Some topological aspects of interval spaces

1.   Introduction

Convexity is a fundamental feature in many fields of mathematics. However, in vector spaces, it is not the best environment for understanding the basic characteristic of convex sets. As a remedy, abstract convex structures ^[40] came into existence and have many applications in different areas of mathematics, including topology, graph theory and lattice theory (see ^[39], ^[35] and ^[32]). Convex structures can be determined in several different ways, including through the use of the algebraic closure operator and hull operators. In 1971, Calder ^[17] introduced the concept of Interval operators which is a natural generalization of intervals and it also provides a natural and frequent method of constructing convex structures. Interval operators have many applications in planer geometry such as Pasch-Peano (PP) spaces.

In 1921, Sierpinski ^[20] introduced zero-dimensional topological space consisting of a basis that is clopen and it has been utilized to construct several well-known classes of topological spaces such as Lusin spaces ^[16], non-Archimedean spaces ^[19] and stone spaces ^[21]. Recently, Stine put forward this notion to an arbitrary topological category ^[36,37].

Classical separation axioms of topology have been put forward for topological categories by numerous authors ^[2,18] using different approaches. In 1991, Baran ^[2,18] introduced $T_{0}$, $T_{1}$ and $T_{2}$ objects and (strongly) closed objects in a set-based topological category by using initial, final lifts and (in) discrete objects. Further, he introduced the concept of pre-$T_{2}$ in topological space and later on, extended it to a set-based topological category ^[2,9]. $T_{0}$ objects and the notion of closedness are widely used to define and characterize various forms of Hausdorff objects ^[5], connectedness ^[8] and sobriety ^[11] in some topological categories ^[11,23,33].

In 1994, Mielke ^[30] showed the important role of pre-$T_{2}$ objects in the general theory of geometric realization, their associated intervals and corresponding homotopic structures. Also, in 1999, Mielke ^[31] used pre-$T_{2}$ objects of topological categories to characterize decidable objects in topos theory, where $X \in Obj(\mathcal{E})$ with $\mathcal{E}$ as a topos ^[21], is called decidable if the diagonal $\Delta \subset X^{2}$ is a complemented subobject.

Other uses of pre-$T_{2}$ objects include defining various forms of Hausdorff objects ^[5], $T_{3}$ and $T_{4}$ objects ^[7] in some well-known topological categories ^[14,25]. There is also, a relationship between pre-$T_{2}$ objects and partitions, as well as equivalence relations in case of Top see ^[36] in the some other categories see ^{[10,12,13,15]}.

The salient objectives of the paper are stated as follows:

$(1)$ To characterize separated, $T_{0}$, ${\bf{T_{0}}}$, $T_{1}$, pre-$T_{2}$, $T_{2}$, $ST_{2}$ and $NT_{2}$ interval spaces, and examine their mutual relationship;

$(2)$ To give the characterization of closedness of singleton sets and $D$-connectedness in the category IS (i.e., the category of interval spaces and interval preserving mappings);

$(3)$ To examine the zero-dimensionality and study its relation to $D$-connectedness in the category of interval spaces and interval preserving mappings.

2.   Preliminaries

Let $X$ be a non-empty set and $\{B_i\}_{i \in I} \overset{dir}{\subseteq} P(X)$ denotes the directed subset of $X$, which means that, for any $E, F \in \{B_i\}_{i \in I}$, there exists $G \in \{B_i\}_{i \in I}$ such that $E \subseteq G$ and $F \subseteq G$. For any non-empty sets $X$ and $Y$, and $f : X \longrightarrow Y$ be any mapping. Define forward mapping $f^\rightarrow :P(X) \longrightarrow P(Y)$ and backward mapping $f^\leftarrow :P(Y) \longrightarrow P(X)$ by $f^\rightarrow (E) = \{f(x) \mid x \in E\}$ and $f^\leftarrow(G) = \{x \mid f(x) \in G\}$ for any $E \in P(X)$ and $G \in P(Y)$, respectively.

Definition 2.1. (cf. ^[40,41]) A convex structure $\mathfrak{C}$ on the set X is a subset of $P(X)$ satisfying the following:

(1) $\emptyset, X \in \mathfrak{C}$;

(2) $\{B_i\}_{i \in I} \subseteq \mathfrak{C}$ implies $\bigcap\limits_{i \in I} B_{i} \in \mathfrak{C}$;

(3) $\{B_i\}_{i \in I} \overset{dir}{\subseteq} \mathfrak{C}$ implies $\bigcup\limits_{i \in I} B_{i} \in \mathfrak{C}.$

The pair $(X, \mathfrak{C})$ is called convexity space. The members of $\mathfrak{C}$ are called convex sets and their complements are called concave sets.

A mapping $g :(X, \mathfrak{C}_X) \longrightarrow (Y, \mathfrak{C}_Y)$ is called convexity preserving mapping provided that $E \in \mathfrak{C}_Y$ implies $g^\leftarrow(E) \in \mathfrak{C}_X.$ Let ${\bf CS}$ denotes the category of convexity spaces $(X, \mathfrak{C})$ and convexity preserving mappings.

The smallest convex set including a set E is defined as $co(E) = \bigcap\{F : E \subseteq F \in {\mathfrak{C}}\}$ is called the convex hull of E. A set of type $co(E)$ with E is finite, and it is called polytope ^[40].

Definition 2.2. (cf. ^[40,41]) A closure operator cl on X is a mapping $cl : P(X) \longrightarrow P(X)$ satisfying:

(1) $cl(\emptyset) = \emptyset$;

(2) $E \subseteq cl(E)$;

(3) $E \subseteq F$ implies $cl(E) \subseteq cl(F)$;

(4) $cl(cl(F)) = cl(F)$.

The pair $(X, cl)$ is called a closure space. Further, the closure space $(X, cl)$ is said to be an algebraic closure space if $cl(E) = \cup \{cl(F) \mid F$ is a finite subset of $E \}$ is satisfied.

A mapping $g : (X, cl_X) \longrightarrow (Y, cl_Y)$ between two closure spaces is called a closure preserving mapping such that $g^\rightarrow (cl_X(E)) \subseteq cl_Y(g^\rightarrow (E))$, $\forall E\in P(X).$ Let CLS denotes the category of closure spaces and closure preserving mappings, and ACLS (the category of algebraic closure spaces and algebraic closure preserving mappings) is the full subcategory of CLS. Note that ACLS $\cong$ CS ^[40,41].

Definition 2.3. (cf. ^[40,41]) The mapping $J: X\times X \rightarrow P(X)$ is called an interval operator satisfying the following:

(1) For all $x, y \in X$, $x, y \in J(x, y)$ (Extensive Law);

(2) $J(x, y) = J(y, x)$ (Symmetry Law).

The pair $(X, J)$ is called an interval space, and $J(x, y)$ is the interval between $x$ and $y$.

The mapping $f: (E, J_E) \longrightarrow (F, J_F)$ is called a interval preserving mapping, if

Let IS denotes the category of interval spaces and interval preserving mappings. Note that IS is the full subcategory of CS.

Example 2.1. (cf. ^[41]) Let ${\mathbb{R}}$ be the set of real numbers, and define a mapping $J_ {\mathbb {R}}:{\mathbb{R} }\times {\mathbb{R}} \longrightarrow P(\mathbb{R})$ by

where $J_{\mathbb{R}}$ indicates the interval operator on $\mathbb{R}$.

Example 2.2. (cf. ^[40,41]) Let $d$ be a metric on $X$, and define a mapping $J_d:X \times X \longrightarrow P(X)$ as follows:

where $J_d$ indicates the geodesic interval operator on $X$.

Example 2.3. (cf. ^[40]) Let $V$ be a vector space and define a mapping $J_V:V \times V \longrightarrow P(V)$ by $J_{V}(x, y) = \{xt+(1-t)y \mid 0 \leq t \leq 1\}$, where $J_V$ indicates the standard interval operator on the vector space $V.$

Example 2.4. (cf. ^[40]) Let $(X, \leq)$ be a partially ordered set and define a mapping $J_{\leq}:X \times X \longrightarrow P(X)$ as follows:

where $J_{\leq}$ indicates the ordered interval operator on $X$.

Example 2.5. (cf. ^[40]) Let $(M, m)$ be a median algebra and define a mapping $J_{m}:M \times M \longrightarrow P(M)$ as follows:

where $J_{m}$ indicates the median interval operator on $M$.

For any interval space $(X, J)$, if for any $x, y, z \in X$ and $w \in J(y, z)$, $t \in J(x, w)$, and then there exists $k \in J(x, y)$ such that $t \in J(z, k)$. This property is known as the Peano Property. Further, if for any $p, x, y \in X$, $z \in J(p, x)$ and $w \in J(p, y)$, then the intervals $J(x, w)$ and $J(z, y)$ intersect. This property is known as the Pasch property ^[40].

Any interval space $(X, J)$ satisfying the Pasch and Peano properties is called a PP space. Note that every vector space over a totally ordered field is a PP space ^[40].

Definition 2.4. (cf. ^[40,41]) A convex space $(X, \mathfrak{C})$ is called an arity 2 convex space satisfying the following: for all $B \in P(X)$ and all $x, y \in B$, $co(\{x, y\}) \subseteq B$ implies $B \in \mathfrak{C}$.

Let ${\bf CS(2)}$ denotes the category of arity 2 convex spaces $(X, \mathfrak{C})$ and convexity preserving mappings. Note that ${\bf CS(2)}$ can be embedded in ${\bf IS}$ as a reflexive subcategory ^[40,41].

Proposition 2.1. (cf. ^[40,41]) Suppose $(X, \mathfrak{C})$ is a convex space and define $J^{\mathfrak{C}} : X \times X \longrightarrow P(X)$ by

Then $J^{\mathfrak {C}}$ represents the interval operator on $X$.

Proposition 2.2. (cf. ^[40,41]) Suppose $(X, J)$ is interval space and define ${\mathfrak{C}}^{J}$ by

Then, $(X, {\mathfrak{C}}^{J})$ is an arity 2 convex space.

A functor $\mathcal{U}: \mathcal{E} \longrightarrow {\bf Set}$ (the category of sets and functions) is called topological if $(1)$ $\mathcal{U}$ is concrete $(2)$ $\mathcal{U}$ consists of small fibers and $(3)$ every $\mathcal{U}$-source has a unique initial lift, i.e., if for every source $(f_{i}: X \rightarrow (X_{i}, \zeta_{i}))_{i \in I}$ there exists a unique structure $\zeta$ on $X$ such that $g: (Y, \eta) \rightarrow (X, \zeta)$ is a morphism iff for each $i \in I$, $f_{i} \circ g : (Y, \eta) \rightarrow (X_{i}, \zeta_{i})$ is a morphism or equivalently, each $\mathcal{U}$-sink has a unique final lift ^[1,38].

Note that a topological functor $\mathcal{U}: \mathcal{E} \longrightarrow {\bf Set}$ has a left adjoint $\mathcal{D}: {\bf Set} \longrightarrow \mathcal{E}$, called the discrete functor. An object of the form $X = \mathcal{U}\mathcal{D}(X)$ is called a discrete object in $\mathcal{E}$, i.e., the $\mathcal{E}$-objects $X$ such that every $f: \mathcal{U}X \longrightarrow \mathcal{U}Y$, $Y \in \mathcal{E}$, is an $\mathcal{E}$-morphism.

Also, the functor $\mathcal{U}$ is called a normalized topological functor if the subterminals have a unique structure ^[1,38].

Lemma 2.1. (cf. ^[41]) Let $(X_i, J_i)$ be the collection of interval space and $(f_i : (X, J_{*}) \longrightarrow (X_i, J_i))_{i\in I}$ be a source. Then, for any $x, y \in X$,

is the initial interval structure on $X$.

Lemma 2.2. (cf. ^[41]) Let $(X, J)$ be an interval space. Then, we have the following:

(1) The discrete interval structure on $X$ is defined by $J_{dis}(x, y) = \{x, y\}$ for any distinct $x, y \in X$.

(2) The indiscrete interval structure on $X$ is given by $J_{ind}(x, y) = X$ for any distinct $x, y \in X$.

Remark 2.1. The topological functor $\mathcal{U}: {\bf IS} \longrightarrow {\bf Set}$ is normalized since a unique structure exists on $\emptyset$, the empty set or $X = \{x\}$, i.e., a one-point set for $X \in Obj({\bf IS})$ ^[41].

3.   Separated, pre-Hausdorff and Hausdorff interval spaces

Let $X$ be a set and the wedge $X^2 \bigvee_\Delta X^2$ be two any disjoint copies of $X^2$ intersecting diagonally. In other words, the pushout of $\Delta:X\longrightarrow X^{2}$ along itself. A point $(x, y)$ in $X^2 \bigvee_\Delta X^2$ is denoted by $(x, y)_1$ (resp. $(x, y)_2$) if it is in the first (resp. second) component.

Definition 3.1. (cf. ^[2]) The mapping $A: X^2 \bigvee_\Delta X^2 \longrightarrow X^3$ is said to be the principal axis mapping provided that

Definition 3.2. (cf. ^[2]) The mapping $S: X^2 \bigvee_\Delta X^2 \longrightarrow X^3$ is said to be a skewed axis mapping provided that

Definition 3.3. (cf. ^[2]) The mapping $\nabla : X^2 \bigvee_\Delta X^2 \longrightarrow X^2$ is said to be a fold mapping provided that $\nabla(x, y)_j = (x, y)$ for $j = 1, 2$.

Definition 3.4. Let $\mathcal{U}:\mathcal{E} \longrightarrow {\bf Set}$ be a topological functor and $X \in Obj(\mathcal{E})$ with $\mathcal{U}(X) = Y$.

(1) $X$ is called separated provided that every initial morphism with the domain $X$ is a monomorphism ^[42].

(2) $X$ is called $T_{0}$ provided that the initial lift of the $\mathcal{U}$-source {$A : Y^2 \bigvee_ \Delta Y^2\rightarrow \mathcal{U}(X^3) = Y^3$ and $\nabla : Y^2 \bigvee_\Delta Y^2\rightarrow \mathcal{U}\mathcal{D}(Y^2) = Y^2$} is discrete, where $\mathcal{D}$ is the discrete functor which is the left adjoint of $\mathcal{U}$ ^[2].

(3) $X$ is called $T_{1}$ provided that the initial lift of the $\mathcal{U}$-source {$S : Y^2 \bigvee_ \Delta Y^2\rightarrow \mathcal{U}(X^3) = Y^3$ and $\nabla : Y^2 \bigvee_\Delta Y^2\rightarrow \mathcal{U}\mathcal{D}(Y^2) = Y^2$} is discrete ^[2].

(4) $X$ is called ${\bf{T_{0}}}$ if $X$ does not contain an indiscrete subspace with at least two points ^[29].

(5) $X$ is called pre-$T_2$ iff initial lifts of the $\mathcal{U}$-sources $\{A :Y^2 \bigvee_\Delta Y^2 \rightarrow \mathcal{U}(X^3) = Y^3$ and $S :Y^2 \bigvee_\Delta Y^2 \rightarrow \mathcal{U}(X^3) = Y^3\}$ coincide ^[2].

(6) $X$ is called $ST_2$ provided that $X$ is separated and pre-$T_2$.

(7) $X$ is called $T_2$ provided that $X$ is $T_0$ and pre-$T_2$ ^[2].

(8) $X$ is called $NT_2$ provided that $X$ is ${\bf T_0}$ and pre-$T_2$ ^[2].

Remark 3.1. (1) In the category ${\bf Top}$, separated, $T_{0}$ and ${\bf{T_{0}}}$ (resp. $T_{1}$) reduce to the usual $T_{0}$ (resp. $T_{1}$) of topological spaces. Similarly, $ST_{2}$, $T_{2}$ and $NT_{2}$ reduce to a classical Hausdorff topological space ^[4,6,42].

(2) If $\mathcal{U}:\mathcal{E} \longrightarrow \mathcal{B}$ is a topological functor, where $\mathcal{B}$ is an elementary topos, then Definition 3.4 is still valid ^[2].

(3) In any arbitrary topological category, every ${\bf T_0}$ object is separated but converse is not in general ^[42]. Further, the $T_{0}$ object and separated object, and $T_{0}$ and ${\bf T_0}$ objects are independent of each other ^[4].

(4) In any arbitrary topological category, there is no relation among $ST_2$, $T_2$ and $NT_2$ ^[5]. However, for any topological functor $\mathcal{U}:$ pre-$T_{2}(\mathcal{E}) \longrightarrow {\bf Set}$, where pre-$T_{2}(\mathcal{E})$ is the full subcategory of all pre-$T_{2}$ objects in $\mathcal{E}$, all $T_{0}$, $T_{1}$, $ST_2$, $T_2$ and $NT_2$ objects are equivalent ^[9].

(5) Let $\mathcal{U}:\mathcal{E} \longrightarrow {\bf Set}$ be a topological functor and $X \in Obj(\mathcal{E})$. If $X$ is an indiscrete object, then $X$ is pre-$T_2$ ^[9].

Theorem 3.1. An interval space $(X, J)$ is separated iff $X$ has at most one point.

Proof. Suppose $(X, J)$ is separated, $X \neq \emptyset$ and $X \neq \{a\}$. Then, there exists $b \in X$ with $a \neq b$. If $X = \{a, b\}$, then $J(a, b) = X$, an indiscrete structure. Let $f: (X, J) \longrightarrow (X, J)$ be a mapping defined by $f(a) = a = f(b)$. Since $(X, J)$ is an indiscrete interval space, $f$ is initial (i.e., $f^{\leftarrow}(J(f(a), f(b))) = f^{\leftarrow}(J(a, a)) = X = J(a, b)$) but it is not mono. Hence, $(X, J)$ is not a separated interval space.

Note that every subspace of a separated interval space is separated since the composition of initial lifts is initial and the composition of monomorphisms is a monomorphism. If Card$X \geq 3$, for any $a, b \in X$ with $a \neq b$ and $M = \{a, b\} \subset X$, then the subinterval structure $J_{M}$ on $M$ is indiscrete. By Definition 1.1 of ^[42], a separated interval space can not have an indiscrete subspace with at least two points, which is a contradiction. Hence, $X$ must be the empty set or a one-point set.

Conversely, if $X = \emptyset$ or $X = \{a\}$, then clearly $(X, J)$ is separated.

Theorem 3.2. Every interval space $(X, J)$ is $T_{0}$.

Proof. Let $(X, J)$ be an interval space. We show that $(X, J)$ is $T_{0}$. Let $\overline{J}$ be an initial structure on $X^2 \bigvee_\Delta X^2$ induced by $A :X^2 \bigvee_\Delta X^2 \longrightarrow (X^3, J^3)$ and $\nabla :X^2 \bigvee_\Delta X^2 \longrightarrow (X^2, J^2_{dis})$, where $J^3$ and $J^2_{dis}$ are products and discrete interval structures on $X^3$ and $X^2$, respectively. Let $m, n \in X^2 \bigvee_\Delta X^2$.

Case Ⅰ: If $m = n$, then $\nabla m = \nabla n$ and $pr_{k}Am = pr_{k}An$, $k = 1, 2, 3$, where $pr_{k}$ is the projection mapping $pr_{k} :X^3 \longrightarrow X$ for $k = 1, 2, 3$.

On the other hand,

and

It follows that $m \in pr_{k}A^\leftarrow(J(pr_{k}Am, pr_{k}Am))$ for $k = 1, 2, 3.$

By Lemma 2.1, we obtain $\overline{J}(m, m) = \{m\}$, a discrete structure.

Case Ⅱ: Let $m\neq n$ and $\nabla m = \nabla n$. If $\nabla m = (x, y) = \nabla n$ for some $(x, y) \in X^{2}$ given that $m\neq n$, consequently, it follows that $m = (x, y)_{i}$ and $n = (x, y)_{j}$ with $i \neq j$ and $i, j = 1, 2$.

Suppose $m = (x, y)_{1}$ and $n = (x, y)_{2}$. By Lemma 2.2 (1),

and

Similarly,

and

Since $x = pr_{1}A(x, y)_{1} = pr_{1}A(x, y)_{2} \in J(x, x)$, consequently, $(x, y)_{1}, (x, y)_{2} \in pr_{1}A^\leftarrow(J(x, x))$. Similarly, $x = pr_{2}A(x, y)_{2} = pr_{3}A(x, y)_{1} \in J(x, y)$ and $y = pr_{2}A(x, y)_{1} = pr_{3}A(x, y)_{2} \in J(x, y)$, and it follows that $(x, y)_{1}, (x, y)_{2} \in pr_{k}A^\leftarrow(J(x, y))$ for $k = 2, 3$.

By Lemma 2.1,

In a similar way, if $m = (x, y)_{2}$ and $n = (x, y)_{1}$, then $\overline{J}(m, n) = \{(x, y)_1, (x, y)_2\}$.

Case Ⅲ: Let $m\neq n$ and $\nabla m \neq\nabla n$. Note that

and $pr_{k}Am, pr_{k}An\in J(pr_{k}Am, pr_{k}An)$ for $k = 1, 2, 3$, and consequently, $m, n \in pr_{k}A^\leftarrow(J(pr_{k}Am, pr_{k}An))$. By Lemma 2.1, we have

Hence $\overline{J}$ is the discrete structure and by Definition 3.4 (ⅱ), $(X, J)$ is $T_{0}$.

Theorem 3.3. Every interval space $(X, J)$ is $T_{1}$.

Proof. The proof is similar to Theorem 3.2. So the proof is omitted.

Theorem 3.4. An interval space $(X, J)$ is ${\bf{T_{0}}}$ iff $X$ has at most one point.

Proof. Suppose $(X, J)$ is ${\bf{T_{0}}}$, $X \neq \emptyset$ and $X \neq \{a\}$. Then, there exists $b \in X$ with $a \neq b$. Let $M = \{a, b\}$ and $J_{M}$ be an interval structure induced by the inclusion mapping $i: M \longrightarrow (X, J)$. By Lemma 2.1, $J_{M}(a, b) = i^{\leftarrow}(J(i(a), i(b))) = M \cap J(a, b) = M$, i.e., the indiscrete structure on $M$, which is a contradiction. Thus, $X$ has at most one point.

Conversely, if $X = \emptyset$ or $X = \{a\}$, then clearly $(X, J)$ is ${\bf{T_{0}}}$.

Corollary 3.1. Let $(X, J)$ be an interval space. The following statements are equivalent:

(1) $(X, J)$ is separated.

(2) $(X, J)$ is ${\bf{T_{0}}}.$

(3) $X$ has at most one point.

Proof. The proof can be deduced from Theorems 3.1 and 3.4.

Theorem 3.5. An interval space $(X, J)$ is pre-$T_{2}$ iff $(X, J)$ is an indiscrete interval space.

Proof. Suppose that $(X, J)$ is pre-$T_{2}$. If $X = \emptyset$, $X = \{x\}$ or $X = \{x, y\}$, then $J_{dis} = J_{ind} = J$. Now, consider $CardX = 3$, i.e., $X = \{x, y, z\}$. Then, by Definition 2.3, $X$ carries only discrete and indiscrete structures. Assume, on the contrary, that $(X, J)$ is not an indiscrete interval space. It follows that for all $x, y \in X$ with $x \neq y$, $J(x, y) = \{x, y\}$. Let $J_{A}$ and $J_{S}$ be initial structures on $X^2 \bigvee_\Delta X^2$ induced by $A :X^2 \bigvee_\Delta X^2 \longrightarrow (X^3, J^3)$ and $S :X^2 \bigvee_\Delta X^2 \longrightarrow (X^3, J^3)$, respectively. Here, $J^3$ is the product structure on $X^3$. Also, $pr_{k}$ is the projection mapping $pr_{k} :X^3 \longrightarrow X$ for $k = 1, 2, 3$. We show that $(X, J)$ is not pre-$T_{2}$, i.e., $J_{A}(m, n) \neq J_{S}(m, n)$ for some $m, n \in X^2 \bigvee_\Delta X^2$.

Suppose $m = (x, y)_{1}$ and $n = (z, y)_{2} \in X^2 \bigvee_\Delta X^2$ for all $x, y, z \in X$ with $x \neq y \neq z$. Note that

and

By Lemma 2.1,

Similarly,

Therefore, $J_{A}((x, y)_{1}, (z, y)_{2}) \neq J_{S}((x, y)_{1}, (z, y)_{2})$, and consequently, $(X, J)$ is not pre-$T_{2}$.

Now, consider $CardX > 3$. Assume, on the contrary, that $(X, J)$ is not an indiscrete interval space. Then, there exists $M \subset X$ such that $J(x, y) = M$ for all $x, y \in X$ with $\{x, y\} \subset M \neq X$ and $x \neq y$. Then, there exists a point $z \in X$ but $z \neq M$ whenever $J(x, y) = M$ for all $x, y \in X$ with $x \neq y$. Similar to the above, consider $(z, y)_{1} \in X^2 \bigvee_\Delta X^2$ for any $z, y \in X$ with $y \neq z$. Since $pr_{1}S(z, y)_{1} = z \in J(x, z)$, consequently, $(z, y)_{1} \in pr_{1}S^{\leftarrow}(J(x, z))$. Similarly, $pr_{2}S(z, y)_{1} = y \in J(y, z)$, and, consequently, $(z, y)_{1} \in pr_{2}S^{\leftarrow}(J(y, z))$ and $(z, y)_{1} \in pr_{3}S^{\leftarrow}(J(x, y))$. Thus, by Lemma 2.1, $(z, y)_{1} \in J_{S}((x, y)_{1}, (z, y)_{2})$ for any $(x, y)_{1}, (z, y)_{2} \in X^2 \bigvee_\Delta X^2$. However $(z, y)_{1} \notin J_{A}((x, y)_{1}, (z, y)_{2})$ since $pr_{3}A(z, y)_{1} = z \notin J(x, y)$ and it follows that $(z, y)_{1} \notin pr_{3}A^{\leftarrow}(J(x, y))$. Thus, $J_{A}((x, y)_{1}, (z, y)_{2}) \neq J_{S}((x, y)_{1}, (z, y)_{2})$. Consequently, an interval space $(X, J)$ is not pre-$T_{2}$.

Conversely, let $(X, J)$ be an indiscrete interval space. Then, by Remark 3.1 (5), $(X, J)$ is pre-$T_{2}$.

Theorem 3.6. An interval space $(X, J)$ is $T_{2}$ iff $(X, J)$ is an indiscrete interval space.

Proof. The proof follows from Theorems 3.2 and 3.5.

Remark 3.2. (1) In O-REL (the category of ordered relative spaces and relative mappings) ^[27] as well as in b-UFIL (the category of b-UFIL spaces and buc mappings) ^[27,28], $T_{1} \implies T_{0} \implies {\bf{T_{0}}}$ ^[22,34].

(2) In $\mathcal{V}$-Cls (the category of $\mathcal{V}$-closure spaces and continuous mappings) with $\mathcal{V}$ as an integral quantale ^[26], $T_{2} = T_{1} \implies T_{0} \implies {\bf{T_{0}}}$ ^[33].

(3) In Born (the category of bornological spaces and bounded mappings), all objects are $T_{0}$, $T_{1}$ and $T_{2}$ ^[4], and $X$ is separated or ${\bf{T_{0}}}$ iff $X$ is either empty or a singleton ^[4]. However, in Prox (the category of proximity spaces and proximity mappings), all objects are not $T_{0}$, $T_{1}$ and $T_{2}$ but they are all equal ^[24].

(4) In IS, by Theorems 3.2, 3.3 and 3.6, and Corollary 3.1, we conclude that ${\bf{T_{0}}} \implies T_{0} = T_{1}$ and $T_{2} \implies T_{0} = T_{1}$ but the converse is not true in general.

Corollary 3.2. An interval space $(X, J)$ is $NT_{2}$ iff $(X, J)$ is $ST_{2}$ iff $X$ has a cardinality 1.

Proof. It follows from Theorems 3.1, 3.4 and 3.5.

4.   Notion of closedness and D-connectedness in interval spaces

Let $X$ be any set and $p \in X$. Let the infinite wedge product of $X$ at $p$ be the infinitely countable disjoint copies of $X$ identifying at $p$ and denoted by $\bigvee _{p}^{\infty } X$.

For a point $x \in \bigvee _{p}^{\infty }X$, we write it as $x_{j}$ if it belongs to the $j^{th}$ component of the infinite wedge product.

Definition 4.1. (cf. ^[2]) Let $X^\infty = X \times X \times X \times......$ be the countable Cartesian product of $X$.

(1) The mapping $A_{p}^{\infty} \; :\; \bigvee _{p}^{\infty } X \longrightarrow X^{\infty }$ is said to be an infinite principal $p$-axis mapping provided that

(2) The mapping $\nabla _{p}^{\infty } \; :\; \bigvee _{p}^{\infty } X \longrightarrow X$ is said to be an infinite fold mapping at $p$ provided that

Definition 4.2. (cf. ^[2,3]) Let $\mathcal{U}:\mathcal{E} \longrightarrow {\bf Set}$ be a topological functor and $X \in Obj(\mathcal{E})$ with $\mathcal{U}(X) = Y$ and $p \in Y$. $\{p\}$ is closed provided that the initial lift of the $\mathcal{U}$-source $\lbrace \bigvee _{p}^{\infty }Y \xrightarrow[]{A_{p}^{\infty }}\mathcal{U}X^{\infty } = Y^{\infty }\; and\; \bigvee _{p}^{\infty }Y\xrightarrow[]{\nabla _{p}^{\infty } }\mathcal{U}DY = Y\rbrace$ is discrete, where $\mathcal{D}$ is the discrete functor which is left adjoint of $\mathcal{U}$.

Remark 4.1. In Top, the closedness of $\{p\}$ reduces to the usual closedness of the singleton set $\{p\}$ ^[2,3]. Also, for any $X \in obj(\textbf{Top})$, $X$ is $T_{1}$ iff all points of $X$ are closed. However, in an arbitrary topological category, this is not true in general ^[3].

Theorem 4.1. Every singleton set $\{p\}$ in an interval space $(X, J)$ is closed.

Proof. Let $(X, J)$ be an interval space, $p \in X$. We show that $\{p\}$ is closed. Let $\bar{J}$ be an initial structure on $\bigvee_p^\infty X$ induced by $A_p^\infty : \bigvee_p^\infty X \rightarrow (X^\infty, J^\infty)$ and $\nabla_p^\infty : \bigvee_p^\infty X \rightarrow (X, J_{dis}),$ where $J^\infty$ and $J_{dis}$ are the product interval structures and discrete interval structures on $X^\infty$ and $X$, respectively. Let $m, n \in \bigvee_p^\infty X$.

If $m = n$, then $\nabla_p^\infty m = \nabla_p^\infty n$ and also $pr_k A_p^\infty m = pr_k A_p^\infty n$ for $k \in I$. Here, $pr_k$ are the projection mappings $pr_k :X^\infty \rightarrow X$, where $k \in I.$

By Lemma 2.2 (1),

and it follows that $\nabla_p^{\infty \leftarrow}(\{\nabla_p^\infty m\}) = \{m\}$ and

Since $pr_k A_p^{\infty}m \in J(pr_k A_p^\infty m, pr_k A_p^\infty m)$ for each $k \in I$, consequently, $m \in pr_k A_p^{\infty \leftarrow}(J(pr_{k}A_{p}^{\infty}m, pr_{k}A_{p}^{\infty}m))$.

By Lemma 2.1,

Let $m \neq n$ and $\nabla_p^\infty m = \nabla_p^\infty n$. If $\nabla_p^\infty m = p = \nabla_p^\infty n$, consequently, $m = (p, p, p, ..., p, ...) = p_{i} = p_{j} = n$ for all $i, j \in I$, which is a contradiction.

Suppose $\nabla_p^\infty m = x = \nabla_p^\infty n$, so it follows easily that $m = x_i$ and $n = x_j$ for some $i, j\in I$ with $i \neq j$. Note that

Since $x = pr_k A_{p}^{\infty} m\in J(x, p)$, consequently, $m \in pr_k A_{p}^{\infty \leftarrow}(J(x, p))$ for $k = i$ or $k = j$, and $p = pr_k A_{p}^{\infty} n \in J(x, p)$ for any $k \in I$; it follows that $n \in pr_k A_p^{\infty \leftarrow}(J(x, p))$. Thus, $m, n \in pr_k A_p^{\infty \leftarrow}(J(pr_k A_p^\infty m, pr_k A_p^\infty n))$ for any $k \in I$. On the other hand,

By Lemma 2.1,

Suppose that $m \neq n$ and $\nabla_p^\infty m \neq \nabla_p^\infty n$.

By Lemma 2.2 (1), $\nabla_p^{\infty \leftarrow}(J_{dis}(\nabla_p^\infty m, \nabla_p^\infty n)) = \nabla_p^\leftarrow(\{\nabla_p^\infty m, \nabla_p^\infty n\}) = \{m, n\}$ and $m, n \in J(pr_k A_p^\infty m, pr_k A_p^\infty n)$ for any $k \in I$. By Lemma 2.1, $\bar{J}(m, n) = \{m, n\}$, which is a discrete structure. Thus, by Definition 4.2, $\{p\}$ is closed.

Definition 4.3. (cf. ^[8,38]) Let $\mathcal{U}:\mathcal{E} \longrightarrow {\bf Set}$ be a topological functor and $X \in Obj(\mathcal{E})$. $X$ is said to be $D$-connected provided that any morphism from $X$ to any discrete object is constant.

Remark 4.2. In Top, the $D$-connectedness reduces to the usual connectedness ^[8,38].

Theorem 4.2. An interval space $(X, J)$ is $D$-connected iff there exists a proper subset $N$ of $X$ such that $\{x, y\} \subset J(x, y)$ for some $x \in N$ and $y \in N^c$.

Proof. Let $(X, J)$ be $D$-connected and there exists a nonempty subset $N$ of $X$, $J(x, y) = \{x, y\}$ for all $x \in N$ and $y \in N^c$. Suppose $(Y, J_{dis})$ is the discrete interval space with $Card Y > 1$. Define the mapping $f: (X, J) \rightarrow (Y, J_{dis})$ by

Let $x, y \in X$. If $x, y \in N$ then

and

and consequently,

Thus $f$ is an interval preserving mapping. Similarly, if $x, y \in N^c$, then $f$ is also an interval preserving mapping.

Now, let $x \in N$ and $y \in N^c$ (resp. $y \in N$ and $x \in N^c$). Note that

and $J_{dis}(f(x), f(y)) = \{f(x), f(y)\} = \{a, b\}$. Thus, $f^\rightarrow ((J(x, y)) = J_{dis}(f(x), f(y)).$ Hence, $f$ is an interval preserving mapping, but it is not constant, which is a contradiction.

Conversely, suppose that the condition holds. Let $(Y, J_{dis})$ be a discrete interval space and $f : (X, J) \rightarrow (Y, J_{dis})$ be an interval preserving mapping.

If $CardY = 1$, then $f$ is constant. Suppose that $CardY > 1$, and $f$ is not constant. Then there exist $x, y \in X$ with $x \neq y$ such that $f(x) \neq f(y)$ and let $N = f^\leftarrow \{f(x)\}.$ Note that $N$ is a proper subset of $X$. By our assumption $\{x, y\} \subset J(x, y)$ for some $x \in N$ and $y\notin N$, we have

By Lemma 2.2 (1), it follows that $f$ is not an interval-preserving mapping, which is a contradiction. Thus $f$ must be constant and, by Definition 4.3, $(X, J)$ is $D$-connected.

Theorem 4.3. Let $(X, J_{X})$ and $(Y, J_{Y})$ be interval spaces, and let $f: (X, J_{X}) \longrightarrow (Y, J_{Y})$ be an interval preserving mapping. If $(X, J_{X})$ is $D$-connected and $f$ is surjective, then $(Y, J_{Y})$ is $D$-connected.

Proof. Let $f(x), f(y) \in f(X)$ with $f(x) \neq f(y)$. Since $f$ is an interval preserving mapping, it follows that $f^\rightarrow (J_X(x, y)) \subseteq J_Y((f(x), f(y))$. The assumption that there exists a proper subset $N$ of $X$ such that $\{x, y\} \subset J(x, y)$ for some $x \in N$ and $y \notin N$ implies that

and consequently, $\{f(x), f(y)\} \subset J_Y((f(x), f(y))$ for some $f(x) \in f(N)$ and $f(y) \notin f(N)$. Therefore, $f(X)$ is $D$-connected. Since $f$ is surjective, it follows that $f(X) = Y$ is $D$-connected.

5.   Zero-dimensionality in interval spaces

In 1997, Stine ^[36] gave an alternative characterization of the zero-dimensional space $(X, \tau)$ that is, $(X, \tau)$ is a zero-dimensional space provided that for all $i \in I$, there exists a family $(X_{i}, \tau_{i_{dis}})$ and there exists $f_{i}: (X, \tau) \longrightarrow (X_{i}, \tau_{i_{dis}})$ such that $\tau$ is the initial topology by $(X_{i}, \tau_{i_{dis}})$ via $f_{i}$, where $(X_{i}, \tau_{i_{dis}})$ is the family of discrete topological spaces. Considering the categorical counterparts, we have the following definition, as given in ^[37].

Definition 5.1. (cf. ^[37]) Let $\mathcal{U}: \mathcal{C} \longrightarrow \mathcal{E}$ be a topological and $\mathcal{D}: \mathcal{E} \longrightarrow \mathcal{C}$ be a discrete functor. Any object $X \in Obj(\mathcal{C})$ is called a zero-dimensional object provided that for all $i \in I$, there exists $A_{i} \in Obj(\mathcal{E})$ and the morphisms $f_{i}: \mathcal{U}(X) \longrightarrow A_{i}$ such that $(\overline{f}_{i}: X \longrightarrow \mathcal{D}(A_{i}))_{i \in I}$ is the initial lift of $(f_{i}: \mathcal{U}(X) \longrightarrow \mathcal{U}(\mathcal{D}(A_{i})) = A_{i})_{i \in I}$.

Remark 5.1. (1) For $\mathcal{C} = \textbf{Top}$ and $\mathcal{E} = \textbf{Set}$, by Theorem 4.3.1 of ^[36], Definition 5.1 reduces to the usual zero-dimensional topological space.

(2) If $\; \mathcal{U}: \mathcal{C} \longrightarrow \mathcal{E}$ is a normalized topological functor, by Theorem 4.3.4 and 5.3.1 of ^[37], then every indiscrete object in $\mathcal{C}$ is a zero-dimensional object.

Theorem 5.1. Every discrete and indiscrete interval space $(X, J)$ is zero-dimensional.

Proof. Suppose $(X, J)$ is an interval space and $X = \{x\}$ or $X = \{x, y\}$. Then $J_{dis} = J_{ind} = J$. By Remarks 2.1 and 5.1, it is zero-dimensional.

Let $Card X \geq 3$, and $J = J_{dis}$. Consider $f_{i}(x) = x$ (identity mapping) and $X = X_{i}$ for all $i \in I$. Clearly, $f_{i}: X \longrightarrow X_{i}$ is an interval preserving mapping and $f_{i}: X \longrightarrow X_{i}$ is the initial lift of $f_{i}: (X, J) \longrightarrow (X_i, J_{idis})$. Thus, by Definition 5.1, $(X, J)$ is zero-dimensional.

Now, let $J = J_{ind}$ and take $f_{i}(x) = c$ (constant mapping) for all $i \in I$. Clearly, $f_{i}: X \longrightarrow X_{i}$ is an interval preserving mapping which is the initial lift of $f_{i}: (X, J) \longrightarrow (X_i, J_{idis})$. Therefore, by Definition 5.1, $(X, J)$ is zero-dimensional.

Corollary 5.1. Every interval space (except for a discrete interval space) is $D$-connected.

Corollary 5.2. Every $D$-disconnected (not $D$-connected) interval space with cardinality greater than 2 is zero-dimensional.

Proof. Let $(X, J)$ be a $D$-disconnected interval space with cardinality greater than 2. By Theorem 4.2, for all $x, y \in X$ with $x \neq y$, $J(x, y) = \{x, y\}$ and consequently, $(X, J)$ is discrete. Thus, by Theorem 5.1, $(X, J)$ is zero-dimensional.

6.   Conclusions

First, we characterized separated, ${\bf{T_{0}}}$, $T_{0}$, $T_{1}$, pre-$T_{2}$, $T_{2}$, $NT_{2}$ and $ST_{2}$ interval spaces and showed that separated $= {\bf{T_{0}}} \implies T_{0} = T_{1}$ and $T_{2} \implies T_{0} = T_{1}$ but the converse is not true in general. Also, we proved that in any interval space with cardinality at most one point, $NT_{2} = ST_{2}$. Further, we showed that every singleton set is closed and every interval space (except for a discrete interval space) is $D$-connected. Finally, we characterized zero-dimensionality in interval spaces and showed that every discrete and indiscrete interval space is zero-dimensional. Considering these results, the followings can be treated as open research problems:

$(1)$ How can one characterize sobriety, ultraconnectedness and irreducibility in the category IS?

$(2)$ Can one characterize pre-$T_{2}$, zero-dimensionality and separatedness for quantale generalization of interval spaces, and what would be their relation to the classical ones?

Acknowledgments

We would like to thank the referees for their valuable and helpful suggestions that improved the paper radically.

Conflict of interest

The authors declare that they have no conflicts of interest.