# Two Classes of Infrasoft Separation Axioms

Authors: Tareq M. Al-shami; Jia-Bao Liu
Journal: Journal of Mathematics
 (2021)

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## Abstract

One of the considerable topics in the soft setting is the study of soft topology which has enticed the attention of many researchers. To contribute to this scope, we devote this work to investigate two classes of separation axioms with respect to the distinct ordinary elements through one of the generalizations of soft topology called infrasoft topology. We first formulate the concepts of infra-tp-soft Tj using total belong and partial nonbelong relations and then introduce the concepts of infra-tt-soft Tj-spaces using total belong and partial nonbelong relations. To illustrate the relationships between them, we provide some examples. We discuss their fundamental properties and study their behaviors under some special types of infrasoft topologies. An extensive discussion is given for the transmission of these two classes between infrasoft topology and its parametric infratopologies. In the end, we demonstrate which ones have topological and hereditary properties, and we show their behaviors under the finite product of soft spaces.


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## Body

## 1. Introduction

One of the powerful mathematical approaches to cope with uncertain problems is the soft set. It was proposed in 1999 by Molodtsov [1] who explained its applications to different areas such as smoothness of functions, theory of measurement, game theory, Riemann integration, operations research, etc. In 2002, soft sets were successfully applied to handle decision-making problems by Maji et al. [2]. They introduced some operations and operators on soft sets such as their soft intersection and union and the complement of a soft set in [3]. Despite the weakness of some concepts and results in these early references, they form the essential start point of soft set theory. To keep some properties and results of crisp set theory, Ali et al. [4] displayed new types of these operations and operators. Novel types of them were established such as lower and upper soft equality [5], gf-soft union [6], and T-soft subset and T-soft equality [7]. Since the advent of soft sets, many authors applied successfully to address problems in some disciplines such as computer science [8], decision-making [9, 10], and medical science [11, 12]. These applications prove the adequacy of soft sets to treat and model a lot of real-life issues.In 2011, Shabir and Naz [13] defined soft topologies as hybridization of classical topology and soft sets. Then, numerous researchers have discussed the topological concepts, properties, and results in soft topologies; see [14–17]. Soft point [18, 19] is an essential concept in soft set theory; it represents the soft version of an ordinary element. Some types of soft topologies called enriched and extended soft topologies are studied [19].Some topological properties are kept under conditions weaker than topology’s conditions; also, some phenomena are described under structures relaxing a topology. The structures of suprasoft topology [20] and infrasoft topology [21] have been born with these goals. They have become two of the most interesting developments of soft topology in recent years. Supratopology is a class of subsets that extend the concept of topological space by dispensing with the postulate that the class is closed under finite intersections, whereas infratopology is a class of subsets that extend the concept of topological space by dispensing with the postulate that the class is closed under arbitrary unions. Generalized soft topology [22] and soft weak structure are other generalizations of soft topology. Recently, the concepts of compactness and connectedness have been studied through the frame of infrasoft topology in [23, 24], respectively.Another technique of generalizing soft topology was introduced by combining other structures with a soft topology. In this regard, Ittanagi [25] initiated the concept of soft bitopology.Our contribution to this field concerns the analysis of what type of “separation axioms” are meaningful in the study of infrasoft topology. As was the case of classical topology, soft separation axioms are among the most interesting and substantial ideas in soft topology. They form a tool to establish more restricted (and wider) classes of well-behaved soft spaces. It should be noted that a large variety of separation axioms in the soft structures is attributed to two factors. One is the distinguished objects that we intend to separate: they can be either ordinary points or soft points. The other is the type of belongingness and nonbelongingness relations that we require in the definitions: they can be either partial or total. For more details about this subject, see [26], which investigated new sorts of separation axioms on suprasoft topologies, and [10, 27], which studied two kinds of soft Ti-spaces on soft topologies. In [28], the researchers presented soft separation axioms with respect to distinct soft points. Terepeta [29] discussed soft separation axioms using their counterparts in classical topological spaces. Singh and Noorie [30] conducted a comparative study among some soft separation axioms published in the literature.In this article, we note that many properties of soft topological spaces are still valid on infrasoft topological spaces and initiating examples that show some relationships between certain topological concepts is easier on infrasoft topological spaces. Therefore, we aim in this paper to perform an exhaustive analysis of infrasoft topological spaces.The organization of this manuscript is as follows. Section2 mentions some definitions and properties related to soft set theory and infrasoft topology. Section 3 initiates the concepts of infra-tp-soft Ti-spaces i=0,1,2,3,4 and studies basic properties. Section 4 defines the concepts of infra-tt-soft Ti-spaces i=0,1,2,3,4 and discusses main properties. Elucidative examples are supplied to validate the obtained relationships and results. Ultimately, Section 5 outlines the paper’s achievements and proposes some future works.


## 2. Preliminaries

The concepts and results that we need in this paper are recalled in this section.


### 2.1. Soft Set Theory

The concepts and results of soft set theory that we need in this paper are mentioned in this part.Definition 1.
(see [1]). A map E from Σ which is set of parameters to the power set 2Y of Y is called a soft set over Y. It is denoted by EΣ and identified as EΣ=σ,Eσ:σ∈ΣandEσ∈2Y.
The set of all soft sets overY with Σ as a set of parameters is symbolized by SYΣ.Definition 2 (see [4]).
The complement of a soft setEΣ, symbolized by EΣc, is given by EΣc=EcΣ, where Ec:Σ⟶2Y is a map defined by Ecσ=Y,Eσ for each σ∈Σ.Definition 3 (see [2]).
If the image of each parameter ofΣ under a map E:Σ⟶Y is the empty set, then EΣ is called the null soft set over Y. Its complement is called the absolute soft set. The notations Φ and Y˜, respectively, denote the null and absolute soft sets.Definition 4 (see [18, 19]).
If the image of one parameter, sayσ, under a map P:Σ⟶Y is a singleton set, say ν, and the image of each parameter σ′∈Σ,σ is the empty set, then a soft set PΣ is called a soft point over Y. It is briefly symbolized by Pσν.Definition 5 (see [4]).
The intersection of two soft setsEΩ and FΓ over Y, symbolized by EΩ∩˜FΓ, is a soft set HΣ, where Σ=Ω∩Γ≠∅, and a map H:Σ⟶2Y is given by Hσ=Eσ∩Fσ for each σ∈Σ.Definition 6 (see [4]).
The union of two soft setsEΩ and FΓ over Y, symbolized by EΩ∪˜FΓ, is a soft set HΣ, where Σ=Ω∪Γ and a map H:Σ⟶2Y is given as follows:(1)Hσ=Eσ,σ∈Ω,Γ,Fσ,σ∈Γ,Ω,Eσ∪Fσ,σ∈Ω∩Γ.Definition 7 (see [31]).
A soft setEΣ is a subset of a soft set FΩ, symbolized by EΣ⊆˜FΩ, if Σ⊆Ω and for all σ∈Σ, we have Eσ⊆Fσ. The soft sets EΣ and EΩ are called soft equal if each is a subset of the other.Definition 8 (see [13]).
A soft setEΣ over Y which is defined by Eσ=ν for each σ∈Σ is called a singleton soft set. It is denoted by νΣ.Definition 9 (see [27]).
If all components of a soft set are equal, then we called it a stable soft set.Definition 10 (see [16]).
The Cartesian product ofEΣ and HΩ, symbolized by G×HΣ×Ω, is defined as G×Hσ,ω=Eσ×Hω for each σ,ω∈Σ×Ω.
The definition of soft maps given in [32] was reformulated in a way that reduces calculation burden and gives a justification (logical explanation) for some soft concepts such as why we determine that fη is injective or surjective according to its two crisp maps f and η.Definition 11 (see [33]).
Letf:X⟶Y and η:Σ⟶Ω be two crisp maps. A soft map fη of SXΣ into SYΩ is a relation such that each soft point in SXΣ is related to one and only one soft point in SYΩ such that fηPσν=Pησfν for each Pσν∈SXΣ.
In addition,fη−1Pωy=⊔σ∈η−1ων∈f−1yPσν for each Pωy∈SYΩ.Definition 12 (see [32]).
A soft mapfη:SYΣ⟶SℤΩ is said to be injective (resp. surjective, bijective) if both f and η are injective (resp., surjective, bijective).Definition 13 (see [13, 27]).
For a soft setEΣ over Y and ν∈Y, we say that(i)
ν∈EΣ (it reads as ν totally belongs to EΣ) if ν∈Eσ for each σ∈Σ;(ii)
ν∉EΣ (it reads as ν does not partially belong to EΣ) if ν∉Eσ for some σ∈Σ;(iii)
ν⋐EΣ (it reads as ν partially belongs to EΣ) if ν∈Eσ for some σ∈Σ;(iv)
ν⋐EΣ (it reads as ν does not totally belong to EΣ) if ν∉Eσ for each σ∈Σ.Proposition 1 (see [27]).
Consider a soft mapfη:SYΣ⟶SℤΩ and let EΣ and HΩ be soft sets in SYΣ and SℤΩ, respectively. Then the next results hold true:(i)
Ifη is surjective and ν∈EΣ, then fν∈fηEΣ.(ii)
Ifν⋐EΣ, then fν⋐fηEΣ provided that f is injective.(iii)
Iffη is injective and ν∉EΣ, then fν∉fηEΣ.(iv)
Ifμ∈HΩ, then ν∈fη−1HΩ for each ν∈f−1μ.(v)
Ifμ⋐HΩ, then ν⋐fη−1HΩ for each ν∈f−1μ.(vi)
Ifη is surjective and μ∉HΩ, then ν∉fη−1HΩ for each ν∈f−1μ.


### 2.2. Infrasoft Topological Spaces

The concepts and results of infrasoft topology that we need in this paper are mentioned in this part.Definition 14 (see [21]).
The collectionδ of soft sets over Y under a parameters set Σ is said to be an infrasoft topology on Y if it is closed under finite soft intersection and Φ∈δ.
The tripleY,δ,Σ is called an infrasoft topological space. Every member of δ is called an infrasoft open set and its relative complement is called an infrasoft closed set. We called Y,δ,Σ stable if all its infrasoft open sets are stable.Definition 15 (see [21]).
We define the infrainterior points and infraclosure points of a soft subsetEΣ of Y,δ,Σ which are, respectively, denoted by IntδEΣ and ClδEΣ as follows:(i)
IntδEΣ is the union of all infrasoft open sets contained in EΣ.(ii)
ClδEΣ is the intersection of all infrasoft closed sets containing EΣ.Theorem 1 (see [21]).
LetEΣ and FΣ be two soft subsets of Y,δ,Σ. Then,(i)
IfEΣ⊆~FΣ, then ClδEΣ⊆~ClδFΣ;(ii)
Pσν∈ClδEΣ if and only if EΣ∩˜HΣ≠Φ for each infrasoft open set FΣ containing Pσν.Proposition 2 (see [21]).
LetY,δ,Σ be an infrasoft topological space. Then the collection δσ=Eσ:EΣ∈δ forms an infratopology on Y for each σ∈Σ.
We calledδσ a parametric infratopology.Proposition 3 (see [21]).
Suppose thatδσσ∈Σ is a family of crisp infratopologies on Y. Then the family(2)σ,Eσ:σ∈Σ∈SYΣsuch thatEσ∈δσfor eachσ∈Σ,defines an infrasoft topology on Y.The infrasoft topological space given in the above proposition is called an extended infrasoft topology.Definition 16 (see [21]).
LetY,δ,Σ be an infrasoft topological space and ℤ be a nonempty subset of Y. Then δℤ=ℤ˜∩˜EΣ:EΣ∈δ is called an infrasoft relative topology on ℤ and ℤ,δℤ,Σ is called an infrasoft subspace of Y,δ,Σ.Theorem 2 (see [21]).
Letℤ,δℤ,Σ be an infrasoft subspace of Y,δ,Σ. Then HΣ is an infrasoft closed subset of ℤ,δℤ,Σ iff there exists an infrasoft closed subset FΣ of Y,δ,Σ such that HΣ=ℤ˜∩˜FΣ.Definition 17 (see [33]).
A soft mappingfφ:Y,δ,Σ⟶ℤ,θ,Ω is said to be(i)
Infrasoft continuous if the preimage of any infrasoft open set is an infrasoft open set;(ii)
Infrasoft open (resp., infrasoft closed) if the image of any infrasoft open (resp., infrasoft closed) set is an infrasoft open (resp., infrasoft closed) set;(iii)
An infrasoft homeomorphism if it is bijective, infrasoft continuous, and infrasoft open.


## 2.1. Soft Set Theory

The concepts and results of soft set theory that we need in this paper are mentioned in this part.Definition 1.
(see [1]). A map E from Σ which is set of parameters to the power set 2Y of Y is called a soft set over Y. It is denoted by EΣ and identified as EΣ=σ,Eσ:σ∈ΣandEσ∈2Y.
The set of all soft sets overY with Σ as a set of parameters is symbolized by SYΣ.Definition 2 (see [4]).
The complement of a soft setEΣ, symbolized by EΣc, is given by EΣc=EcΣ, where Ec:Σ⟶2Y is a map defined by Ecσ=Y,Eσ for each σ∈Σ.Definition 3 (see [2]).
If the image of each parameter ofΣ under a map E:Σ⟶Y is the empty set, then EΣ is called the null soft set over Y. Its complement is called the absolute soft set. The notations Φ and Y˜, respectively, denote the null and absolute soft sets.Definition 4 (see [18, 19]).
If the image of one parameter, sayσ, under a map P:Σ⟶Y is a singleton set, say ν, and the image of each parameter σ′∈Σ,σ is the empty set, then a soft set PΣ is called a soft point over Y. It is briefly symbolized by Pσν.Definition 5 (see [4]).
The intersection of two soft setsEΩ and FΓ over Y, symbolized by EΩ∩˜FΓ, is a soft set HΣ, where Σ=Ω∩Γ≠∅, and a map H:Σ⟶2Y is given by Hσ=Eσ∩Fσ for each σ∈Σ.Definition 6 (see [4]).
The union of two soft setsEΩ and FΓ over Y, symbolized by EΩ∪˜FΓ, is a soft set HΣ, where Σ=Ω∪Γ and a map H:Σ⟶2Y is given as follows:(1)Hσ=Eσ,σ∈Ω,Γ,Fσ,σ∈Γ,Ω,Eσ∪Fσ,σ∈Ω∩Γ.Definition 7 (see [31]).
A soft setEΣ is a subset of a soft set FΩ, symbolized by EΣ⊆˜FΩ, if Σ⊆Ω and for all σ∈Σ, we have Eσ⊆Fσ. The soft sets EΣ and EΩ are called soft equal if each is a subset of the other.Definition 8 (see [13]).
A soft setEΣ over Y which is defined by Eσ=ν for each σ∈Σ is called a singleton soft set. It is denoted by νΣ.Definition 9 (see [27]).
If all components of a soft set are equal, then we called it a stable soft set.Definition 10 (see [16]).
The Cartesian product ofEΣ and HΩ, symbolized by G×HΣ×Ω, is defined as G×Hσ,ω=Eσ×Hω for each σ,ω∈Σ×Ω.
The definition of soft maps given in [32] was reformulated in a way that reduces calculation burden and gives a justification (logical explanation) for some soft concepts such as why we determine that fη is injective or surjective according to its two crisp maps f and η.Definition 11 (see [33]).
Letf:X⟶Y and η:Σ⟶Ω be two crisp maps. A soft map fη of SXΣ into SYΩ is a relation such that each soft point in SXΣ is related to one and only one soft point in SYΩ such that fηPσν=Pησfν for each Pσν∈SXΣ.
In addition,fη−1Pωy=⊔σ∈η−1ων∈f−1yPσν for each Pωy∈SYΩ.Definition 12 (see [32]).
A soft mapfη:SYΣ⟶SℤΩ is said to be injective (resp. surjective, bijective) if both f and η are injective (resp., surjective, bijective).Definition 13 (see [13, 27]).
For a soft setEΣ over Y and ν∈Y, we say that(i)
ν∈EΣ (it reads as ν totally belongs to EΣ) if ν∈Eσ for each σ∈Σ;(ii)
ν∉EΣ (it reads as ν does not partially belong to EΣ) if ν∉Eσ for some σ∈Σ;(iii)
ν⋐EΣ (it reads as ν partially belongs to EΣ) if ν∈Eσ for some σ∈Σ;(iv)
ν⋐EΣ (it reads as ν does not totally belong to EΣ) if ν∉Eσ for each σ∈Σ.Proposition 1 (see [27]).
Consider a soft mapfη:SYΣ⟶SℤΩ and let EΣ and HΩ be soft sets in SYΣ and SℤΩ, respectively. Then the next results hold true:(i)
Ifη is surjective and ν∈EΣ, then fν∈fηEΣ.(ii)
Ifν⋐EΣ, then fν⋐fηEΣ provided that f is injective.(iii)
Iffη is injective and ν∉EΣ, then fν∉fηEΣ.(iv)
Ifμ∈HΩ, then ν∈fη−1HΩ for each ν∈f−1μ.(v)
Ifμ⋐HΩ, then ν⋐fη−1HΩ for each ν∈f−1μ.(vi)
Ifη is surjective and μ∉HΩ, then ν∉fη−1HΩ for each ν∈f−1μ.


## 2.2. Infrasoft Topological Spaces

The concepts and results of infrasoft topology that we need in this paper are mentioned in this part.Definition 14 (see [21]).
The collectionδ of soft sets over Y under a parameters set Σ is said to be an infrasoft topology on Y if it is closed under finite soft intersection and Φ∈δ.
The tripleY,δ,Σ is called an infrasoft topological space. Every member of δ is called an infrasoft open set and its relative complement is called an infrasoft closed set. We called Y,δ,Σ stable if all its infrasoft open sets are stable.Definition 15 (see [21]).
We define the infrainterior points and infraclosure points of a soft subsetEΣ of Y,δ,Σ which are, respectively, denoted by IntδEΣ and ClδEΣ as follows:(i)
IntδEΣ is the union of all infrasoft open sets contained in EΣ.(ii)
ClδEΣ is the intersection of all infrasoft closed sets containing EΣ.Theorem 1 (see [21]).
LetEΣ and FΣ be two soft subsets of Y,δ,Σ. Then,(i)
IfEΣ⊆~FΣ, then ClδEΣ⊆~ClδFΣ;(ii)
Pσν∈ClδEΣ if and only if EΣ∩˜HΣ≠Φ for each infrasoft open set FΣ containing Pσν.Proposition 2 (see [21]).
LetY,δ,Σ be an infrasoft topological space. Then the collection δσ=Eσ:EΣ∈δ forms an infratopology on Y for each σ∈Σ.
We calledδσ a parametric infratopology.Proposition 3 (see [21]).
Suppose thatδσσ∈Σ is a family of crisp infratopologies on Y. Then the family(2)σ,Eσ:σ∈Σ∈SYΣsuch thatEσ∈δσfor eachσ∈Σ,defines an infrasoft topology on Y.The infrasoft topological space given in the above proposition is called an extended infrasoft topology.Definition 16 (see [21]).
LetY,δ,Σ be an infrasoft topological space and ℤ be a nonempty subset of Y. Then δℤ=ℤ˜∩˜EΣ:EΣ∈δ is called an infrasoft relative topology on ℤ and ℤ,δℤ,Σ is called an infrasoft subspace of Y,δ,Σ.Theorem 2 (see [21]).
Letℤ,δℤ,Σ be an infrasoft subspace of Y,δ,Σ. Then HΣ is an infrasoft closed subset of ℤ,δℤ,Σ iff there exists an infrasoft closed subset FΣ of Y,δ,Σ such that HΣ=ℤ˜∩˜FΣ.Definition 17 (see [33]).
A soft mappingfφ:Y,δ,Σ⟶ℤ,θ,Ω is said to be(i)
Infrasoft continuous if the preimage of any infrasoft open set is an infrasoft open set;(ii)
Infrasoft open (resp., infrasoft closed) if the image of any infrasoft open (resp., infrasoft closed) set is an infrasoft open (resp., infrasoft closed) set;(iii)
An infrasoft homeomorphism if it is bijective, infrasoft continuous, and infrasoft open.


## 3. Infra-tp-Soft Ti-Spaces i=0,1,2,3,4

In this section, we display new family of separation axioms, namely, infra-tp-soft Ti-spaces i=0,1,2,3,4, where t and p are the first letters of total belong and partial nonbelong relations that are used to define these spaces. We explore main properties and supply various examples to elucidate the relationships between them.Definition 18.
Y,δ,Σ is said to be(i)
An infra-tp-soft T0-space if for every ν≠μ∈Y, there exists an infrasoft open set EΣ such that ν∈EΣ and μ∉EΣ, or μ∈EΣ and ν∉EΣ;(ii)
An infra-tp-soft T1-space if for every ν≠μ∈Y, there exist infrasoft open sets EΣ and FΣ such that ν∈EΣ,μ∉EΣ and μ∈FΣ,ν∉FΣ;(iii)
An infra-tp-soft T2-space (or an infra-tp-soft Hausdorff space) if for every ν≠μ∈Y, there exist disjoint infrasoft open sets EΣ and FΣ such that ν∈EΣ,μ∉EΣ and μ∈FΣ,ν∉FΣ;(iv)
An infra-tp-soft regular space if for every infrasoft closed set HΣ such that ν∉HΣ, there exist disjoint infrasoft open sets EΣ and FΣ such that HΣ⊆˜EΣ and ν∈FΣ;(v)
An infrasoft normal space if disjoint infrasoft closed sets are separated by disjoint infrasoft open sets;(vi)
An infra-tp-soft T3 (resp., infra-tp-soft T4) -space if it is infra-tp-soft regular (resp., infrasoft normal) and infra-tp-soft T1.
Before we investigate the relationships between the above infrasoft axioms, we need the following auxiliary result.Lemma 1.
Every infrasoft open (infrasoft closed) subset of an infra-tp-soft regular space is stable; i.e., every infra-tp-soft regular space is stable.Proof.
LetEΣ be an infrasoft open subset of an infra-tp-soft regular space Y,δ,Σ. Suppose that EΣ is unstable; i.e., there exist ν∈Y and σ,σ′∈Σ such that ν∈Eσ and ν∉Eσ′. Now, EΣc is an infrasoft closed set such that ν∉Ecσ and ν∈Ecσ′. It is clear that for any infrasoft open set FΣ containing EΣc we obtain ν⋐FΣ; this implies we cannot find an infrasoft open set whose intersection with FΣ is the null soft set such that ν totally belongs to it. Hence, EΣ must be stable.
One can similarly prove the lemma for an infrasoft closed set.Corollary 1.
The number of infrasoft open subsets of a finite infra-tp-soft regular space Y,δ,Σ is greater than or equal to 2Y.Proposition 4.
Every infra-tp-soft Ti-space is an infra-tp-soft Ti−1-space for i=1,2,3.Proof.
The proof is clear for the casesi=1,2. To prove the case of i=3, let ν,μ be two distinct elements in an infra-tp-soft T3-space. Then there exists an infrasoft open set EΣ such that ν∈EΣ and μ∉EΣ. Now, EΣc is infrasoft closed such that ν∉EΣc. By hypothesis, there exist disjoint infrasoft open sets UΣ and VΣ such that EΣc⊆˜UΣ and ν∈VΣ. It follows from the above lemma that μ⋐EΣ. Then μ∈UΣ; hence, we obtain the desired result.
Examples below explain that we cannot reverse the above proposition in general.Example 1.
The collectionsδ1=Φ,Y˜,E1Σ,E3Σ, δ2=Φ,Y˜,E4Σ,E5Σ,E6Σ, and δ3=Φ,Y˜,E1Σ,E2Σ,E3Σ are three infrasoft topologies on the universal set Y=ν1,ν2 with a set of parameters Σ=σ1,σ2, where(3)E1Σ=σ1,ν1,σ2,ν1;E2Σ=σ1,ν2,σ2,ν2;E3Σ=σ1,ν2,σ2,∅;E4Σ=σ1,ν1,σ2,Y;E5Σ=σ1,ν2,σ2,Y;E6Σ=σ1,∅,σ2,Y;
Obviously,Y,δ1,Σ, Y,δ2,Σ, and Y,δ3,Σ are, respectively, infra-tp-soft T0, infra-tp-soft T1, and infra-tp-soft T2. On the other hand, Y,δ1,Σ is not infra-tp-soft T1, Y,δ2,Σ is not infra-tp-soft T2, and Y,δ3,Σ is not infra-tp-soft T3.Remark 1.
In the above example,Y,δ3,Σ is an infra-tp-soft T4-space. As we know, the soft topology and general topology (infrasoft topology and general infratopology) are identical if a set of parameters is a singleton. Then we can say that there is an infrasoft topology which is infra-tp-soft T3, but not infra-tp-soft T4. Hence, the concepts of infra-tp-soft T3 and infra-tp-soft T4-spaces are independent of each other.
Now, we explore and discuss some properties of infra-tp-soft Ti-spaces.Proposition 5.
IfPσν is an infrasoft closed subset of Y,δ,Σ for each ν∈Y, then Y,δ,Σ is an infra-tp-soft T1-space.Proof.
Letν≠μ∈Y. By assumption, Pσνc and Pσμc are infrasoft open sets. Obviously, μ∈Pσνc,ν∉Pσνc, and ν∈Pσμc,μ∉Pσμc. Hence, Y,δ,Σ is infra-tp-soft T1.
In Example1, Y,δ2,Σ is infra-tp-soft T1. However, Pσ2ν1 and Pσ2ν2 are not infrasoft closed sets. So, the converse of the above proposition need not be true in general.Proposition 6.
Y,δ,Σ is an infra-tp-soft T2-space iff for each ν≠μ, there exists an infrasoft open set EΣ such that ν∈EΣ and μ⋐ClδEΣ, where μ⋐ClδEΣ if there is an infrasoft open set GΣ such that μ∈GΣ and EΣ∩˜GΣ=Φ.Proof.
Letν≠μ in an infra-tp-soft T2-space. Then there exist disjoint infrasoft open sets EΣ and FΣ such that ν∈EΣ and μ∈FΣ. Now, EΣ⊆˜FΣc which means ClδEΣ⊆˜FΣc. Since μ∉Fcσ for each σ∈Σ, μ⋐ClδEΣ. Conversely, let ν≠μ∈Y and EΣ be an infrasoft open set such that ν∈EΣ and μ⋐ClδEΣ. Then there is an infrasoft open set GΣ such that μ∈GΣ and EΣ∩˜GΣ=Φ. This ends the proof that Y,δ,Σ is infra-tp-soft T2.Proposition 7.
IfY,δ,Σ is an infra-tp-soft regular space, then for each ν∈Y and infrasoft open set HΣ partially containing ν, there exists an infrasoft open set GΣ such that ν∈GΣ⊆˜ClδGΣ⊆˜HΣ.Proof.
LetHΣ be an infrasoft open set partially containing ν. Then HΣc is an infrasoft closed set and ν∉HΣc. By assumption, there are disjoint infrasoft open sets EΣ and GΣ such that HΣc⊆˜EΣ and ν∈GΣ. Obviously, GΣ⊆˜EΣc⊆˜HΣ. Thus, ClδGΣ⊆˜EΣc⊆˜HΣ.Theorem 3.
IfY,δ,Σ is an infra-tp-soft regular space, then the following concepts are identical.(i)
Y,δ,Σ is infra-tp-soft T2.(ii)
Y,δ,Σ is infra-tp-soft T1.(iii)
Y,δ,Σ is infra-tp-soft T0.Proof.
The implicationsi⟶ii⟶iii follow from Proposition 4.
To proveiii⟶i, let ν≠μ∈Y. Since Y,δ,Σ is infra-tp-soft T0, then we have an infrasoft open set EΣ such that ν∈EΣ and μ∉EΣ, or μ∈EΣ and ν∉EΣ. Say, ν∈EΣ and μ∉EΣ. It is clear that ν∉EΣc and μ∈EΣc. Since Y,δ,Σ is infra-tp-soft regular, there exist two disjoint infrasoft open sets UΣ and VΣ such that ν∈UΣ and μ∈EΣc⊆˜VΣ. Since UΣ and VΣ are disjoint, μ∉UΣ and ν∉VΣ. This ends the proof that Y,δ,Σ is infra-tp-soft T2.Proposition 8.
A finite infra-tp-soft T2-space Y,δ,Σ contains at least Y+2 infrasoft open sets, where Y>1.Proof.
We prove the proposition using mathematical induction. WhenY=2, we need two infrasoft open sets beside the null and absolute soft sets. Then the claim is true for i=2. Suppose that the claim is true for Y=n; i.e., δ=n+2. Without loss of generality, consider Y,δ,Σ as the smallest infra-tp-soft T2-space. Now, let Y=n+1; i.e., Y′=Y∪ξ, where ξ∉Y. Then δ′=δ∪ξ is the smallest infra-tp-soft T2-space on Y′. Hence, we obtain the desired result.
In the following result we will study the properties and relationships between infrasoft topology and its parametric infratopologies.Proposition 9.
IfY,δ,Σ is an infra- tp-soft Ti-space, then Y,δσ is infra-Ti for each σ∈Σ, where i=2,3.Proof.
Letν≠μ in an infra-tp-soft T2-space Y,δ,Σ. Then there exist disjoint infrasoft open sets EΣ and FΣ totally containing ν and μ, respectively. Therefore, Eσ and Fσ are disjoint infraopen sets containing ν and μ, respectively. Hence, Y,δσ is an infra-T2-space.
To prove the proposition in case ofi=3, it suffices to prove the axiom of infraregularity. To do that, let G be an infraclosed subset of Y,δσ such that ν∉G. Since δ is infrasoft tp-regular, a soft set HΣ=σ,G:for eachσ∈Σ is an infrasoft closed subset of Y,δ,Σ such that ν∉HΣ. By assumption, there exist disjoint infrasoft open sets EΣ and FΣ such that HΣ⊆˜EΣ and ν∈FΣ. Thus, Eσ and Fσ are disjoint infraopen subsets of Y,δσ containing G and ν, respectively. Hence, Y,δσ is infraregular, as required.
To demonstrate that the converse of the proposition above fails, we supply the next example.Example 2.
ConsiderY and Σ as the same as in Example 1. Then δ=Φ,Y˜,EΣ,FΣ is an infrasoft topology on Y, where(4)EΣ=σ1,ν,σ2,μ,FΣ=σ1,μ,σ2,ν.
It is clear thatY,δ,Σ is not an infra-tp-soft T0-space. On the other hand, Y,δσ1 and Y,δσ2 are infra-T3 and infra-T4.
From the above example and the next two examples, we elucidate that there is not a relationship between infra-tp-soft Ti-spaces and their parametric infratopological spaces when i=0,1,4.Example 3.
In Example1, note that Y,δ2σ2 is not infra-T0 in spite of the fact that Y,δ2,Σ is infra-tp-soft T1.Example 4.
The collectionδ=Φ,Y˜,E1Σ,E3Σ,E3Σ,E4Σ is an infrasoft topology on Y=ν1,ν2,ν3 with Σ=σ1,σ2 as a set of parameters, where(5)E1Σ=σ1,ν1,σ2,Y;E2Σ=σ1,ν2,σ2,Y;E3Σ=σ1,ν3,σ2,Y,E4Σ=σ1,∅,σ2,Y.
One can check thatY,δ,Σ is infra-tp-soft T4. But Y,δσ2 is not infra-T4.Theorem 4.
LetY,δ,Σ be stable. Then, Y,δ,Σ is infra-tp-soft Ti iff Y,δσ is infra-Ti for each i=0,1,2,3,4.Proof.
SinceY,δ,Σ is stable, then E is an infraopen subset of Y,δσ if and only if σ,E:σ∈Σ is an infrasoft open subset of Y,δ,Σ. Hence, the desired result is proved.Remark 2.
Any extended infrasoft topological space is not infra-tp-soft regular; hence, it is not infra-tp-soft T3.Theorem 5.
LetY,δ,Σ be extended. If there exists σ∈Σ such that Y,δσ is infra-Ti, then Y,δ,Σ is infra-tp-soft Ti for i=0,1.Proof.
Wheni=1, let ν≠μ∈Y. Since Y,δσ is infra-T1, there are two infraopen sets U,V in δσ such that ν∈U,V and μ∈V,U. Since δ is extended, there are two infrasoft open sets GΣ,HΣ in δ such that(6)Gσ=UandGσ′=Y,for eachσ′≠σ,Hσ=VandGσ′=Y,for eachσ′≠σ.
Now,ν∈GΣ, μ∉GΣ, and μ∈HΣ, ν∉HΣ. Hence, X,δ,Σ is infra-tp-soft T1.
The casei=0 can be proved similarly.Theorem 6.
LetY,δ,Σ be extended. If all Y,δσ are infra-Ti, then X,δ,Σ is infra-tp-soft Ti for i=0,1,2,4.Proof.
We only prove the theorem wheni=4.
The case ofi=1 was proved in the above theorem. So, it suffices to prove the axioms of infrasoft normality. To do this, let HΣ,LΣ be two disjoint infrasoft closed sets. Then Hσ and Lσ are two disjoint infraclosed sets for each σ∈Σ. Since X,δσ is infranormal, there are two disjoint infraopen sets Uσ and Vσ such that Hσ⊆Uσ and Lσ⊆Vσ. Since Y,δ,Σ is extended, there are disjoint infrasoft open subsets GΣ,FΣ of Y,δ,Σ such that(7)Gσ=Uσ,for eachσ∈Σ,Fσ=Vσ,for eachσ∈Σ.
Thus,X,δ,Σ is infrasoft normal. Hence, it is infra-tp-soft T4.Theorem 7.
Every soft subspaceℤ,δℤ,Σ of an infra-tp-soft Ti-space Y,δ,Σ is an infra-tp-soft Ti-space, for i=0,1,2,3.Proof.
We give a proof fori=3. The other cases are made similarly.
First, we prove thatℤ,δℤ,Σ is infra-tp-soft T1. To do this, let ν≠μ∈ℤ⊆Y. Since Y,δ,Σ is infra-tp-soft T1, δ contains infrasoft open sets EΣ and FΣ such that ν∈EΣ,μ∉EΣ and μ∈FΣ,ν∉FΣ. Now, UΣ=Y˜∩˜EΣ and VΣ=Y˜∩˜FΣ are infrasoft open subsets of ℤ,δℤ,Σ such that ν∈UΣ,μ∉UΣ and μ∈VΣ,ν∉VΣ. Thus, ℤ,δℤ,Σ is infra-tp-soft T1.
Second, we prove thatℤ,δℤ,Σ is infra-tp-soft regular. To do this, let LΣ be an infrasoft closed subset of ℤ,δℤ,Σ and μ∈ℤ such that μ∉LΣ. Then there exists an infrasoft closed subset HΣ of Y,δ,Σ such that LΣ=Y˜∩˜HΣ. By assumption, there exist disjoint infrasoft open sets EΣ and FΣ such that HΣ⊆˜EΣ and μ∈FΣ. Now, Y˜∩˜EΣ and Y˜∩˜FΣ are infrasoft open subsets of ℤ,δℤ,Σ such that LΣ⊆˜Y˜∩˜EΣ and μ∈Y˜∩˜FΣ. Thus, ℤ,δℤ,Σ is infra-tp-soft regular. This ends the proof that ℤ,δℤ,Σ is infra-tp-soft T3.
Following similar technique given above, one can prove the following result.Theorem 8.
Every infrasoft closed subspaceℤ,δℤ,Σ of an infra-tp-soft T4-space Y,δ,Σ is an infra-tp-soft T4-space.Theorem 9.
The concepts of infra-tp-soft Ti-spaces are preserved under finite product spaces, where i=0,1,2.Proof.
We give a proof for the theorem whenj=2. The remaining two cases follow similar lines.
Without loss of generality, we considerY1,δ1,Σ1 and Y2,δ2,Σ2 as two infra-tp-soft T2-spaces. Let ν1,μ1≠ν2,μ2 in Y1×Y2. Then ν1≠ν2 or μ1≠μ2. Say, ν1≠ν2. By assumption, δ1 contains two disjoint infrasoft open sets EΣ1,FΣ1 such that ν1∈EΣ1 and ν2∉EΣ1, and ν2∈FΣ1 and ν1∉FΣ1. Now, EΣ1×Y2˜ and FΣ1×Y2˜ are disjoint infrasoft open sets such that ν1,μ1∈EΣ1×Y2˜ and ν2,μ2∉EΣ1×Y2˜, and ν2,μ2∈FΣ1×Y2˜ and ν1,μ1∉FΣ1×Y2˜. Hence, Y1×Y2 is infra-tp-soft T2.
We close this section by studying the behaviors of infra-tp-soft Ti-spaces under some types of soft maps.Proposition 10.
The inverse image of infra-tp-soft Ti-spaces under an infrasoft continuous map fη is infra-tp-soft Ti for i=0,1,2 provided that f is injective and η is surjective.Proof.
Wheni=2, consider fη:Y,δ,Σ⟶ℤ,θ,Ω is an infrasoft continuous map. Let ν≠μ∈Y. By the injectivity of f, there are only two distinct points ζ,ξ∈ℤ such that fν=ζ and fμ=ξ. By assumption, there are two disjoint infrasoft open sets EΩ and GΩ such that ζ∈EΩ, ξ∉EΩ and ξ∈GΩ, ζ∉GΩ. Now, fη−1EΩ and fη−1GΩ are disjoint infrasoft open sets. By the surjectivity of η we obtain from Proposition 1 that ν∈fη−1EΩ, μ∉fη−1EΩ and μ∈fη−1GΩ, ν∉fη−1GΩ. Hence, we get the coveted result.
In a similar manner, one can prove the next results.Proposition 11.
The inverse image of infra-tp-soft Ti-spaces under a bijective infrasoft continuous map fη is infra-tp-soft Ti for i=0,1,2,3,4.Proposition 12.
The image of infra-tp-soft Ti-spaces under a bijective infrasoft open map fη is infra-tp-soft Ti for i=0,1,2,3,4.Proposition 13.
The property of being an infra-tp-soft Ti-space i=0,1,2,3,4 is preserved under an infrasoft homeomorphism map.


## 4. Infra-tt-Soft Ti-Spaces i=0,1,2,3,4

In this part, we apply the relations of total belong and total nonbelong to define the concepts of infra-tt-soft Ti-spaces i=0,1,2,3,4, where tt is the abbreviation for the first letter of total belong and total nonbelong relations that are used to define these spaces. We provide some counterexamples to illustrate the relationships between these concepts and the interrelations between them and infra-tp-soft Ti-spaces.Definition 19.
Y,δ,Σ is said to be(i)
An infra-tt-soft T0-space if for every ν≠μ∈Y, there is an infrasoft open set EΣ such that ν∈EΣ,μ⋐EΣ, or μ∈EΣ,ν⋐EΣ;(ii)
An infra-tt-soft T1-space if for every ν≠μ∈Y, there are infrasoft open sets EΣ and FΣ such that ν∈EΣ,FΣ and μ∈FΣ,EΣ. In other words, ν∈EΣ,μ⋐EΣ and μ∈FΣ,ν⋐FΣ;(iii)
An infra-tt-soft T2-space if for every ν≠μ∈Y, there are two disjoint infrasoft open sets EΣ and FΣ such that ν∈EΣ\FΣ and μ∈FΣ\EΣ;(iv)
An infra-tt-soft regular space if for every infrasoft closed set HΣ such that ν⋐HΣ, there are disjoint infrasoft open sets EΣ and FΣ such that HΣ⊆˜EΣ and ν∈FΣ;(v)
An infra-tt-soft T3 (resp., infra-tt-soft T4) -space if it is infra-tt-soft regular (resp., infrasoft normal) and infra-tt-soft T1.
This section begins with the next result which describes the relationships between infra-tt-soft Ti-spaces as well as the interrelations between them and infra-tp-soft Ti-spaces.Proposition 14.
(i)
Every infra-tt-soft Ti-space is infra-tt-soft Ti−1 for i=1,2,3.(ii)
The concepts of infra-tt-soft T2-space and infra-tp-soft T2-space are identical.(iii)
Every infra-tt-soft Ti-space is infra-tp-soft Ti for i=0,1,4.(iv)
Every infra-tp-soft T3-space is infra-tt-soft T3.Proof.
(i)
The proof of the casesi=1,2 follows directly from the above definition. To prove case i=3, let ν,μ be two distinct elements in an infra-tt-soft T3-space. Then there exists an infrasoft open set EΣ such that ν∈EΣ and μ⋐EΣ. Now, EΣc is infrasoft closed such that ν⋐EΣc. By hypothesis, there exist disjoint infrasoft open sets UΣ and VΣ such that EΣc⊆˜UΣ and ν∈VΣ. Obviously, μ∈UΣ; hence, we obtain the desired result.(ii)
LetEΣ and FΣ be two disjoint soft sets such that ν∈EΣ and μ∈FΣ. Then μ⋐EΣ iff μ∉EΣ, and ν⋐FΣ iff ν∉FΣ. This proves the required equivalence.(iii)
obvious.(iv)
It is clear that every infra-tp-soft regular space is infra-tt-soft regular. According to Lemma 1, the concepts of infra-tt-soft T1-space and infra-tp-soft T1-space are identical. Hence, we get the coveted result.
To elucidate that the converses of the above proposition fail, we provide the next three counterexamples.Example 5.
LetY be any infinite set and ν,μ∈Y. We define the following three infrasoft topologies on Y with Σ as set of parameters such that Σ≠1.(8)δ1=Y˜∪EΣ⊆˜Y˜:ν∉EΣ;δ2=Φ∪EΣ⊆˜Y˜:EΣcis finite,δ3=Y˜∪EΣ⊆˜Y˜:ν∉EΣorμ∉EΣ.
Now, we can note thatY,δ1,Σ, Y,δ2,Σ, and Y,δ3,Σ are, respectively, infra-tt-soft T0, infra-tt-soft T1, and infra-tt-soft T2. On the other hand, we can check that Y,δ1,Σ is not infra-tt-soft T1 and Y,δ2,Σ is not infra-tt-soft T2. Also, Y,δ3,Σ is not infra-tt-soft T3 because there does not exist an infrasoft open set (except for the absolute soft set) containing an infrasoft closed set EΣ=σ,ν,μ:for eachσ∈Σ.Example 6.
ConsiderY,δ,Σ as the same as given in Example 4. Then Y,δ,Σ is an infra-tp-soft T4-space but it is not infra-tt-soft T0.Example 7.
Any discrete soft topological space is infra-tt-soft T3. On the other hand, it contains some infrasoft open sets which are not stable, so it is not infra-tp-soft regular. Hence, it is not infra-tp-soft T3.Remark 3.
We show thatY,δ3,Σ given in Example 5 is not infra-tt-soft T3. In contrast, one can check that it is infra-tt-soft T4. According to Remark 1 there is an infra-tt-soft T3-space which is not infra-tt-soft T4. Hence, the concepts of infra-tt-soft T3 and infra-tt-soft T4-spaces are independent of each other.Theorem 10.
LetY,δ,Σ be a stable space. Then the following statements hold.(i)
Y,δ,Σ is infra-tt-soft T0⇔Y,δ,Σ is infra-tp-soft T0.(ii)
Y,δ,Σ is infra-tt-soft T1⇔Y,δ,Σ is infra-tp-soft T1.(iii)
Y,δ,Σ is infra-tt-soft T2⇔Y,δ,Σ is infra-tp-soft T2.(iv)
Y,δ,Σ is infra-tt-soft T3⇔Y,δ,Σ is infra-tp-soft T3.(v)
Y,δ,Σ is infra-tt-soft T4⇔Y,δ,Σ is infra-tp-soft T4.Proof.
It comes from the fact that partial nonbelong and total nonbelong relations are identical with respect to the stable soft sets.
In what follows, we study and investigate some properties of infra-tt-soft Ti-spaces.Proposition 15.
If every singleton soft subset ofY,δ,Σ is infrasoft closed, then Y,δ,Σ is an infra-tt-soft T1-space.Proof.
Letν≠μ∈Y. By assumption, νΣc and μΣc are infrasoft open sets. Obviously, μ∈νΣc,ν⋐νΣc, and ν∈μΣc,μ⋐μΣc. Hence, Y,δ,Σ is infra-tt-soft T1.
The converse of the above proposition is not always true as the next counterexample explains.Example 8.
Letδ=ℕ˜∪FΣ⊏˜ℕ˜:FΣis finite be an infrasoft topology on the set of natural numbers ℕ with Σ as any set of parameters. It is clear that ℕ,δ,Σ is infra-tt-soft T1. But every singleton soft set is not infrasoft closed.Proposition 16.
LetY be a finite set. Then, Y,δ,Σ is infra-tt-soft T1 iff every singleton soft set is infrasoft open.Proof.
To prove the necessary condition, letνΣ be a singleton soft set. Then for each μi≠ν∈Y there exists an infrasoft open set EΣ such that ν∈EiΣ and μi⋐EiΣ. Now, ∩˜i=1n−1EiΣ=νΣ is an infrasoft open set, where n is the cardinality number of Y.
The sufficient condition is obvious.Corollary 2.
IfY,δ,Σ is a finite infra-tt-soft T1-space, then δ≥Y+2.Corollary 3.
IfY,δ,Σ is an infra-tt-soft T1-space, then νΣ=∩˜EΣ:ν∈EΣ∈δ for all ν∈Y.Proposition 17.
Y,δ,Σ is an infra-tt-soft T2-space iff for every ν≠μ, there exists an infrasoft open set EΣ such that ν∈EΣ and μ⋐ClδEΣ.Proof.
It comes from Proposition6 and item (ii) of Proposition 14.Proposition 18.
A finite infra-tp-soft T2-space Y,δ,Σ contains at least Y+2 infrasoft open sets, where Y>1.Proof.
It comes from Proposition8 and item (ii) of Proposition 14.Proposition 19.
IfY,δ,Σ is an infra-tt-soft regular space, then for each ν∈Y and infrasoft open set HΣ totally containing ν, there exists an infrasoft open set GΣ such that ν∈GΣ⊆˜ClδGΣ⊆˜HΣ.Proof.
It follows similar arguments given in the proof of Proposition7.Theorem 11.
LetY,δ,Σ be infra-tp-soft regular. Then Y,δ,Σ is infra-tt-soft T2 iff it is infra-tp-soft T0.Proof.
It comes from Theorem3 and item (ii) of Proposition 14.Corollary 4.
The following concepts are equivalent under an infra-tp-soft regular space Y,δ,Σ.(i)
Y,δ,Σ is infra-tt-soft T2;(ii)
Y,δ,Σ is infra-tt-soft T1;(iii)
Y,δ,Σ is infra-tt-soft T0;(iv)
Y,δ,Σ is infra-tp-soft T0;(v)
Y,δ,Σ is infra-tp-soft T1.Now, we determine the conditions under which the property infra-tt-soft Ti-spaces is transmitted from infrasoft topology to crisp infratopology and vice versa.Proposition 20.
IfY,δ,Σ is an infra-tt-soft Ti-space, then Y,δσ is infra-Ti for each σ∈Σ, where i=0,1,2.Proof.
Letν≠μ in an infra-tt-soft T0-space Y,δ,Σ. Without loss of generality, there is an infrasoft open set EΣ such that ν∈EΣ and μ⋐EΣ. Therefore, Eσ is an infraopen set such that ν∈Eσ and μ∉Eσ. Hence, Y,δσ is an infra-T0-space.
The casesi=1,2 are achieved similarly.
Example2 illustrates that the converse of the above proposition fails.
The next remark and example elucidate that there is not a relationship between infra-tt-soft Ti-spaces and their parametric infratopological spaces when i=3,4.Remark 4.
Y,δ3,Σ given in Example 5 is not infra-tt-soft T3 but all its parametric infratopological spaces are infra-T3. Also, Y,δ1,Σ given in Example 5 is not infra-tt-soft T4 but all its parametric infratopological spaces are infra-T4.Example 9.
LetY=ν1,ν2 be the universal set and Σ=σ1,σ2 be a parameters set. We define the following soft sets EiΣ over Y as follows:(9)E1Σ=σ1,ν1,σ2,Y;E2Σ=σ1,ν2,σ2,Y,E3Σ=σ1,∅,σ2,Y.
Thenδ=Φ,Y˜,E1Σ,E2Σ,E3Σ is an infrasoft topology on Y. Obviously, Y,δ,Σ is not infra-tt-soft T0. But its parametric infratopological space Y,δσ1 is infra-T4 and infra-T3.Theorem 12.
LetY,δ,Σ be stable (infra-tp-soft regular). Then, Y,δσ is infra-Ti iff Y,δ,Σ is infra-tt-soft Ti for each i=0,1,2,3,4.Proof.
It is straightforward.Theorem 13.
LetY,δ,Σ be extended. If all Y,δσ are infra-Ti, then X,δ,Σ is infra-tt-soft Ti for i=0,1,2,3,4.Proof.
The cases ofi=0,1,2 were proved in Theorem 1. The case of i=4 can be made similar to the proof of Theorem 6. It remains to prove case i=3. It suffices to prove the axiom of infra-tt-soft regularity. To do this, let HΣ be an infrasoft closed set such that ν⋐HΣ. Then ν∉Hσ for each σ∈Σ. Since Hσ is an infraclosed set and Y,δσ is infraregular, there exist disjoint infraopen sets Uσ and Vσ such that Hσ⊆Uσ and ν∈Vσ. Since Y,δ,Σ is extended, then there exist infrasoft open sets GΣ and FΣ such that Gσ=Uσ and Fσ=Vσ for each σ∈Σ. Obviously, HΣ⊆˜GΣ and ν∈FΣ. The disjointness of GΣ and FΣ proves that Y,δ,Σ is infra-tt-soft regular.
The following findings can be proved in a similar way to their counterparts in Section3.Theorem 14.
Every soft subspaceℤ,δℤ,Σ of an infra-tt-soft Ti-space Y,δ,Σ is an infra-tt-soft Ti-space for i=0,1,2,3.Theorem 15.
Every infrasoft closed subspaceℤ,δℤ,Σ of an infra-tt-soft T4-space Y,δ,Σ is an infra-tt-soft T4-space.Theorem 16.
The concepts of infra-tt-soft Ti-spaces are preserved under finite product spaces, where i=0,1,2.Proposition 21.
The inverse image of infra-tt-soft Ti-spaces under an infrasoft continuous map fη is infra-tt-soft Ti for i=0,1,2 provided that f is injective.Proposition 22.
The inverse image of infra-tt-soft Ti-spaces under a bijective infrasoft continuous map fη is infra-tt-soft Ti for i=0,1,2,3,4.Proposition 23.
The image of infra-tt-soft Ti-spaces under a bijective infrasoft open map fη is infra-tt-soft Ti for i=0,1,2,3,4.Proposition 24.
The property of being an infra-tt-soft Ti-space i=0,1,2,3,4 is preserved under an infrasoft homeomorphism map.In the end of this work, we summarize the relationships between the two introduced infrasoft separation axioms in Figure1.Figure 1
The relationships between infrasoft separation axioms.


## 5. Conclusion and Future Work

In this paper, we have introduced two types of separation axioms in the frame of infrasoft topological spaces with respect to distinct ordinary points. The reasons beyond this study are the following.(1)
Construct new classes of soft spaces and explore the (classical) topological properties that are valid in these spaces.(2)
Investigate these soft spaces in the frame of rough sets models as those given [34–36] and show whether they have an influence on the rough approximations and accuracy measures.(3)
Obtain new results induced from the interaction between these soft spaces and some celebrated concepts such as infrasoft connectedness and infrasoft compactness which we will introduce later.We outline the achievements of this work in the following.(1)
Formulate the concepts of infra-tp-soft Ti-spaces for i=0,1,2,3,4 using total belong and partial nonbelong relations.(2)
Formulate the concepts of infra-tt-soft Ti-spaces for i=0,1,2,3,4 using total belong and total nonbelong relations.(3)
Provide some counterexamples to show the relationships between the spaces in each type of them as well as the interrelationships between the two types.(4)
Study the transmission property of these two types of soft spaces between infrasoft topology and its parametric infratopologies.(5)
Discuss the image and preimage of these two types of soft spaces under some infrasoft maps.In upcoming papers, we plan to do the following.(1)
Establish novel types of soft spaces using the two pairs of relations⋐,∉ and ⋐,⋐.(2)
Define the celebrated generalizations of open sets in the context of infrasoft topological spaces such as infrasoftα-open, infrasoft semiopen, infrasoft preopen, infrasoft b-open, and infrasoft β-open sets.(3)
Carry out further investigations in the areas of rough set theory and infrasoft topologies.



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