Pre-Hausdorffness and Hausdorffness in Quantale-Valued Gauge Spaces

Abstract

In this paper, we characterize each of $T_0$, $T_1$, Pre-Hausdorff and Hausdorff separation properties for the category $\mathbf{L}\text{-}\mathbf{GS}$ of quantale-valued gauge spaces and $\mathcal{L}$-gauge morphisms. Moreover, we investigate how these concepts are related to each other in this category. We show that $T_0$, $T_1$ and $T_2$ are equivalent in the realm of Pre-Hausdorff quantale-valued gauge spaces. Finally, we compare our results with the ones in some other categories.

1. Introduction

In 1989, Lowen [1,2] introduced approach spaces as a common framework for both metric and topological spaces. More precisely, let X be a set and let $pqMe{t}^{\infty }(X)$ be the set of all extended pseudo-quasi metrics (pseudo-reflexive property and triangular inequality) on X, $\mathfrak{D}\subseteq pqMe{t}^{\infty }(X)$ and $d\in pqMe{t}^{\infty }(X)$, then

(i)

$\mathfrak{D}$ is called $\mathrm{ideal}$ if it is closed under the formation of finite suprema and if it is closed under the operation of taking smaller function.

(ii)

$\mathfrak{D}$ dominates d if $\forall x\in X,ϵ>0$ and $\omega <\infty$ there exists $e\in \mathfrak{D}$ such that $d(x,.)\wedge \omega \le e(x,.)+ϵ$ and if $\mathfrak{D}$ dominates d, then $\mathfrak{D}$ is called saturated.

If $\mathfrak{D}$ is an ideal in $pqMe{t}^{\infty }(X)$ and saturated, then $\mathfrak{D}$ is called gauge. The pair $(X,\mathfrak{D})$ is called a gauge-approach space [2]. Approach spaces can be defined by various distinct structures such as gauges, approach distances, approach systems or limit operators. Although these structures are conceptually different, they are equivalent, see [2].

Note that $\mathbf{Top}$, the category of topological spaces and continuous maps, and $\mathbf{Met}$, the category of metric spaces and non-expansive maps, can be embedded as a full and isomorphism-closed subcategory of $\mathbf{App}$, the topological category of approach spaces and contractions. Therefore, metric and topological spaces are mostly studied in $\mathbf{App}$.

Approach spaces are closely related to various disciplines and have several applications in practically all branches of mathematics, such as fixed point theory [3], convergence theory [4], domain theory [5] and probability theory [6]. Due to the widely recognized usefulness of approach spaces in research, several generalizations of approach spaces have emerged recently, including quantale-valued gauge spaces [7] and probabilistic approach spaces [8]. Quantale-valued bounded strong topological spaces and bounded interior spaces, which are frequently used by fuzzy mathematicians, have recently been used to characterize some quantale-valued approach spaces [9]. Although the classical approach structures (gauges, approach distances and approach systems) are equivalent, their arbitrary quantale generalizations are different, see Example 5.11 of [7,10].

Classical $T_0$ separation of topology has been extended to the topological category [11,12,13]. In 1991, Weck-Schwarz [14] and in 1995, Mehmet Baran and Hüseyin Altındiş [15] analyzed the relationship among these various generalizations of $T_0$ objects. $T_0$ objects are widely used to define and characterize various forms of Hausdorff [11] and sober [16] objects in topological categories.

Recall that a topological space $(B,\tau )$ is called a Pre-Hausdorff space if for each distinct pair $x,y\in B$, the subspace $(\{x,y\},{\tau }_{\{x,y\}})$ is not indiscrete; then there exist disjoint neigbourhoods of x and y [11].

In 1994, Mielke [17] showed the important role of Pre-$T_2$ objects in general theory of geometric realization, their associated intervals and corresponding homotopic structures. In addition, in 1999, Mielke [18] used Pre-$T_2$ objects of topological categories to characterize decidable objects in Topos theory [19]. Another uses of Pre-Hausdorff objects is to define Hausdorff objects [11] in an arbitrary topological category. There is also a relationship between Pre-$T_2$ structures and partitions in some categories [20,21].

Note that there is no relationship between $T_0$ property and Pre-$T_2$ property. For example, let B be a set with at least two elements and $\tau$ be the indiscrete topology on B, then $(B,\tau )$ is Pre-$T_2$, but it is not $T_0$. If we take the cofinite topology ${\tau }_c$ on the set of real numbers $\mathbb{R}$, then $(\mathbb{R},{\tau }_c)$ is $T_0$, but it is not Pre-$T_2$. However, if $(B,\tau )$ is a Pre-Hausdorff space, then by Theorem 3.5 of [22], all of $T_0$, $T_1$ and $T_2$ are equivalent.

The salient objectives of the paper are stated:

(i)

To explicitly characterize each of $T_0$, $\overline{T_0}$ and $T_1$ separation properties in the category $\mathbf{L}\text{-}\mathbf{GS}$ of quantale-valued gauge spaces and $\mathcal{L}$-gauge morphisms;

(ii)

To give the characterization of each of Pre-$\overline{T_2}$, $\overline{T_2}$ and $N{T}_2$ in the category $\mathbf{L}\text{-}\mathbf{GS}$;

(iii)

To examine the mutual relationship among all these separation axioms;

(iv)

To compare our results with the ones in some other categories.

2. Preliminaries

In order theory, the join of a subset A of a partially ordered set $(L,\le )$ where ≤ is any order on L, is the least upper bound (supremum) of A, denoted $\bigvee A$, and the meet of A is the greatest lower bound (infimum), denoted $\bigwedge A$. A complete lattice is a partially ordered set in which all subsets have both a join (⋁) and a meet (⋀). For any complete lattice, the top and bottom elements are denoted by ⊤ and ⊥, respectively. A complete lattice in which arbitrary joins distribute over arbitrary meets is said to be completely distributive.

Definition 1

([23]). A quantale $\mathcal{L}=(L,\le ,\ast )$ is a complete lattice $(L,\le )$ endowed with a binary operation * satisfying the following:

(i) 

$(L,\ast )$ is a semi group;

(ii) 

$({\bigvee }_{i\in I}{a}_i)\ast b={\bigvee }_{i\in I}(a_i\ast b)$ and $b\ast ({\bigvee }_{i\in I}{a}_i)={\bigvee }_{i\in I}(b\ast a_i)$ for all $a_i,b\in L$ and index-set I, i.e., ∗ is distributive over arbitrary joins.

Definition 2.

Let $(L,\le )$ be a complete lattice, then the well-below relation ⊲ and the well-above relation ≺ are defined by

(i) 

$a⊲b$ if $\forall K\subseteq L$ such that $b\le \bigvee K$ there exists $k\in K$ such that $a\le k$;

(ii) 

$a\prec b$ if $\forall K\subseteq L$ such that $\bigwedge K\le a$ there exists $k\in K$ such that $k\le b$.

Definition 3

([23]). A quantale $\mathcal{L}=(L,\le ,\ast )$ is said to be

(i) 

a commutative quantale if $(L,\ast )$ is a commutative semi-group;

(ii) 

an integral quantale if $a\ast \top =\top \ast a=a$ for all $a\in L$;

(iii) 

a value quantale if $\mathcal{L}$ is commutative and integral quantale with an underlying completely distributive lattice $(L,\le )$ such that $\perp ⊲\top$ and $a\vee b⊲\top$ for all $a,b⊲\top$;

(iv) 

a linearly ordered quantale if either $a\le b$ or $b\le a$ for all $a,b\in L$.

Example 1.

(i) 

Lawvere’s quantale, $L=[0,\infty ]$ with the opposite order and addition as the quantale operation, where $c+\infty =\infty +c=\infty$ for all $c\in L$, is a linearly ordered value quantale [23,24].

(ii) 

Let $\mathcal{L}=([0,1],\le ,\ast )$ be a triangular norm with a binary operation * defined as $\forall a,b\in [0,1],a\ast b=a.b$ and named as a product triangular norm [25]. The triple $\mathcal{L}=([0,1],\le ,.)$ is a commutative and integral quantale.

(iii) 

Let $\mathcal{L}=({▵}^{+},\le ,\ast )$ (a probabilistic quantale) where $\phi \ast \Psi =\phi .\Psi$ for all $\phi ,\Psi \in {▵}^{+}$, then $\mathcal{L}$ is not linearly ordered quantale [7].

In this sequel, we consider only integral and commutative quantales $\mathcal{L}$ with underlying completely distributive lattices.

Definition 4

(cf. [7]). Let X be a nonempty set. A map $m:X\times X\to \mathcal{L}=(L,\le ,\ast )$ is called an $\mathcal{L}$-metric on X if it satisfies for all $s\in X$, $m(s,s)=\top$, and for all $s,t,y\in X$, $m(s,t)\ast m(t,y)\le m(s,y)$. The pair $(X,m)$ is called an $\mathcal{L}$-metric space.

A map $f:(X,m_X)\to (Y,m_Y)$ is called an $\mathcal{L}$-metric morphism if $m_X(s_1,s_2)\le m_Y(f(s_1),f(s_2))$ for all $s_1,s_2\in X$.

The category whose objects are $\mathcal{L}$-metric spaces and morphisms are $\mathcal{L}$-metric morphisms is denoted by $\mathbf{L}\text{-}\mathbf{MET}$. Furthermore, we define $\mathbf{L}\text{-}\mathbf{MET}(\mathbf{X})$ as the set of all $\mathcal{L}$-metrics on X.

Example 2.

(i) 

If $\mathcal{L}=(\{0,1\},\le ,\wedge )$, then an $\mathcal{L}$-metric space is a preordered set.

(ii) 

If $\mathcal{L}$ is a Lawvere’s quantale, then an $\mathcal{L}$-metric space is an extended pseudo-quasi metric space.

(iii) 

If $\mathcal{L}=({▵}^{+},\le ,\ast )$, then an $\mathcal{L}$-metric space is a probabilistic quasi metric space [23].

Definition 5

(cf. [7]). Let $\mathcal{H}\subseteq \mathit{L}\text{-}\mathit{MET}(\mathit{X})$ and $m\in \mathit{L}\text{-}\mathit{MET}(\mathit{X})$.

(i) 

m is called locally supported by $\mathcal{H}$ if for all $s\in X$, $a⊲\top$, $\perp \prec \omega$, there is $n\in \mathcal{H}$ such that $n(s,.)\ast a\le m(s,.)\vee \omega$.

(ii) 

$\mathcal{H}$ is called locally directed if for all finite subsets ${\mathcal{H}}_0\subseteq \mathcal{H}$, ${\bigwedge }_{m\in {\mathcal{H}}_0}m$ is locally supported by $\mathcal{H}$.

(iii) 

$\mathcal{H}$ is called locally saturated if for all $m\in \mathit{L}\text{-}\mathit{MET}(\mathit{X})$ we have $m\in \mathcal{H}$ whenever m is locally supported by $\mathcal{H}$.

(iv) 

The set $\tilde{\mathcal{H}}=\{m\in \mathit{L}\text{-}\mathit{MET}(\mathit{X}):m$ is locally supported by $\mathcal{H}\}$ is called local saturation of $\mathcal{H}$.

Definition 6

(cf. [7]). Let X be a set. $\mathcal{G}\subseteq \mathit{L}\text{-}\mathit{MET}(\mathit{X})$ is called an $\mathcal{L}$-gauge if $\mathcal{G}$ satisfies the following:

(i) 

$\mathcal{G}\ne \varnothing$.

(ii) 

$m\in \mathcal{G}$ and $m\le n$ implies $n\in \mathcal{G}$.

(iii) 

$m,n\in \mathcal{G}$ implies $m\wedge n\in \mathcal{G}$.

(iv) 

$\mathcal{G}$ is locally saturated.

The pair $(X,\mathcal{G})$ is called an $\mathcal{L}$-gauge space.

A map $f:(X,\mathcal{G})\to (X^{\prime },{\mathcal{G}}^{\prime })$ is called an $\mathcal{L}$-gauge morphism if $m^{\prime }\circ (f\times f)\in \mathcal{G}$ whenever $m^{\prime }\in {\mathcal{G}}^{\prime }$.

The category whose objects are $\mathcal{L}$-gauge spaces and morphisms are $\mathcal{L}$-gauge morphisms is denoted by $\mathbf{L}\text{-}\mathbf{GS}$ (cf. [7]).

Definition 7

(cf. [7]). Let $(X,\mathcal{G})$ be an $\mathcal{L}$-gauge space and let $\mathcal{H}\subseteq \mathit{L}\text{-}\mathit{MET}(\mathit{X})$. If $\tilde{\mathcal{H}}=\mathcal{G}$, then $\mathcal{H}$ is called a basis for the gauge $\mathcal{G}$.

Proposition 1

(cf. [7]). Let $\mathcal{L}=(L,\le ,\ast )$ be a value quantale. If $\varnothing \ne \mathcal{H}\subseteq \mathit{L}\text{-}\mathit{MET}(\mathit{X})$ is locally directed, then $\mathcal{G}=\tilde{\mathcal{H}}$ is a gauge with $\mathcal{H}$ as a basis.

Lemma 1

(pcf. [7]). Let $\mathcal{L}=(L,\le ,\ast )$ be a value quantale, $(X_i,{\mathfrak{B}}_i)$ be the collection of $\mathcal{L}$-approach spaces and let $f_i:X\to (X_i,{\mathfrak{B}}_i)$ be a source. A basis for the initial $\mathcal{L}$-gauge on X is given by

$$\mathcal{H}=\{\underset{i\in K}{\bigwedge }{m}_i\circ (f_i\times f_i):K\subseteq Ifinite,m_i\in {\mathcal{G}}_i,\forall i\in I\}$$

Lemma 2.

Let X be a nonempty set and $(X,\mathcal{G})$ be an $\mathcal{L}$-gauge space.

(i) 

The discrete $\mathcal{L}$-gauge structure on X is given by ${\mathcal{G}}_{dis}=\mathit{L}\text{-}\mathit{MET}(\mathit{X})$ [26].

(ii) 

The indiscrete $\mathcal{L}$-gauge structure on X is given by ${\mathcal{G}}_{ind}=\{\top \}$ [7].

Note that for a value quantale $\mathcal{L}$, the category $\mathbf{L}\text{-}\mathbf{GS}$ is a topological category [27,28] over $\mathbf{Set}$ (the category of sets and functions) [7].

3. $T_0$ and $T_1$ Quantale-Valued Approach Spaces

Let X be a non-empty set and the wedge $X^2{\mathbb{V}}_{▵}{X}^2$ be the pushout of the diagonal $▵:X\to X^2$ along itself [11].

A point $(s,t)$ in $X^2{\mathbb{V}}_{▵}{X}^2$ is denoted as ${(s,t)}_1$ if it lies in the first component and as ${(s,t)}_2$ if it lies in the second component. Note that ${(s,t)}_1={(s,t)}_2$ if $s=t$.

Definition 8

(cf. [11]). $A:X^2{\mathbb{V}}_{▵}{X}^2\to X^3$, the principal axis map is defined by

$$\begin{array}{ccc}\hfill A{(s,t)}_i=\left\{\begin{array}{cc}(s,t,s),\hfill & i=1\hfill \\ (s,s,t),\hfill & i=2,\hfill \end{array}\right.& \hfill & \end{array}$$

$S:X^2{\mathbb{V}}_{▵}{X}^2\to X^3$, the skewed axis map is defined by

$$\begin{array}{ccc}\hfill S{(s,t)}_i=\left\{\begin{array}{cc}(s,t,t),\hfill & i=1\hfill \\ (s,s,t),\hfill & i=2,\hfill \end{array}\right.& \hfill & \end{array}$$

and $\nabla :X^2{\mathbb{V}}_{▵}{X}^2\to X^2$, the fold map is defined by $\nabla {(s,t)}_i=(s,t)$ for $i=1,2$.

Definition 9.

Let $U:\mathcal{E}\to \mathbf{Set}$ be a topological functor and $X\in Ob(\mathcal{E})$ with $U(X)=B$.

(i) 

X is $\overline{T_0}$ if the initial lift of the U-source $\{A:B^2{\mathbb{V}}_{▵}{B}^2\to U(X^3)=B^3$ and $\nabla :B^2{\mathbb{V}}_{▵}{B}^2\to UD(B^2)=B^2\}$ is discrete, where D is the discrete functor [11].

(ii) 

X is $T_0$ if X does not contain an indiscrete subspace with at least two points [13].

(iii) 

X is $T_1$ if the initial lift of the U-source $\{S:B^2{\mathbb{V}}_{▵}{B}^2\to U(X^3)=B^3$ and $\nabla :B^2{\mathbb{V}}_{▵}{B}^2\to UD(B^2)=B^2\}$ is discrete [11].

In $\mathbf{Top}$, both $\overline{T_0}$ and $T_0$ are equivalent, and they reduce to the usual $T_0$ separation property [11,13]. Similarly, $T_1$ reduces to classical $T_1$ property [11].

Theorem 1.

An $\mathcal{L}$-gauge space $(X,\mathcal{G})$ is $\overline{T_0}$ if for all $s,t\in X$ with $s\ne t$, there exists $m\in \mathcal{G}$ such that $m(s,t)\wedge m(t,s)=\perp$.

Proof. 

Suppose $(X,\mathcal{G})$ is $\overline{T_0}$ and $s,t\in X$ with $s\ne t$. Let ${\mathcal{H}}_{dis}=\{m_{dis}\}$ be a basis for the discrete $\mathcal{L}$-gauge where $m_{dis}$ is the discrete $\mathcal{L}$-metric on $X^2{\mathbb{V}}_{▵}{X}^2$. For ${(s,t)}_1,{(s,t)}_2\in X^2{\mathbb{V}}_{▵}{X}^2$ with ${(s,t)}_1\ne {(s,t)}_2$. Note that

$$\begin{array}{ccc}\hfill m_{dis}(\nabla {(s,t)}_1,\nabla {(s,t)}_2)& =& m_{dis}((s,t),(s,t))=\top \hfill \\ \hfill m({\pi }_{1}A{(s,t)}_1,{\pi }_{1}A{(s,t)}_2)& =& m({\pi }_1(s,t,s),{\pi }_1(s,s,t))=m(s,s)=\top \hfill \\ \hfill m({\pi }_{2}A{(s,t)}_1,{\pi }_{2}A{(s,t)}_2)& =& m({\pi }_2(s,t,s),{\pi }_2(s,s,t))=m(t,s)\hfill \\ \hfill m({\pi }_{3}A{(s,t)}_1,{\pi }_{3}A{(s,t)}_2)& =& m({\pi }_3(s,t,s),{\pi }_3(s,s,t))=m(s,t)\hfill \end{array}$$

Since ${(s,t)}_1\ne {(s,t)}_2$ and $(X,\mathcal{G})$ is $\overline{T_0}$, by Lemma 1 and Definition 9 (i),

$$\begin{array}{ccc}\hfill \perp & =& \bigwedge \{m_{dis}(\nabla {(s,t)}_1,\nabla {(s,t)}_2),m({\pi }_{k}A{(s,t)}_1,{\pi }_{k}A{(s,t)}_2)(k=1,2,3)\}\hfill \\ & =& \bigwedge \{\top ,m(s,t),m(t,s)\}\hfill \\ & =& m(s,t)\wedge m(t,s)\hfill \end{array}$$

Conversely, let $\overline{\mathcal{H}}$ be the initial $\mathcal{L}$-gauge basis on $X^2{\mathbb{V}}_{▵}{X}^2$ induced by $A:X^2{\mathbb{V}}_{▵}{X}^2\to U(X^3,{\mathcal{G}}^3)=X^3$ and $\nabla :X^2{\mathbb{V}}_{▵}{X}^2\to U(X^2,{\mathcal{G}}_{dis})=X^2$, where, by Lemma 2 (i), ${\mathcal{G}}_{dis}=\mathbf{L}\text{-}\mathbf{MET}(\mathbf{X})$ is the discrete $\mathcal{L}$-gauge on $X^2$, and ${\mathcal{G}}^3$ is the product structure on $X^3$ induced by the projection maps ${\pi }_k:X^3\to X$ for $k=1,2,3$.

Suppose for all $s,t\in X$ with $s\ne t$, there exists $m\in \mathcal{G}$ such that $m(s,t)\wedge m(t,s)=\perp$. Let $\overline{m}\in \overline{\mathcal{H}}$ and $u,v\in X^2{\mathbb{V}}_{▵}{X}^2$.

Case I: If $u=v$, then $\overline{m}(u,v)=\overline{m}(u,u)=\top$

Case II: If $u\ne v$ and $\nabla u\ne \nabla v$, then $m_{dis}(\nabla u,\nabla v)=\perp$ since $m_{dis}$ is discrete. By Lemma 1,

$$\begin{array}{ccc}\hfill \overline{m}(u,v)& =& \bigwedge \{m_{dis}(\nabla u,\nabla v),m({\pi }_{k}Au,{\pi }_{k}Av)(k=1,2,3)\}\hfill \\ & =& \bigwedge \{\perp ,m({\pi }_{1}Au,{\pi }_{1}Av),m({\pi }_{2}Au,{\pi }_{2}Av),m({\pi }_{3}Au,{\pi }_{3}Av)\}=\perp \hfill \end{array}$$

Case III: Suppose $u\ne v$ and $\nabla u=\nabla v$. If $\nabla u=(s,t)=\nabla v$ for some $s,t\in X$ with $s\ne t$, then $u={(s,t)}_1$ and $v={(s,t)}_2$ or $u={(s,t)}_2$ and $v={(s,t)}_1$ since $u\ne v$.

If $u={(s,t)}_1$ and $v={(s,t)}_2$, then

$$\begin{array}{ccc}\hfill m_{dis}(\nabla u,\nabla v)& =& m_{dis}(\nabla {(s,t)}_1,\nabla {(s,t)}_2)\hfill \\ & =& m_{dis}((s,t),(s,t))=\top \hfill \end{array}$$

$$\begin{array}{ccc}\hfill m({\pi }_{1}Au,{\pi }_{1}Av)& =& m({\pi }_{1}A{(s,t)}_1,{\pi }_{1}A{(s,t)}_2)\hfill \\ & =& m(s,s)=\top \hfill \end{array}$$

$$\begin{array}{ccc}\hfill m({\pi }_{2}Au,{\pi }_{2}Av)& =& m({\pi }_{2}A{(s,t)}_1,{\pi }_{2}A{(s,t)}_2)=m(t,s)\hfill \\ \hfill m({\pi }_{3}Au,{\pi }_{3}Av)& =& m({\pi }_{3}A{(s,t)}_1,{\pi }_{3}A{(s,t)}_2)=m(s,t)\hfill \end{array}$$

It follows that

$$\begin{array}{ccc}\hfill \overline{m}(u,v)& =& \overline{m}({(s,t)}_1,{(s,t)}_2)\hfill \\ & =& \bigwedge \{m_{dis}(\nabla {(s,t)}_1,\nabla {(s,t)}_2),m({\pi }_{k}A{(s,t)}_1,{\pi }_{k}A{(s,t)}_2)(k=1,2,3)\}\hfill \\ & =& \bigwedge \{\top ,m(s,t),m(t,s)\}\hfill \\ & =& m(s,t)\wedge m(t,s)\hfill \end{array}$$

By the assumption $m(s,t)\wedge m(t,s)=\perp$, and we have $\overline{m}(u,v)=\perp$.

Similarly, if $u={(s,t)}_2$ and $v={(s,t)}_1$, then $\overline{m}(u,v)=\perp$.

Therefore, for all $u,v\in X^2{\mathbb{V}}_{▵}{X}^2$, we have

$$\begin{array}{c}\hfill \overline{m}(u,v)=\left\{\begin{array}{cc}\top ,\hfill & u=v\hfill \\ \perp ,\hfill & u\ne v\hfill \end{array}\right.\end{array}$$

and by Lemma 2 (i), $\overline{m}$ is the discrete $\mathcal{L}$-metric on $X^2{\mathbb{V}}_{▵}{X}^2$. Hence, by Definition 9 (i), $(X,\mathcal{G})$ is $\overline{T_0}$. □

Note that in a quantale $(L,\le ,\ast )$, if $p\in L$ and $p\ne \top$, then p is called a prime element if $a\wedge b\le p$ implies $a\le p$ or $b\le p$ for all $a,b\in L$.

Corollary 1.

Let $(X,\mathcal{G})$ be an $\mathcal{L}$-gauge space where $\mathcal{L}$ has a prime bottom element. $(X,\mathcal{G})$ is $\overline{T_0}$ if for all $s,t\in X$ with $s\ne t$, there exists $m\in \mathcal{G}$ such that $m(s,t)=\perp$ or $m(t,s)=\perp$.

Proof. 

It follows from the definition of the prime bottom element and Theorem 1. □

Theorem 2.

An $\mathcal{L}$-gauge space $(X,\mathcal{G})$ is $T_0$ if for all $s,t\in X$ with $s\ne t$, there exists $m\in \mathcal{G}$ such that $m(s,t)<\top$ or $m(t,s)<\top$.

Proof. 

Let $(X,\mathcal{G})$ be $T_0$, $B=\{s,t\}\subset X$ and ${\mathcal{H}}_B$ be the initial $\mathcal{L}$-gauge basis induced by $i:B\to (X,\mathcal{L})$ and $m_B\in {\mathcal{H}}_B$. For all $s,t\in X$ with $s\ne t$, $m_B(s,t)=m(i(s),i(t))=m(s,t)$ or $m_B(t,s)=m(i(t),i(s))=m(t,s)$. It follows that $m(s,t)<\top$ or $m(t,s)<\top$; otherwise $m(s,t)=\top =m(t,s)$, and X contains an indiscrete subspace with at least two elements.

Conversely, suppose the condition holds. Let B be an indiscrete subspace of X with at least two elements $s,t\in B$ with $s\ne t$. Let ${\mathcal{H}}_B$ be the initial $\mathcal{L}$-gauge basis induced by $i:B\to (X,\mathcal{L})$ and $m_B\in {\mathcal{H}}_B$. It follows that $T=m_B(s,t)=m(i(s),i(t))=m(s,t)$ and $T=m_B(t,s)=m(i(t),i(s))=m(t,s)$ and consequently, $m(s,t)=\top =m(t,s)$, a contradiction to our assumption. Therefore, X does not contain an indiscrete subspace with at least two elements. Hence, by Definition 9 (ii), $(X,\mathcal{G})$ is $T_0$.

□

Theorem 3.

An $\mathcal{L}$-gauge space $(X,\mathcal{G})$ is $T_1$ if for all $s,t\in X$ with $s\ne t$, there exists $m\in \mathcal{G}$ such that $m(s,t)=\perp =m(t,s)$.

Proof. 

Suppose that $(X,\mathcal{G})$ is $T_1$ and $s,t\in X$ with $s\ne t$. Let $u={(s,t)}_1$, $v={(s,t)}_2\in X^2{\mathbb{V}}_{▵}{X}^2$. Note that

$$\begin{array}{ccc}\hfill m_{dis}(\nabla u,\nabla v)& =& m_{dis}((s,t),(s,t))=\top \hfill \\ \hfill m({\pi }_{1}Su,{\pi }_{1}Sv)& =& m({\pi }_1(s,t,t),{\pi }_1(s,s,t))=m(s,s)=\top \hfill \\ \hfill m({\pi }_{2}Su,{\pi }_{2}Sv)& =& m({\pi }_2(s,t,t),{\pi }_2(s,s,t))=m(t,s)\hfill \\ \hfill m({\pi }_{3}Su,{\pi }_{3}Sv)& =& m({\pi }_3(s,t,t),{\pi }_3(s,s,t))=m(t,t)=\top \hfill \end{array}$$

where $m_{dis}$ is the discrete $\mathcal{L}$-metric on $X^2{\mathbb{V}}_{▵}{X}^2$ and ${\pi }_k:X^3\to X$ are the projection maps for $k=1,2,3$. Since $u\ne v$ and $(X,\mathcal{G})$ is $T_1$, by Lemma 1 and Definition 9 (iii),

$$\begin{array}{ccc}\hfill \perp & =& \bigwedge \{m_{dis}(\nabla u,\nabla v),m({\pi }_{k}Su,{\pi }_{k}Sv)(k=1,2,3)\}=\bigwedge \{\top ,m(t,s)\}=m(t,s)\hfill \end{array}$$

Similarly, if $u={(s,t)}_2$, $v={(s,t)}_1\in X^2{\mathbb{V}}_{▵}{X}^2$, then

$$\begin{array}{ccc}\hfill \perp & =& \bigwedge \{m_{dis}(\nabla u,\nabla v),m({\pi }_{k}Su,{\pi }_{k}Sv)(k=1,2,3)\}=\bigwedge \{\top ,m(s,t)\}=m(s,t)\hfill \end{array}$$

Conversely, let $\overline{\mathcal{H}}$ be the initial $\mathcal{L}$-gauge basis on $X^2{\mathbb{V}}_{▵}{X}^2$ induced by $S:X^2{\mathbb{V}}_{▵}{X}^2\to U(X^3,{\mathcal{G}}^3)=X^3$ and $\nabla :X^2{\mathbb{V}}_{▵}{X}^2\to U(X^2,{\mathcal{G}}_{dis})=X^2$ where, by Proposition 2, ${\mathcal{G}}_{dis}=\mathbf{L}\text{-}\mathbf{MET}(\mathbf{X})$ is the discrete $\mathcal{L}$-gauge on $X^2$ and ${\mathcal{G}}^3$ is the product structure on $X^3$ induced by the projection maps ${\pi }_k:X^3\to X$ for $k=1,2,3$.

Suppose for all $s,t\in X$ with $s\ne t$, there exists $m\in \mathcal{G}$ such that $m(s,t)=\perp =m(t,s)$. Let $\overline{m}\in \overline{\mathcal{H}}$ and $u,v\in X^2{\mathbb{V}}_{▵}{X}^2$.

Case I: If $u=v$, then $\overline{m}(u,v)=\overline{m}(u,u)=\top$

Case II: If $u\ne v$ and $\nabla u\ne \nabla v$, then $m_{dis}(\nabla u,\nabla v)=\perp$ since $m_{dis}$ is a discrete structure on $X^2$. By Lemma 1,

$$\begin{array}{ccc}\hfill \overline{m}(u,v)& =& \bigwedge \{m_{dis}(\nabla u,\nabla v),m({\pi }_{k}Su,{\pi }_{k}Sv)(k=1,2,3)\}\hfill \\ & =& \bigwedge \{\perp ,m({\pi }_{1}Su,{\pi }_{1}Sv),m({\pi }_{2}Su,{\pi }_{2}Sv),m({\pi }_{3}Su,{\pi }_{3}Sv)\}=\perp \hfill \end{array}$$

Case III: Suppose $u\ne v$ and $\nabla u=\nabla v$. If $\nabla u=(s,t)=\nabla v$ for some $s,t\in X$ with $s\ne t$, then $u={(s,t)}_1$ and $v={(s,t)}_2$ or $u={(s,t)}_2$ and $v={(s,t)}_1$ since $u\ne v$.

If $u={(s,t)}_1$ and $v={(s,t)}_2$, then by Lemma 1,

$$\begin{array}{ccc}\hfill \overline{m}(u,v)& =& \overline{m}({(s,t)}_1,{(s,t)}_2)\hfill \\ & =& \bigwedge \{m_{dis}(\nabla {(s,t)}_1,\nabla {(s,t)}_2),m({\pi }_{k}S{(s,t)}_1,{\pi }_{k}S{(s,t)}_2)(k=1,2,3)\hfill \\ & =& \bigwedge \{\top ,m(t,s)\}=m(t,s)=\perp \hfill \end{array}$$

since $s\ne t$ and $m(t,s)=\perp$.

Similarly, if $u={(s,t)}_2$ and $v={(s,t)}_1$, then $m(s,t)=\perp$.

Hence, for all $u,v\in X^2{\mathbb{V}}_{▵}{X}^2$, we obtain

$$\begin{array}{c}\hfill \overline{m}(u,v)=\left\{\begin{array}{cc}\top ,\hfill & u=v\hfill \\ \perp ,\hfill & u\ne v\hfill \end{array}\right.\end{array}$$

and it follows that $\overline{m}$ is the discrete $\mathcal{L}$-metric on $X^2{\mathbb{V}}_{▵}{X}^2$. By Definition 9 (iii), $(X,\mathcal{G})$ is $T_1$.

□

4. (Pre-)Hausdorff $\mathcal{L}$-Gauge Spaces

Definition 10.

Let $U:\mathcal{E}\to \mathbf{Set}$ be a topological functor and $X\in Ob(\mathcal{E})$ with $U(X)=B$.

(i) 

X is Pre-$\overline{T_2}$ if the initial lifts of U-sources $\{A:B^2{\mathbb{V}}_{▵}{B}^2\to U(X^3)=B^3$ and $\{S:B^2{\mathbb{V}}_{▵}{B}^2\to U(X^3)=B^3$ coincide [11].

(ii) 

X is $\overline{T_2}$ if X is $\overline{T_0}$ and Pre-$\overline{T_2}$ [11].

(iii) 

X is $N{T}_2$ if X is $T_0$ and Pre-$\overline{T_2}$ [29].

In $\mathbf{Top}$, both $\overline{T_2}$ and $N{T}_2$ are equivalent, and they reduce to the usual $T_2$ [11,13]. By Theorem 2.1 of [30], a topological space $(B,\tau )$ is Pre-Hausdorff if the initial topologies on $B^2{\mathbb{V}}_{▵}{B}^2$ induced by the maps A and S agree.

Theorem 4.

An $\mathcal{L}$-gauge space $(X,\mathcal{G})$ is Pre-$\overline{T_2}$ if there exists $m\in \mathcal{G}$ such that the following conditions are satisfied.

(I) 

For all $s,t\in X$ with $s\ne t$, $m(s,t)\wedge m(t,s)=m(s,t)=m(t,s)$.

(II) 

For any three distinct points $s,t,y\in X$, $m(t,s)\wedge m(y,s)\wedge m(t,y)=m(t,s)\wedge m(y,s)=m(s,t)\wedge m(y,t)=m(y,s)\wedge m(t,y)$.

(III) 

For any four distinct points $s,t,y,z\in X$, $m(s,y)\wedge m(t,y)\wedge m(t,z)=m(s,y)\wedge m(t,y)\wedge m(s,z)=m(s,z)\wedge m(t,y)\wedge m(t,z)=m(s,y)\wedge m(t,z)\wedge m(s,z)$.

Proof. 

Suppose that $(X,\mathcal{G})$ is Pre-$\overline{T_2}$ and $s,t\in X$ with $s\ne t$. Let ${\pi }_k:X^3\to X$, $k=1,2,3$ be the projection maps.

Suppose $u={(s,t)}_1,v={(s,t)}_2\in X^2{\mathbb{V}}_{▵}{X}^2$. Note that

$$\begin{array}{ccc}\hfill m({\pi }_{1}Au,{\pi }_{1}Av)& =& m({\pi }_1(s,t,s),{\pi }_1(s,s,t))=m(s,s)=\top \hfill \\ \hfill m({\pi }_{2}Au,{\pi }_{2}Av)& =& m({\pi }_2(s,t,s),{\pi }_2(s,s,t))=m(t,s)\hfill \\ \hfill m({\pi }_{3}Au,{\pi }_{3}Av)& =& m({\pi }_3(s,t,s),{\pi }_3(s,s,t))=m(s,t)\hfill \end{array}$$

and

$$\begin{array}{ccc}\hfill m({\pi }_{1}Su,{\pi }_{1}Sv)& =& m({\pi }_1(s,t,t),{\pi }_1(s,s,t))=m(s,s)=\top \hfill \\ \hfill m({\pi }_{2}Su,{\pi }_{2}Sv)& =& m({\pi }_2(s,t,t),{\pi }_2(s,s,t))=m(t,s)\hfill \\ \hfill m({\pi }_{3}Su,{\pi }_{3}Sv)& =& m({\pi }_3(s,t,t),{\pi }_3(s,s,t))=m(t,t)=\top \hfill \end{array}$$

Since $(X,\mathcal{G})$ is Pre-$\overline{T_2}$ and by Definition 10 (i), we have

$$\begin{array}{ccc}\hfill \bigwedge \{m({\pi }_{k}Au,{\pi }_{k}Av):k=1,2,3\}& =& \bigwedge \{m({\pi }_{k}Su,{\pi }_{k}Sv):k=1,2,3\}\hfill \\ \hfill \bigwedge \{\top ,m(s,t),m(t,s)\}& =& \bigwedge \{\top ,m(t,s)\}\hfill \\ \hfill m(s,t)\wedge m(t,s)& =& m(t,s)\hfill \end{array}$$

Let $u={(s,t)}_2,v={(s,t)}_1\in X^2{\mathbb{V}}_{▵}{X}^2$. Similarly, since $(X,\mathcal{G})$ is Pre-$\overline{T_2}$ and by Definition 10 (i), we have $m(s,t)\wedge m(t,s)=m(s,t)$, and consequently $m(s,t)\wedge m(t,s)=m(s,t)=m(t,s)$.

Let $s,t,y$ be any three distinct points of X. Since $(X,\mathcal{G})$ is Pre-$\overline{T_2}$ and by Definition 10 (i), we have

$$\begin{array}{ccc}\hfill \bigwedge \{m({\pi }_{k}A{(t,y)}_1,{\pi }_{k}A{(s,y)}_2):k=1,2,3\}& =& \bigwedge \{m({\pi }_{k}S{(t,y)}_1,{\pi }_{k}S{(s,y)}_2):k=1,2,3\}\hfill \\ \hfill \bigwedge \{m(t,s),m(y,s),m(t,y)\}& =& \bigwedge \{\top ,m(t,s),m(y,s)\},\hfill \end{array}$$

$$\begin{array}{ccc}\hfill \bigwedge \{m({\pi }_{k}A{(s,y)}_1,{\pi }_{k}A{(t,y)}_2):k=1,2,3\}& =& \bigwedge \{m({\pi }_{k}S{(s,y)}_1,{\pi }_{k}S{(t,y)}_2):k=1,2,3\}\hfill \\ \hfill \bigwedge \{m(s,t),m(y,t),m(s,y)\}& =& \bigwedge \{\top ,m(s,t),m(y,t)\},\hfill \end{array}$$

and

$$\begin{array}{ccc}\hfill \bigwedge \{m({\pi }_{k}A{(s,t)}_1,{\pi }_{k}A{(y,t)}_2):k=1,2,3\}& =& \bigwedge \{m({\pi }_{k}S{(s,t)}_1,{\pi }_{k}S{(y,t)}_2):k=1,2,3\}\hfill \\ \hfill \bigwedge \{m(s,y),m(t,y),m(s,t)\}& =& \bigwedge \{\top ,m(s,y),m(t,y)\}.\hfill \end{array}$$

By condition (I), we have $m(t,s)\wedge m(y,s)\wedge m(t,y)=m(t,s)\wedge m(y,s)=m(s,t)\wedge m(y,t)=m(y,s)\wedge m(t,y)$.

Let $s,t,y,z$ be any four distinct points of X. Since $(X,\mathcal{G})$ is Pre-$\overline{T_2}$ and by Definition 10 (i), we have

$$\begin{array}{ccc}\hfill \bigwedge \{m({\pi }_{k}A{(s,t)}_1,{\pi }_{k}A{(y,z)}_2):k=1,2,3\}& =& \bigwedge \{m({\pi }_{k}S{(s,t)}_1,{\pi }_{k}S{(y,z)}_2):k=1,2,3\}\hfill \\ \hfill \bigwedge \{m(s,y),m(t,y),m(s,z)\}& =& \bigwedge \{m(s,y),m(t,y),m(t,z)\},\hfill \end{array}$$

$$\begin{array}{ccc}\hfill \bigwedge \{m({\pi }_{k}A{(s,t)}_1,{\pi }_{k}A{(z,y)}_2):k=1,2,3\}& =& \bigwedge \{m({\pi }_{k}S{(s,t)}_1,{\pi }_{k}S{(z,y)}_2):k=1,2,3\}\hfill \\ \hfill \bigwedge \{m(s,z),m(t,z),m(s,y)\}& =& \bigwedge \{m(s,z),m(t,z),m(t,y)\},\hfill \end{array}$$

and

$$\begin{array}{ccc}\hfill \bigwedge \{m({\pi }_{k}A{(z,y)}_1,{\pi }_{k}A{(t,s)}_2):k=1,2,3\}& =& \bigwedge \{m({\pi }_{k}S{(z,y)}_1,{\pi }_{k}S{(t,s)}_2):k=1,2,3\}\hfill \\ \hfill \bigwedge \{m(z,t),m(y,t),m(z,s)\}& =& \bigwedge \{m(z,t),m(y,t),m(y,s)\}.\hfill \end{array}$$

By condition (I), we have $m(s,y)\wedge m(t,y)\wedge m(t,z)=m(s,y)\wedge m(t,y)\wedge m(s,z)=m(s,z)\wedge m(t,y)\wedge m(t,z)=m(s,y)\wedge m(t,z)\wedge m(s,z)$.

Conversely, suppose that the conditions hold. Then, we will show that $(X,\mathcal{G})$ is Pre-$\overline{T_2}$. Let $\overline{\mathcal{H}}$ and ${\mathcal{H}}^{\prime }$ be two initial $\mathcal{L}$-gauge bases on $X^2{\mathbb{V}}_{▵}{X}^2$ induced by $A:X^2{\mathbb{V}}_{▵}{X}^2\to U(X^3,{\mathcal{G}}^3)=X^3$ and $S:X^2{\mathbb{V}}_{▵}{X}^2\to U(X^3,{\mathcal{G}}^3)=X^3$, respectively, and ${\mathcal{G}}^3$ be the product structure on $X^3$ induced by ${\pi }_k:X^3\to X$ the projection map for $k=1,2,3$. Let $\overline{m}$ and $m^{\prime }$ be any two $\mathcal{L}$-metrics in $\overline{\mathcal{H}}$ and ${\mathcal{H}}^{\prime }$, respectively. We need to show that $\overline{m}=m^{\prime }$.

First, note that $\overline{m}$ and $m^{\prime }$ are symmetric by assumption (I), $m(s,t)\wedge m(t,s)=m(s,t)=m(t,s)$.

Suppose u and v are any two points in $X^2{\mathbb{V}}_{▵}{X}^2$.

If $u=v$, then $\overline{m}(u,v)=\overline{m}(u,u)=\top =m^{\prime }(u,u)=m^{\prime }(u,v)$.

If $u\ne v$, and they are in the same component of $X^2{\mathbb{V}}_{▵}{X}^2$, i.e., $u={(s,t)}_i$ and $v={(y,z)}_i$ for $i=1,2$, then

$$\begin{array}{ccc}\hfill \overline{m}(u,v)& =& \bigwedge \{m({\pi }_{k}Au,{\pi }_{k}Av):k=1,2,3\}\hfill \\ & =& \bigwedge \{m(s,y),m(t,z)\}\hfill \\ & =& \bigwedge \{m({\pi }_{k}Su,{\pi }_{k}Sv):k=1,2,3\}\hfill \\ & =& m^{\prime }(u,v)\hfill \end{array}$$

Suppose $u\ne v$, and they are in the different component of $X^2{\mathbb{V}}_{▵}{X}^2$. We have:

Case I: $u={(s,t)}_1$ or ${(t,s)}_1$ and $v={(s,t)}_2$ or ${(t,s)}_2$ for $s\ne t$.

If $u={(s,t)}_1$ and $v={(s,t)}_2$ (resp. $v={(t,s)}_2$), then

$$\overline{m}(u,v)=\bigwedge \{m({\pi }_{k}Au,{\pi }_{k}Av):k=1,2,3\}=m(s,t)\wedge m(t,s)(resp.m(s,t)),$$

$$m^{\prime }(u,v)=\bigwedge \{m({\pi }_{k}Su,{\pi }_{k}Sv):k=1,2,3\}=m(t,s)(resp.m(s,t)\wedge m(t,s))$$

Consequently, we have $\overline{m}(u,v)=m^{\prime }(u,v)$ by assumption (I).

Similarly, if $u={(t,s)}_1$ and $v={(s,t)}_2$ (resp. $v={(t,s)}_2$), then $\overline{m}(u,v)=m^{\prime }(u,v)$.

Case II: $u={(s,t)}_1$, ${(s,y)}_1$, ${(t,y)}_1$, ${(t,s)}_1$, ${(y,s)}_1$ or ${(y,t)}_1$ and $v={(s,t)}_2$, ${(s,y)}_2$, ${(t,y)}_2$, ${(t,s)}_2$, ${(y,s)}_2$ or ${(y,t)}_2$ for three distinct points $s,t,y$ of X.

If $u={(s,t)}_1$ or ${(t,s)}_1$ and $v={(s,t)}_2$ or ${(t,s)}_2$, $u={(s,y)}_1$ or ${(y,s)}_1$ and $v={(s,y)}_2$ or ${(y,s)}_2$, $u={(t,y)}_1$ or ${(y,t)}_1$ and $v={(t,y)}_2$ or ${(y,t)}_2$, then by case I, we have $\overline{m}(u,v)=m^{\prime }(u,v)$.

If $u={(s,t)}_1$ and $v={(s,y)}_2$ or ${(t,y)}_2$ (resp. $u={(t,s)}_1$ and $v={(s,y)}_2$ or ${(t,y)}_2$), then by assumption (I),

$$\overline{m}(u,v)=\bigwedge \{m({\pi }_{k}Au,{\pi }_{k}Av):k=1,2,3\}=m(t,s)\wedge m(s,y)(resp.m(t,s)\wedge m(t,y)),$$

$$m^{\prime }(u,v)=\bigwedge \{m({\pi }_{k}Su,{\pi }_{k}Sv):k=1,2,3\}=m(t,s)\wedge m(t,y)(resp.m(t,s)\wedge m(s,y)),$$

and by assumption (II), we have $\overline{m}(u,v)=m^{\prime }(u,v)$.

If $u={(s,t)}_1$ and $v={(y,s)}_2$ or $u={(t,s)}_1$ and $v={(y,t)}_2$ (resp. $u={(s,t)}_1$ and $v={(y,t)}_2$ or $u={(t,s)}_1$ and $v={(y,s)}_2$), then by assumption (I),

$$\overline{m}(u,v)=\bigwedge \{m({\pi }_{k}Au,{\pi }_{k}Av):k=1,2,3\}=m(s,y)\wedge m(t,y)(resp.m(s,y)\wedge m(t,y)\wedge m(s,t)),$$

$$m^{\prime }(u,v)=\bigwedge \{m({\pi }_{k}Su,{\pi }_{k}Sv):k=1,2,3\}=m(s,y)\wedge m(t,y)\wedge m(s,t)(resp.m(s,y)\wedge m(t,y)),$$

and by assumption (II), we have $\overline{m}(u,v)=m^{\prime }(u,v)$.

If $u={(s,y)}_1$ and $v={(s,t)}_2$ or ${(y,t)}_2$ (resp. $u={(y,s)}_1$ and $v={(s,t)}_2$ or ${(y,t)}_2$), then by assumption (I),

$$\overline{m}(u,v)=\bigwedge \{m({\pi }_{k}Au,{\pi }_{k}Av):k=1,2,3\}=m(y,s)\wedge m(s,t)(resp.m(y,s)\wedge m(y,t)),$$

$$m^{\prime }(u,v)=\bigwedge \{m({\pi }_{k}Su,{\pi }_{k}Sv):k=1,2,3\}=m(y,s)\wedge m(y,t)(resp.m(y,s)\wedge m(s,t)),$$

and by assumption (II), we have $\overline{m}(u,v)=m^{\prime }(u,v)$.

If $u={(s,y)}_1$ and $v={(t,y)}_2$ or $u={(y,s)}_1$ and $v={(t,s)}_2$ (resp. $u={(s,y)}_1$ and $v={(t,s)}_2$ or $u={(y,s)}_1$ and $v={(t,y)}_2$), then by assumption (I),

$$\overline{m}(u,v)=\bigwedge \{m({\pi }_{k}Au,{\pi }_{k}Av):k=1,2,3\}=m(s,t)\wedge m(y,t)\wedge m(s,y)(resp.m(s,t)\wedge m(y,t)),$$

$$m^{\prime }(u,v)=\bigwedge \{m({\pi }_{k}Su,{\pi }_{k}Sv):k=1,2,3\}=m(s,t)\wedge m(y,t)(resp.m(s,t)\wedge m(y,t)\wedge m(s,y)),$$

and by assumption (II), we have $\overline{m}(u,v)=m^{\prime }(u,v)$.

If $u={(t,y)}_1$ and $v={(s,y)}_2$ or $u={(y,t)}_1$ and $v={(s,t)}_2$ (resp. $u={(t,y)}_1$ and $v={(s,t)}_2$ or $u={(y,t)}_1$ and $v={(s,y)}_2$), then by assumption (I),

$$\overline{m}(u,v)=\bigwedge \{m({\pi }_{k}Au,{\pi }_{k}Av):k=1,2,3\}=m(t,s)\wedge m(y,s)\wedge m(t,y)(resp.m(t,s)\wedge m(y,s)),$$

$$m^{\prime }(u,v)=\bigwedge \{m({\pi }_{k}Su,{\pi }_{k}Sv):k=1,2,3\}=m(t,s)\wedge m(y,s)(resp.m(t,s)\wedge m(y,s)\wedge m(t,y)),$$

and by assumption (II), we have $\overline{m}(u,v)=m^{\prime }(u,v)$.

If $u={(t,y)}_1$ and $v={(t,s)}_2$ or ${(y,s)}_2$ (resp. $u={(y,t)}_1$ and $v={(t,s)}_2$ or ${(y,s)}_2$), then by assumption (I),

$$\overline{m}(u,v)=\bigwedge \{m({\pi }_{k}Au,{\pi }_{k}Av):k=1,2,3\}=m(y,t)\wedge m(t,s)(resp.m(y,t)\wedge m(y,s)),$$

$$m^{\prime }(u,v)=\bigwedge \{m({\pi }_{k}Su,{\pi }_{k}Sv):k=1,2,3\}=m(y,t)\wedge m(y,s)(resp.m(y,t)\wedge m(t,s)),$$

and by assumption (II), we have $\overline{m}(u,v)=m^{\prime }(u,v)$.

Case III: Let $s,t,y,z$ be four distinct points of X.

If $u={(s,t)}_1$ and $v={(y,z)}_2$ (resp. $u={(y,z)}_1$ and $v={(s,t)}_2$), then by assumption (I),

$$\begin{array}{ccc}\hfill \overline{m}(u,v)& =& \bigwedge \{m({\pi }_{k}Au,{\pi }_{k}Av):k=1,2,3\}\hfill \\ & =& m(s,y)\wedge m(t,y)\wedge m(s,z)\hfill \end{array}$$

$$\begin{array}{ccc}\hfill m^{\prime }(u,v)& =& \bigwedge \{m({\pi }_{k}Su,{\pi }_{k}Sv):k=1,2,3\}\hfill \\ & =& m(s,y)\wedge m(t,y)\wedge m(t,z)\hfill \\ & =& (resp.m(s,y)\wedge m(t,z)\wedge m(s,z))\hfill \end{array}$$

and by assumption (III), we have $\overline{m}(u,v)=m^{\prime }(u,v)$.

If $u={(s,t)}_1$ and $v={(z,y)}_2$ (resp. $u={(z,y)}_1$ and $v={(s,t)}_2$), then by assumption (I),

$$\begin{array}{ccc}\hfill \overline{m}(u,v)& =& \bigwedge \{m({\pi }_{k}Au,{\pi }_{k}Av):k=1,2,3\}\hfill \\ & =& m(s,y)\wedge m(t,z)\wedge m(s,z)\hfill \end{array}$$

$$\begin{array}{ccc}\hfill m^{\prime }(u,v)& =& \bigwedge \{m({\pi }_{k}Su,{\pi }_{k}Sv):k=1,2,3\}\hfill \\ & =& m(s,z)\wedge m(t,y)\wedge m(t,z)\hfill \\ & =& (resp.m(s,y)\wedge m(t,y)\wedge m(s,z))\hfill \end{array}$$

and by assumption (III), we have $\overline{m}(u,v)=m^{\prime }(u,v)$.

If $u={(t,s)}_1$ and $v={(y,z)}_2$ (resp. $u={(y,z)}_1$ and $v={(t,s)}_2$), then by assumption (I),

$$\begin{array}{ccc}\hfill \overline{m}(u,v)& =& \bigwedge \{m({\pi }_{k}Au,{\pi }_{k}Av):k=1,2,3\}\hfill \\ & =& m(s,y)\wedge m(t,y)\wedge m(t,z)\hfill \end{array}$$

$$\begin{array}{ccc}\hfill m^{\prime }(u,v)& =& \bigwedge \{m({\pi }_{k}Su,{\pi }_{k}Sv):k=1,2,3\}\hfill \\ & =& m(s,y)\wedge m(t,y)\wedge m(s,z)\hfill \\ & =& (resp.m(s,z)\wedge m(t,y)\wedge m(t,z))\hfill \end{array}$$

and by assumption (III), we have $\overline{m}(u,v)=m^{\prime }(u,v)$.

If $u={(t,s)}_1$ and $v={(z,y)}_2$ (resp. $u={(z,y)}_1$ and $v={(t,s)}_2$), then by assumption (I),

$$\begin{array}{ccc}\hfill \overline{m}(u,v)& =& \bigwedge \{m({\pi }_{k}Au,{\pi }_{k}Av):k=1,2,3\}\hfill \\ & =& m(s,z)\wedge m(t,y)\wedge m(t,z)\hfill \end{array}$$

$$\begin{array}{ccc}\hfill m^{\prime }(u,v)& =& \bigwedge \{m({\pi }_{k}Su,{\pi }_{k}Sv):k=1,2,3\}\hfill \\ & =& m(s,y)\wedge m(t,z)\wedge m(s,z)\hfill \\ & =& (resp.m(s,y)\wedge m(t,y)\wedge m(t,z))\hfill \end{array}$$

and by assumption (III), we have $\overline{m}(u,v)=m^{\prime }(u,v)$.

Similarly, if $u={(s,y)}_1$ and $v={(t,z)}_2$, $u={(t,z)}_1$ and $v={(s,y)}_2$, $u={(s,y)}_1$ and $v={(z,t)}_2$, $u={(z,t)}_1$ and $v={(s,y)}_2$, $u={(y,s)}_1$ and $v={(t,z)}_2$, $u={(t,z)}_1$ and $v={(y,s)}_2$, $u={(y,s)}_1$ and $v={(z,t)}_2$, $u={(z,t)}_1$ and $v={(y,s)}_2$, and if $u={(s,z)}_1$ and $v={(t,y)}_2$, $u={(t,y)}_1$ and $v={(s,z)}_2$, $u={(s,z)}_1$ and $v={(y,t)}_2$, $u={(y,t)}_1$ and $v={(s,z)}_2$, $u={(z,s)}_1$ and $v={(t,y)}_2$, $u={(t,y)}_1$ and $v={(z,s)}_2$, $u={(z,s)}_1$ and $v={(y,t)}_2$, $u={(y,t)}_1$ and $v={(z,s)}_2$, then by assumption (III), we have $\overline{m}(u,v)=m^{\prime }(u,v)$.

Hence, for all points $u,v\in X^2{\mathbb{V}}_{▵}{X}^2$, we obtain $\overline{m}(u,v)=m^{\prime }(u,v)$, and by Lemma 1 and Definition 10 (i), $(X,\mathcal{G})$ is Pre-$\overline{T_2}$. □

Corollary 2.

Let $(X,\mathcal{G})$ be an $\mathcal{L}$-gauge space, where $\mathcal{L}$ is a linearly ordered quantale. $(X,\mathcal{G})$ is Pre-$\overline{T_2}$ if there exists $m\in \mathcal{G}$ such that for any distinct points $s,t,y,z\in X$, the following conditions are satisfied.

(I) 

$m(s,t)=m(t,s).$

(II) 

$m(s,t)=m(s,y)\le m(t,y)$ or $m(s,t)=m(t,y)\le m(s,y)$ or $m(s,y)=m(t,y)\le m(s,t).$

(III) 

$m(s,y)=m(t,y)\le m(s,z),m(t,z)$ or $m(s,y)=m(s,z)\le m(t,y),m(t,z)$ or

$m(s,y)=m(t,z)\le m(t,y),m(s,z)$ or $m(t,y)=m(s,z)\le m(s,y),m(t,z)$ or

$m(t,y)=m(t,z)\le m(s,y),m(s,z)$ or $m(s,z)=m(t,z)\le m(s,y),m(t,y).$

Theorem 5.

An $\mathcal{L}$-gauge space $(X,\mathcal{G})$ is $\overline{T_2}$ if $(X,\mathcal{G})$ is discrete.

Proof. 

By Definition 10 (ii), Theorems 1 and 4, the condition $m(s,t)=m(t,s)=\perp$ for all $s\ne t$ implies that m is the discrete $\mathcal{L}$-metric and if such a $m\in \mathcal{G}$ exists, then $\mathcal{G}$ contains all $\mathcal{L}$-metrics on X, i.e., ${\mathcal{G}}_{dis}=\{d\in \mathbf{L}\text{-}\mathbf{MET}(\mathbf{X}):d\ge m\}$, and consequently, $(X,\mathcal{G})$ is discrete. □

Theorem 6.

An $\mathcal{L}$-gauge space $(X,\mathcal{G})$ is $N{T}_2$ if there exists $m\in \mathcal{G}$ such that the following conditions are satisfied.

(I) 

For all $s,t\in X$ with $s\ne t$, $m(s,t)\wedge m(t,s)=m(s,t)=m(t,s)<\top$.

(II) 

For any three distinct points $s,t,y\in X$, $m(t,s)\wedge m(y,s)\wedge m(t,y)=m(t,s)\wedge m(y,s)=m(s,t)\wedge m(y,t)=m(y,s)\wedge m(t,y)$.

(III) 

For any four distinct points $s,t,y,z\in X$, $m(s,y)\wedge m(t,y)\wedge m(t,z)=m(s,y)\wedge m(t,y)\wedge m(s,z)=m(s,z)\wedge m(t,y)\wedge m(t,z)=m(s,y)\wedge m(t,z)\wedge m(s,z)$.

Proof. 

It follows from Definition 10 (iii), Theorems 2 and 4. □

Corollary 3.

A $(X,\mathcal{G})$, where $\mathcal{L}$ is a linearly ordered quantale, is $N{T}_2$ if there exists $m\in \mathcal{G}$ such that for any distinct points $s,t,y,z\in X$, the following conditions are satisfied.

(I) 

$m(s,t)=m(t,s)<\top .$

(II) 

$m(s,t)=m(s,y)\le m(t,y)$ or $m(s,t)=m(t,y)\le m(s,y)$ or $m(s,y)=m(t,y)\le m(s,t).$

(III) 

$m(s,y)=m(t,y)\le m(s,z),m(t,z)$ or $m(s,y)=m(s,z)\le m(t,y),m(t,z)$ or

$m(s,y)=m(t,z)\le m(t,y),m(s,z)$ or $m(t,y)=m(s,z)\le m(s,y),m(t,z)$ or

$m(t,y)=m(t,z)\le m(s,y),m(s,z)$ or $m(s,z)=m(t,z)\le m(s,y),m(t,y).$

Example 3.

Let X be a set with at least two points and $(X,\mathcal{G})$ be an indiscrete $\mathcal{L}$-gauge space. Then, by Theorem 3.3 of [22], $(X,\mathcal{G})$ is Pre-$\overline{T_2}$, but by Theorems 1, 3 and 5, $(X,\mathcal{G})$ is neither $T_0$, $\overline{T_0}$, $T_1$, $\overline{T_2}$ nor $N{T}_2$.

Theorem 7.

Let $(X,\mathcal{G})$ be a Pre-$\overline{T_2}$$\mathcal{L}$-gauge space, then the following are equivalent.

1. 

$(X,\mathcal{G})$ is $\overline{T_2}.$

2. 

$(X,\mathcal{G})$ is $T_1.$

3. 

$(X,\mathcal{G})$ is $\overline{T_0}.$

Proof. 

Combine Theorems 1 and 3–5. □

5. Comparative Evaluation

In this section, we compare our results with the ones in some other categories.

Let $\mathcal{E}$ be a topological category, and let $\mathbf{T}(\mathcal{E})$ be the full subcategory of $\mathcal{E}$ consisting of all $\mathbf{T}$ objects where $\mathbf{T}$ is $\overline{T_0}$, $T_1$, Pre-$\overline{T_2}$ or $\overline{T_2}$.

By Theorem 3.4 of [22], the full subcategory $\mathbf{Pre}$-${\mathbf{T}}_{\mathbf{2}}(\mathcal{E})$ of $\mathcal{E}$ consisting of all Pre-$\overline{T_2}$ objects in $\mathcal{E}$ is a topological category.

Theorem 8.

The following categories are isomorphic.

1. 

$\overline{T_0}$$(\mathbf{Pre}$-${\mathbf{T}}_{\mathbf{2}}(\mathit{L}\text{-}\mathit{GS})).$

2. 

$T_1$$(\mathbf{Pre}$-${\mathbf{T}}_{\mathbf{2}}(\mathit{L}\text{-}\mathit{GS})).$

3. 

$\overline{T_2}$$(\mathbf{Pre}$-${\mathbf{T}}_{\mathbf{2}}(\mathit{L}\text{-}\mathit{GS})).$

4. 

$T_1$$(\mathit{L}\text{-}\mathit{GS}).$

5. 

$\overline{T_2}$$(\mathit{L}\text{-}\mathit{GS}).$

Proof. 

It follows from Theorem 3.5 of [22] and Theorems 3, 5 and 7. □

We can infer the following:

(1)

In $\mathbf{L}\text{-}\mathbf{GS}$,

(a)

By Theorems 1–3 and 5, $\overline{T_2}=T_1\Rightarrow \overline{T_0}\Rightarrow T_0$.

(b)

By Theorems 4–6, if an $\mathcal{L}$-gauge space $(X,\mathcal{G})$ is $\overline{T_2}$, then $(X,\mathcal{G})$ is both $N{T}_2$ and Pre-$\overline{T_2}$.

(c)

By Theorem 7, $(X,\mathcal{G})$ is a Pre-Hausdorff $\mathcal{L}$-gauge space, then $\overline{T_0}$, $T_1$ and $\overline{T_2}$ are equivalent.

(2)

In the category $\mathbf{App}$ of approach spaces and contraction maps, $T_0$, $\overline{T_0}$ and $T_1$ separation axioms, given in [2,31] are the special forms of our results. For example, if we take Lawvere’s quantale [23,24], then Theorems 1 and 3 reduce to Theorems 3.1.3 and 3.2.3 of [31], respectively.

(3)

For the category $\mathbf{Top}$, $\overline{T_2}=N{T}_2\Rightarrow T_1\Rightarrow \overline{T_0}=T_0$ and $\overline{T_2}=N{T}_2\Rightarrow$ Pre-$\overline{T_2}$ [13,29,30]. Moreover, in the realm of Pre-$T_2$ property, by Theorem 3.5 of [22], all of $T_0$, $T_1$ and $T_2$ are equivalent.

(4)

(a)

In category $\mathbf{Prox}$ of proximity spaces and proximity maps, if a proximity space $(X,z)$ is $\overline{T_0}$ or $T_1$ or $\overline{T_2}$, then $(X,z)$ is Pre-$\overline{T_2}$ [32]. Similarly, in category $\mathbf{CHY}$ of Cauchy spaces and Cauchy continuous maps, $T_0=\overline{T_0}=T_1=\overline{T_2}\Rightarrow$ Pre-$\overline{T_2}$ [33].

(b)

In category $\mathbf{Born}$ of bornological spaces and bounded maps, if a bornological space is $T_0$, then it is $\overline{T_0}$ or $T_1$ or $\overline{T_2}$ or Pre-$\overline{T_2}$ [14,15,29]. However, in category $\mathbf{Lim}$ of limit spaces and filter convergence maps, $T_1\Rightarrow T_0=\overline{T_0}$ [15].

(c)

In $\mathbf{ConFCO}$ (the category of constant filter convergence spaces and continuous maps), $T_0=\overline{T_0}=T_1$ and $\overline{T_2}=N{T}_2\Rightarrow$ Pre-$\overline{T_2}$ [34]. In the realm of Pre-$\overline{T_2}$ property, $T_0=\overline{T_0}=T_1=\overline{T_2}=N{T}_2$ [22,34]. In $\mathbf{ConLFCO}$ (the category of constant local filter convergence spaces and continuous maps), $T_0\Rightarrow \overline{T_0}=T_1$ and $T_0=N{T}_2\Rightarrow \overline{T_2}\Rightarrow$ Pre-$\overline{T_2}$ [34]. In the realm of Pre-$\overline{T_2}$ property, $T_0=N{T}_2\Rightarrow \overline{T_2}=\overline{T_0}=T_1$ [22,34].

(d)

In the category $\infty \mathbf{pqsMet}$ of extended pseudo-quasi-semi-metric spaces and contraction maps, $T_1=\overline{T_2}\Rightarrow \overline{T_0}\Rightarrow T_0$ and $\overline{T_2}\Rightarrow N{T}_2\Rightarrow$ Pre-$\overline{T_2}$ [20,35]. Furthermore, in the realm of Pre-$\overline{T_2}$ property, $\overline{T_0}=T_1=\overline{T_2}$ and $N{T}_2=T_0$ [20,35].

(e)

In category $\mathbf{CP}$ of pair spaces and pair preserving maps, all pair spaces are $\overline{T_0}$, $T_1$, $\overline{T_2}$ and Pre-$\overline{T_2}$ [16]. Moreover, $T_0=N{T}_2\Rightarrow \overline{T_2}=\overline{T_0}=T_1=$ Pre-$\overline{T_2}$ [16].

(5)

(a)

For any arbitrary topological category, there is no relationship between $\overline{T_0}$ and $T_0$ [15]. In addition, it is shown in [29] that the notions of $\overline{T_2}$ and $N{T}_2$ are independent of each other, in general. However, in the realm of Pre-$T_2$ property, by Theorem 3.5 of [22], all of $\overline{T_0}$, $T_1$ and $\overline{T_2}$ are equivalent.

(b)

By Corollary 2.7 of [36], if $U:\mathcal{E}\to \mathbf{Set}$ is normalized (i.e., U is topological and there is only one structure on a one-point set and ∅, the empty set ), then $\overline{T_0}$, $T_1$, Pre-$\overline{T_2}$ and $\overline{T_2}$ imply $\overline{T_0}$ at p, $T_1$ at p, Pre-$\overline{T_2}$ at p and $\overline{T_2}$ at p, respectively. In $\mathbf{L}\text{-}\mathbf{GS}$, by Theorems 3.1–3.4 of [26], if an $\mathcal{L}$-gauge space $(X,\mathcal{G})$ is $\overline{T_0}$ (or $T_1$), then $(X,\mathcal{G})$ is $\overline{T_0}$ at p (or $T_1$ at p).

6. Conclusions

Firstly, we characterized $T_0$, $\overline{T_0}$, $T_1$, Pre-$\overline{T_2}$, $\overline{T_2}$ and $N{T}_2$$\mathcal{L}$-gauge spaces and showed that $\overline{T_2}=T_1\Rightarrow \overline{T_0}\Rightarrow T_0$. Moreover, we obtained that an $\mathcal{L}$-gauge space $(X,\mathcal{G})$ is $\overline{T_2}$, then $(X,\mathcal{G})$ is both $N{T}_2$ and Pre-$\overline{T_2}$, and in the realm of Pre-Hausdorff quantale-valued gauge spaces, $\overline{T_0}$, $T_1$ and $\overline{T_2}$ are equivalent. Finally, we compared our results with the ones in some other categories. Considering these results, the following can be treated as open research problems:

(i)

Can one characterize each of $T_3$, $T_4$, irreducible, compact, connected, sober and zero-dimensional quantale-valued gauge spaces?

(ii)

Can one present the Urysohn’s Lemma, the Tietze Extension Theorem and the Tychonoff Theorem for the category $\mathbf{L}\text{-}\mathbf{GS}$?

(iii)

How can one characterize $T_0$, $\overline{T_0}$, $T_1$, Pre-$\overline{T_2}$, $\overline{T_2}$ and $N{T}_2$ separation axioms for quantale generalization of other approach structures such as approach distances and approach systems, and what would be their relation to each other?

(iv)

In the category $\mathbf{App}$ of approach spaces and contraction maps, what would be the characterization of Pre-$\overline{T_2}$, $\overline{T_2}$ and $N{T}_2$ properties?