since $(\mathcal{O},\delta ,\mathcal{M})$ is extended, there must be a supra-soft open set $V_{\mathcal{M}}$ that satisfies $V(m)=G(m)$ and $V(m^{\prime })=\varnothing$ for each $m^{\prime }\ne m$. Note that $k⋐V_{\mathcal{M}}$ and $q\cancel{⋐}{V}_{\mathcal{M}}$; hence, $(\mathcal{O},\delta ,\mathcal{M})$ is supra $pt$-soft $T_0$.

Sufficiency: Let $k\ne q$. There must exist a member $G_{\mathcal{M}}$ of $\delta$ such that $k⋐G_{\mathcal{M}}$ and $q\cancel{⋐}{G}_{\mathcal{M}}$, or $q⋐G_{\mathcal{M}}$ and $k\cancel{⋐}{G}_{\mathcal{M}}$. Say, $k⋐G_{\mathcal{M}}$ and $q\cancel{⋐}{G}_{\mathcal{M}}$. If $k\in G(m)$ for all $m\in \mathcal{M}$, then the proof finishes. Otherwise, consider, without loss of generality, that there exists $m\in \mathcal{M}$ such that $k\notin G(m)$ and $k\in G(m^{\prime })$ for each $m^{\prime }\ne m$. Since $(\mathcal{O},\delta ,\mathcal{M})$ is extended, there must be a supra-soft open set $V_{\mathcal{M}}$ that satisfies $V(m)=\mathcal{O}$ and $V(m^{\prime })=G(m^{\prime })$ for each $m^{\prime }\ne m$. Note that $k\in V_{\mathcal{M}}$ and $q\notin V_{\mathcal{M}}$; hence, $(\mathcal{O},\delta ,\mathcal{M})$ is $tp$-soft $T_0$. □

Corollary 4.

Let $(\mathcal{O},\delta ,\mathcal{M})$ be extended. Then, $(\mathcal{O},\delta ,\mathcal{M})$ is supra $tp$-soft $T_4$-space iff it is supra $pt$-soft $T_4$.

Theorem 12.

Suppose that a supra-soft${}_{TS}$$(\mathcal{O},\delta ,\mathcal{M})$ is stable. The following statements hold true:

(i) 

$(\mathcal{O},\delta ,\mathcal{M})$ is supra $tt$-soft $T_0$⇔$(\mathcal{O},\delta ,\mathcal{M})$ is supra $tp$-soft $T_0$⇔$(\mathcal{O},\delta ,\mathcal{M})$ is supra $pt$-soft $T_0$⇔$(\mathcal{O},\delta ,\mathcal{M})$ is supra $pp$-soft $T_0$.

(ii) 

$(\mathcal{O},\delta ,\mathcal{M})$ is supra $tt$-soft $T_1$⇔$(\mathcal{O},\delta ,\mathcal{M})$ is supra $tp$-soft $T_1$⇔$(\mathcal{O},\delta ,\mathcal{M})$ is supra $pt$-soft $T_1$⇔$(\mathcal{O},\delta ,\mathcal{M})$ is supra $pp$-soft $T_1$.

(iii) 

$(\mathcal{O},\delta ,\mathcal{M})$ is supra $tt$-soft $T_2$⇔$(\mathcal{O},\delta ,\mathcal{M})$ is supra $tp$-soft $T_2$⇔$(\mathcal{O},\delta ,\mathcal{M})$ is supra $pt$-soft $T_2$⇔$(\mathcal{O},\delta ,\mathcal{M})$ is supra $pp$-soft $T_2$.

(iv) 

$(\mathcal{O},\delta ,\mathcal{M})$ is supra $tt$-soft $T_3$⇔$(\mathcal{O},\delta ,\mathcal{M})$ is supra $tp$-soft $T_3$⇔$(\mathcal{O},\delta ,\mathcal{M})$ is supra $pt$-soft $T_3$⇔$(\mathcal{O},\delta ,\mathcal{M})$ is supra $pp$-soft $T_3$.

(v) 

$(\mathcal{O},\delta ,\mathcal{M})$ is supra $tt$-soft $T_4$⇔$(\mathcal{O},\delta ,\mathcal{M})$ is supra $tp$-soft $T_4$⇔$(\mathcal{O},\delta ,\mathcal{M})$ is supra $pt$-soft $T_4$⇔$(\mathcal{O},\delta ,\mathcal{M})$ is supra $pp$-soft $T_4$.

Proof. 

By the stability of $G_{\mathcal{M}}$, we obtain $k\in G_{\mathcal{M}}$ iff $k⋐G_{\mathcal{M}}$, and $k\notin G_{\mathcal{M}}$ iff $k\cancel{⋐}{G}_{\mathcal{M}}$. Hence, we obtain the required equivalences. □

Corollary 5.

Let a supra-soft${}_{TS}$$(\mathcal{O},\delta ,\mathcal{M})$ be a supra $tp$-soft regular. The following statements hold true:

(i) 

$(\mathcal{O},\delta ,\mathcal{M})$ is supra $tt$-soft $T_0$⇔$(\mathcal{O},\delta ,\mathcal{M})$ is supra $tp$-soft $T_0$⇔$(\mathcal{O},\delta ,\mathcal{M})$ is supra $pt$-soft $T_0$⇔$(\mathcal{O},\delta ,\mathcal{M})$ is supra $pp$-soft $T_0$.

(ii) 

$(\mathcal{O},\delta ,\mathcal{M})$ is supra $tt$-soft $T_1$⇔$(\mathcal{O},\delta ,\mathcal{M})$ is supra $tp$-soft $T_1$⇔$(\mathcal{O},\delta ,\mathcal{M})$ is supra $pt$-soft $T_1$⇔$(\mathcal{O},\delta ,\mathcal{M})$ is supra $pp$-soft $T_1$.

(iii) 

$(\mathcal{O},\delta ,\mathcal{M})$ is supra $tt$-soft $T_2$⇔$(\mathcal{O},\delta ,\mathcal{M})$ is supra $tp$-soft $T_2$⇔$(\mathcal{O},\delta ,\mathcal{M})$ is supra $pt$-soft $T_2$⇔$(\mathcal{O},\delta ,\mathcal{M})$ is supra $pp$-soft $T_2$.

(iv) 

$(\mathcal{O},\delta ,\mathcal{M})$ is supra $tt$-soft $T_3$⇔$(\mathcal{O},\delta ,\mathcal{M})$ is supra $tp$-soft $T_3$⇔$(\mathcal{O},\delta ,\mathcal{M})$ is supra $pt$-soft $T_3$⇔$(\mathcal{O},\delta ,\mathcal{M})$ is supra $pp$-soft $T_3$.

(v) 

$(\mathcal{O},\delta ,\mathcal{M})$ is supra $tt$-soft $T_4$⇔$(\mathcal{O},\delta ,\mathcal{M})$ is supra $tp$-soft $T_4$⇔$(\mathcal{O},\delta ,\mathcal{M})$ is supra $pt$-soft $T_4$⇔$(\mathcal{O},\delta ,\mathcal{M})$ is supra $pp$-soft $T_4$.

Now, we explain how supra $pt$-soft $T_j$-spaces behave in their parametric spaces and vice versa. In fact, the next remark shows that there is no navigation for these axioms between soft and classical frames.

Remark 3.

In Example 4, we showed that $(\mathcal{O},\delta ,\mathcal{M})$ is supra $pt$-soft $T_j$ for $j=2,3,4$; nevertheless, its parametric supra topological spaces fail to be supra $T_1$. In addition, $(\mathcal{O},\delta ,\mathcal{M})$ defined by Example 2 is supra $pt$-soft $T_0$; however, its parametric supra topological space $(\mathcal{O},{\delta }_{m_2})$ is not supra $T_0$.

In contrast, $(\mathcal{O},\delta ,\mathcal{M})$ defined by Example 3 is not a supra $pt$-soft $T_0$-space; however, its parametric supra topological spaces are supra $T_4$.

Theorem 13.

Let $(\mathcal{O},\delta ,\mathcal{M})$ be an extended supra-soft${}_{TS}$. If there exists $m\in \mathcal{M}$ such that $(\mathcal{O},{\delta }_m)$ is supra $T_j$, then $(\mathcal{O},\delta ,\mathcal{M})$ is supra $pt$-soft $T_j$ for each $j=0,1,2$.

Proof. 

We prove the claim when $j=2$. The other cases follow similar lines.

Let $k\ne q\in \mathcal{O}$ and assume that $(\mathcal{O},{\delta }_m)$ is supra $T_2$. There must exist two disjoint members $U,V$ of $(\mathcal{O},{\delta }_m)$ containing k and q, respectively. Since $(\mathcal{O},\delta ,\mathcal{M})$ is extended, then there exist two members $G_{\mathcal{M}},F_{\mathcal{M}}$ of $(\mathcal{O},\delta ,\mathcal{M})$ such that $G(m)=U$, $F(m)=V$ and $G(m^{\prime })=F(m^{\prime })=\varnothing$ for all $m^{\prime }\ne m$. This implies that $k⋐G_{\mathcal{M}}$, $q\cancel{⋐}{G}_{\mathcal{M}}$, and $q⋐F_{\mathcal{M}}$, $k\cancel{⋐}{F}_{\mathcal{M}}$; hence, $(\mathcal{O},\delta ,\mathcal{M})$ is supra $pt$-soft $T_2$. □

Theorem 14.

Let $(\mathcal{O},\delta ,\mathcal{M})$ be an extended supra-soft${}_{TS}$. If all $(\mathcal{O},{\delta }_m)$ is supra $T_j$, then $(\mathcal{O},\delta ,\mathcal{M})$ is supra $pt$-soft $T_j$ for all j.

Proof. 

The cases $j=0,1,2$ follow from the above theorem.

In the cases of $j=3,4$, we are finished if we demonstrate the properties of supra $pt$-soft regular and supra-soft normal.

We first prove that $(\mathcal{O},\delta ,\mathcal{M})$ is supra $pt$-soft regular. Suppose that $H_{\mathcal{M}}$ is a supra-soft closed set and that $k\notin H_{\mathcal{M}}$. There must exist $m\in \mathcal{M}$ such that $k\cancel{⋐}H(m)$. Since $(\mathcal{O},{\delta }_m)$ is supra regular, there exist disjoint members $U,V$ of $(\mathcal{O},{\delta }_m)$ such that $k\in U$, $H(m)\subseteq V$. Since $(\mathcal{O},\delta ,\mathcal{M})$ is extended, then there are members $G_{\mathcal{M}},F_{\mathcal{M}}$ of $(\mathcal{O},\delta ,\mathcal{M})$ given by

$G(m)=U$ and $G(m^{\prime })=\varnothing$ for all the others $m^{\prime }\ne m$

$F(m)=V$ and $G(m^{\prime })=\mathcal{O}$ for all the others $m^{\prime }\ne m$

Now, $G_{\mathcal{M}}$ and $F_{\mathcal{M}}$ are disjoint as well as $k⋐G_{\mathcal{M}}$, $H_{\mathcal{M}}\tilde{\subseteq }{G}_{\mathcal{M}}$. Thus, $(\mathcal{O},\delta ,\mathcal{M})$ is supra $pt$-soft regular, and consequently, it is supra $pt$-soft $T_3$.

With respect to the space of supra-soft normal, it is proven in Theorem 3; hence, $(\mathcal{O},\delta ,\mathcal{M})$ is supra $pt$-soft $T_4$. □

Remark 4.

Example 3 confirms that the restriction to extended supra-soft${}_{TSs}$ in Theorems 13 and 14 is not redundant.

One can prove the following results in a similar way to those in the previous section:

Theorem 15.

Suppose that $(\mathcal{O},\delta ,\mathcal{M})$ is stable. When $j=0,1,2,3,4$, $(\mathcal{O},{\delta }_m)$ is supra $T_j$ iff $(\mathcal{O},\delta ,\mathcal{M})$ is supra $pt$-soft $T_j$.

Corollary 6.

Let $(\mathcal{O},\delta ,\mathcal{M})$ be supra $pt$-soft regular. When $j=0,1,2,3,4$, $(\mathcal{O},{\delta }_m)$ is supra $T_j$ iff $(\mathcal{O},\delta ,\mathcal{M})$ is supra $pt$-soft $T_j$.

Theorem 16.

If $(Y,{\delta }_Y,\mathcal{M})$ is a soft subspace of a $pt$-soft $T_j$-space $(\mathcal{O},\delta ,\mathcal{M})$, then $(Y,{\delta }_Y,\mathcal{M})$ is supra $pt$-soft $T_j$ for $j\ne 4$.

Theorem 17.

If $(Y,{\delta }_Y,\mathcal{M})$ is a supra-soft closed subspace of a supra $pt$-soft $T_4$-space $(\mathcal{O},\delta ,\mathcal{M})$, then $(Y,{\delta }_Y,\mathcal{M})$ is a supra $pt$-soft $T_4$-space.

Theorem 18.

When $j=0,1,2$, the finite product of supra $pt$-soft $T_j$-spaces is supra $pt$-soft $T_j$.

Proposition 18.

Let $f_{\varphi }:(\mathcal{O},\delta ,\mathcal{M})\to (Y,\upsilon ,\mathcal{N})$ be a soft $S^{🟉}$-continuous mapping such that f is injective and ϕ is surjective. If $(Y,\upsilon ,\mathcal{N})$ is supra $pt$-soft $T_j$, then $(\mathcal{O},\delta ,\mathcal{M})$ is supra $pt$-soft $T_j$ for $j=0,1,2$.

Proposition 19.

Let $f_{\varphi }:(\mathcal{O},\delta ,\mathcal{M})\to (Y,\upsilon ,\mathcal{N})$ be a bijective soft $S^{🟉}$-continuous map. If $(Y,\upsilon ,\mathcal{N})$ is supra $pt$-soft $T_j$, then $(\mathcal{O},\delta ,\mathcal{M})$ is supra $pt$-soft $T_j$ for all j.

Proposition 20.

Let $f_{\varphi }:(\mathcal{O},\delta ,\mathcal{M})\to (Y,\upsilon ,\mathcal{N})$ be a bijective soft $S^{🟉}$-open map. If $(\mathcal{O},\delta ,\mathcal{M})$ is supra $pt$-soft $T_j$, then $(Y,\upsilon ,\mathcal{N})$ is supra $pt$-soft $T_j$ for all j.

Proposition 21.

The property of being a supra $pt$-soft $T_j$-space for all j is preserved under $S^{🟉}$-homeomorphisms.

5. Conclusions and Future Work

Supra-soft topology has become one of the most remarkable developments in soft topology. It was born from the principle that weaker conditions sometimes suffice to ensure some valid properties. Furthermore, it can be connected to intuitionistic fuzzy supratopologies [37], because each soft fuzzy set, that is extension of the soft set is representable by an intuitionistic fuzzy set [38]. Since we may need to relax certain soft topological conditions to model some known phenomena, we expanded this area of expertise with the concepts of supra $pp$-soft $T_j$ and supra $pt$-soft $T_j$-spaces $(j=0,1,2,3,4)$. They afford new classifications of supra-soft${}_{TSs}$. They have been formulated with respect to the distinct ordinary points and they are different in terms of the relations that associate points with soft sets. The first class uses partial belong and partial non-belong relations, whereas the second utilizes partial belong and total non-belong relations. With the help of examples, we have shown that these two types of spaces defined by supra-soft separation axioms are wider than those given in [33]. We have also explained the interrelations between them and their parametric supra topological spaces. Moreover, we have investigated whether subspaces, finite product spaces, and soft $S^{🟉}$-continuous mappings preserve these axioms.

In future works, we plan to compare among the different types of supra-soft separation axioms that have been introduced in the literature. We will also try to benefit from the classification obtained by the current axioms in decision-making problems, following techniques that may be similar to those given in [20]. Moreover, we shall investigate the concepts given herein using some generalizations of supra-soft open sets such as supra-soft $\alpha$-open, supra-soft pre-open, supra-soft semi-open, supra-soft b-open, and supra-soft $\beta$-open sets.

To conclude, we provide Table 1 to show whether some topological properties are valid or invalid via supra-soft and infra-soft topologies. Moreover, we demonstrate in Table 2 how the classical systematic relationships among $T_i$-spaces $(i=0,1,2,3,4)$ behave with respect to supra $pp$-soft $T_i$, $pt$-soft $T_i$, supra $tp$-soft $T_i$ and supra $tt$-soft $T_i$.

Table 1. Description of some properties via soft topology, supra-soft topology, and infra-soft topology.

Properties under Comparison	Soft Topology	Supra-Soft Topology	Infra-Soft Topology
$Int(H_{\mathcal{M}})$	Soft open set	Supra-soft open set	Need not be infra soft open set
$Cl(H_{\mathcal{M}})$	Soft closed set	Supra-soft closed set	Need not be infra soft closed set
$Int(Int(H_{\mathcal{M}}))=Int(H_{\mathcal{M}})$	True	True	True
$Cl(Cl(H_{\mathcal{M}}))=Cl(H_{\mathcal{M}})$	True	True	True
$Int(H_{\mathcal{M}}\tilde{\cap }{G}_{\mathcal{M}})$	$=Int(H_{\mathcal{M}})\tilde{\cap }Int(G_{\mathcal{M}})$	$\subseteq Int(H_{\mathcal{M}})\tilde{\cap }Int(G_{\mathcal{M}})$	$=Int(H_{\mathcal{M}})\tilde{\cap }Int(G_{\mathcal{M}})$
$Cl(H_{\mathcal{M}})\tilde{\cup }Cl(G_{\mathcal{M}})$	$=Cl(H_{\mathcal{M}}\tilde{\cup }{G}_{\mathcal{M}})$	$\subseteq Cl(H_{\mathcal{M}}\tilde{\cup }{G}_{\mathcal{M}})$	$=Cl(H_{\mathcal{M}}\tilde{\cup }{G}_{\mathcal{M}})$
Closed subset of compact space is compact	True	True	True
Finite space is compact	Need not be true	Need not be True	Need not be true

Table 2. Description of some properties via classical supra topology and supra-soft topology.

Classical Properties under Comparison	Supra $\mathit{pp}$-Soft ${\mathit{T}}_{\mathit{i}}$	Supra $\mathit{pt}$-Soft ${\mathit{T}}_{\mathit{i}}$	Supra $\mathit{tp}$-Soft ${\mathit{T}}_{\mathit{i}}$	Supra $\mathit{tt}$-Soft ${\mathit{T}}_{\mathit{i}}$
Supra $T_i$-space is supra $T_{i-1}$-space $(i=1,2)$	True	True	True	True
Supra $T_3$-space is supra $T_2$-space	True	Need not be true	True	True
Supra $T_4$-space is supra $T_3$-space	Need not be true	Need not be true	Need not be true	True