Net-Compact Hausdorff Topologies and Continuous Multi-Utility Representations for Closed Preorders

Abstract

In this paper, we deal with continuous multi-utility representations for closed preorders. We introduce the definition of a net-compact topology, which generalizes the concept of a sequentially compact topology. Indeed, a sequentially compact and first countable topological space is net-compact. First, we show that if every closed preorder on a net-compact Hausdorff topological space has a continuous multi-utility representation, then the topology is normal. Second, we prove that every closed preorder on a normal and net-compact Hausdorff topological space admits a continuous multi-utility representation.

1. Introduction

We recall that a subset $\mathcal{F}$ of the set of all continuous increasing real-valued functions f on a preordered topological space$(X,\precsim ,t)$ is said to be a continuous multi-utility representation of ≾ if, for all $x\in X$ and all $y\in X$, the following equivalence holds:

$x\precsim y\iff \forall f\in \mathcal{F},f\left(x\right)\le f\left(y\right).$

It is worth highlighting that ≾ admits a continuous multi-utility representation if and only if, for any two points $x,y\in X$ such that $not(y\precsim x)$, there exists a continuous increasing real-valued function $f_{xy}$ on the preordered topological space $(X,\precsim ,t)$ that satisfies $f_{xy}\left(x\right)<f_{xy}\left(y\right)$.

A (continuous) multi-utility representation $\mathcal{F}$ of ≾, therefore, fully describes ≾. This property of $\mathcal{F}$ underlines the particular importance of (continuous) multi-utility representations in decision theory and mathematical economics. Indeed, (continuous) multi-utility representations are frequently discussed in mathematical utility theory (cf., for instance, Evren and Ok [1], Minguzzi [2,3], Bosi and Herden [4,5], Nishimura and Ok [6], and Alcantud, Bosi and Zuanon [7]).

The concept of multi-utility representation extends the one of utility representation (see Nishimura and Ok [6]). Indeed, in many situations, it is necessary to treat a “preference relation” as a binary relation that is neither complete nor transitive. Nowadays, a large amount of research explores rational decision-making with incomplete preferences across various domains, from consumption choices to decisions involving risk and uncertainty. An even broader volume of literature examines decision-making based on nontransitive preferences.

Furthermore, if the preorder is not total, it is not fully represented by an order-preserving function (see Herden [8,9,10], Levin [11,12], and Mehta [13]), but the existence of such a function may be considered “sufficient” in the cases when one looks for maximal and minimal elements on compact topological spaces. Herden [8] began a general unification of many different and sparse results on the existence of continuous order-preserving functions. He introduced the fundamental notion of a linear (complete) decreasing separable system on a preordered topological space in order to derive the most general conditions for the existence of continuous order-preserving functions. Moreover, the previous work of Mehta [13] was focalized on normal preorders in the spirit of the seminal work of Nachbin [14], who proved a generalization of the famous Urysohn’s Lemma in normal topological spaces. Mehta showed that a continuous total preorder is normal, and in this way, he generalized many results guaranteeing the existence of continuous order-preserving functions for (continuous) total preorders.

We recall that an order-preserving real-valued function f on a preordered set $(X,\precsim )$ is an increasing function on $(X,\precsim )$, which, in addition, preserves the strict part≺ of ≾ in the sense that, for all $x,y\in X$, $x\prec y$ implies $f\left(x\right)<f\left(y\right)$. Even if, as we noticed above, an order-preserving function does not characterize a preorder unless the preorder is total, its existence may be viewed as sufficient for many purposes (for example, this is the case of determining the maximal elements since a maximum for an order preserving-function necessarily leads to a maximal element).

However, the existence of a continuous multi-utility representation $\mathcal{F}$ for a preorder on a topological space $(X,t)$ does not imply the existence of a continuous order-preserving function for ≾ unless the set $\mathcal{F}$ is countable (see Alcantud, Bosi, and Zuanon [7]). In this case, there exists a continuous multi-utility representation such that every function in the representation is order-preserving. We shall not be concerned with the particular case of countable continuous multi-utility representations in this paper.

The notion of preorder is as important as the case of an order. Indeed, such a notion is the usual one considered in microeconomics (see Bridges and Mehta [15]), as an agent may express indifference between two alternatives within the choice set. Actually, the agent can even lack the ability to compare them (see Aumann [16]).

In this paper, we focus our attention on closed preorders, i.e., preorders that are closed in the product topology. It is known that the closedness is a necessary condition for the existence of a continuous multi-utility representation. Further, a closed preorder is necessarily semiclosed in the sense that, for every point $x\in X$, the sets $d\left(x\right):=\{y\in X\mid y\precsim x\}$ and $i\left(x\right):=\{z\in X\mid x\precsim z\}$ are closed subsets of X. It is clear that semiclosed total preorders are continuous total preorders in the classical sense (i.e., when, for every point $x\in X$, the sets $l\left(x\right):=\{y\in X\mid y\prec x\}$ and $r\left(x\right):=\{z\in X\mid x\prec z\}$ are open subsets of X).

For the previous discussion, the present paper appears as a natural continuation of the results presented by Bosi and Herden [5], who underlined the fact that the identification of general conditions under which a closed preorder admits a continuous multi-utility representation is still the most important problem in mathematical utility theory. Indeed, Bosi and Herden [5] presented several results concerning the existence of continuous multi-utility representations of closed preorders by proving, for example, that on a Hausdorff topological space, every closed preorder admits such a representation if and only if every closed preorder is normal.

Our work represents a step toward the complete solution of the difficult problem of characterizing topologies such that every closed preorder has a continuous multi-utility representation (at least in the case of Hausdorff topological spaces).

In order to identify conditions under which every closed preorder admits a continuous multi-utility representation, we introduce the concept of a net-compact topology. This is the case of a topology t on a set X such that every net has an accumulation point, provided that it is defined on a directed set such that there exists some convergent net on it and the limit does not belong to the values of the net. It is clear that compact Hausdorff spaces are net-compact.

The concept of a net-compact topology generalizes the notion of a sequentially compact topology. Indeed, a topology is net-compact, provided that it is a first countable and sequentially compact topology. Specifically, when a topology is first countable, a net with a convergent subsequence certainly admits an accumulation point, and further, every sequence has a convergent subsequence when the topological space is sequentially compact.

In this paper, we show that the continuous multi-utility representability of every closed preorder is equivalent to the normality of the topological space if we deal with net-compact Hausdorff spaces. This rather general result guarantees that the net-compactness is a very appropriate condition in our framework. We further prove that a Hausdorf space is net-compact provided that every closed preorder is D-I closed or equivalently d-i closed (see Definition 5) and that, for Hausdorff spaces, normality and net-compactness are equivalent to the property that every closed preorder is strongly normal.

The main result of this paper proves that, for net-compact Hausdorff topological spaces, the normality of the topological space is equivalent to the continuous multi-utility representability of every closed preorder defined on such a topological space.

We conclude this introductory part by presenting the structure of the paper. Section 2 is devoted to the notation and preliminaries. Section 3 presents the main results linked to the notions of net-compactness and continuous multi-utility. Section 4 concludes, summing up the results achieved in this paper and showing some future research directions.

2. Notation and Preliminaries

Definition 1.

Given an arbitrary preorder (that is a reflexive and transitive binary relation) ≾ on a nonempty set X, we define, for every point $x\in X$, the sets

$d\left(x\right):=\{y\in X\mid y\precsim x\}\mathit{and}i\left(x\right):=\{z\in X\mid x\precsim z\}.$

Definition 2.

Let $(X,\precsim )$ be a preordered set. For every subset A of X, we define

${\displaystyle d\left(A\right):=\bigcup _{x\in A}d\left(x\right)\mathit{and}i\left(A\right):=\bigcup _{x\in A}i\left(x\right).}$

Definition 3.

Let $(X,\precsim )$ be a preordered set. A subset D of X is said to be decreasing if $d\left(x\right)\subset D$ for every $x\in D$ or, equivalently, $d\left(D\right)=D$. By duality, one can define the concept of an increasing subset I of X.

Now, consider any topology t on X. As usual, we denote by $\overline{E}$ the topological closure of the subset E of X.

Definition 4.

For every subset A of a preordered topological space $(X,\precsim ,t)$, we denote by $D\left(A\right)$ the smallest closed decreasing subset of $(X,t)$ that includes A and by $I\left(A\right)$ the smallest closed increasing subset of $(X,t)$ that includes A.

Let us present some definitions concerning a preorder on a topological space, which are of crucial importance for the sequel.

Definition 5.

A preorder ≾ on a topological space $(X,t)$ is said to be

• 1. 

  closed if the preorder ≾ is a closed subset $X\times X$ with respect to the product topology $t\times t$ on the topological space $X\times X$;

• 2. 

  semi-closed if, for every point $x\in X$, one has that the sets $d\left(x\right):=\{y\in X\mid y\precsim x\}$ and $i\left(x\right):=\{z\in X\mid x\precsim z\}$ are closed subsets of X;

• 3. 

  D-I-closed if the sets $D\left(A\right)$ and $I\left(B\right)$ are disjoint for any two closed subsets A and B of $(X,t)$ such that $not(y\precsim x)$ for every $x\in A$ and $y\in B$;

• 4. 

  d-i-closed if the sets $d\left(A\right)$ and $i\left(A\right)$ are closed for every closed subset A of $(X,t)$;

• 5. 

  strongly normal if, for every two closed subsets A and B of X such that $not(y\precsim x)$ for any $x\in A$ and $y\in B$, there exist two disjoint open subsets U and V of X, with U decreasing and V increasing such that one has $A\subset U$ and $B\subset V$.

Definition 6.

A preorder ≾ on a topological space $(X,t)$ is said to have a continuous multi-utility representation if, for all $x\in X$ and all $y\in X$, the following equivalence holds:

$x\precsim y\iff \forall f\in \mathcal{F},f\left(x\right)\le f\left(y\right).$

The interested reader can find all the previous definitions in Bosi and Herden [5].

Remark 1.

It is well-known that closed preorders are always semi-closed, while the converse is not true. Further, if a preorder ≾ on a topological space $(X,t)$ has a continuous multi-utility representation, then ≾ is necessarily closed (see Bosi and Herden [5] [Proposition 2.1]).

It is worth noting that the notion of d-i-closed preorders is well-known in the theory of ordered topological spaces, where, e.g., in McCartan [17] and Künzi [18], the stronger concept of IC-continuity was studied. Moreover, the problem of the existence of continuous multi-utility representations of d-i-closed preorders was studied by Bosi and Herden [5], where also the concept of a D-I-closed preorder was considered.

We now recall the definition of a normal preorder on a topological space (see the classical book by Nachbin [14]).

Definition 7.

A preorder ≾ on a topological space $(X,t)$ is said to be normal if for any two disjoint closed subsets $F_0$ and $F_1$ of X, with $F_0$ being decreasing and $F_1$ increasing, there exist two disjoint open subsets of X, $G_0$ and $G_1$, with $G_0$ decreasing and containing $F_0$ and $G_1$ increasing and containing $F_1$.

Remark 2.

It is clear that a topological space $(X,t)$ is normal if and only if the preorder $\precsim :==$ on $(X,t)$ is normal.

The following classical theorem characterizing normal preorders was proved by Nachbin [14] [Theorem 1].

Theorem 1.

Let ≾ be a preorder on a topological space $(X,t)$. Then, ≾ is normal if and only if, given a closed decreasing set $F_0$ and a closed increasing set $F_1$ with $F_0\cap F_1=\varnothing$, there exists a real-valued continuous increasing function f on $(X,t)$ with values in $[0,1]$ such that $f\left(F_0\right)=\left\{0\right\}$ and $f\left(F_1\right)=\left\{1\right\}$.

Let us prove the following theorem, which is based on the above Theorem 1.

Theorem 2.

A normal and semi-closed preorder ≾ on a topological space $(X,t)$ admits a continuous multi-utility representation.

Proof. 

Consider a normal and semi-closed preorder ≾ on a topological space $(X,t)$, and denote by $\mathcal{I}$ the set of all pairs $(x,y)\in X\times X$ such that $not(y\precsim x)$. For every pair $(x,y)\in \mathcal{I}$, it happens that $d\left(x\right)\cap i\left(y\right)=\varnothing$, and therefore, since ≾ is semiclosed and normal, by Theorem 1, we have that there exists a continuous increasing real-valued function $f_{xy}$ on $(X,\precsim ,t)$ such that, in particular, $f_{xy}\left(x\right)=0$ and $f_{xy}\left(y\right)=1$. Hence, the family $\mathcal{F}:={\left\{f_{xy}\right\}}_{(x,y)\in \mathcal{I}}$ is a continuous multi-utility representation of ≾. This consideration completes the proof. □

For reader’s convenience, we recall the classical definitions concerning nets.

Definition 8.

A directed set $(I,\le )$ is a preordered set such that, for all points $a,b\in I$, there exists a point $c\in I$ such that $a\le c$ and $b\le c$.

Definition 9.

Let $(X,t)$ be a topological space, and consider any directed set $(I,\le )$. Then, a net ${\left(x_i\right)}_{i\in I}$ is a mapping from I into X.

Definition 10.

Let ${\left(x_i\right)}_{i\in I}$ be a net in a topological space $(X,t)$. Then ${\left(x_i\right)}_{i\in I}$ is said to converge to a point $x\in X$ ($x_i\to x$) if, for every open subset O of X containing x, there exists $i\in I$ such that $x_j\in O$ for every $j\ge i$.

Definition 11.

Let ${\left(x_i\right)}_{i\in I}$ be a net in a topological space $(X,t)$. Then, a point $x\in X$ is said to be an accumulation point of ${\left(x_i\right)}_{i\in I}$ if, for every open subset O of X containing x and for every $i\in I$, there exists $j\in I$ such that $j\ge i$ and $x_j\in O$.

Remark 3.

It is known that, given a topological space $(X,t)$, a point $x\in X$ is an accumulation point of ${\left(x_i\right)}_{i\in I}$ if and only if some directed subset $(J,\le )$ of $(I,\le )$ can be chosen in such a way that, for every $i\in I$, there exists some $j\in J$ such that $j\ge i$ and the net ${\left(y_j\right)}_{j\in J}$ converges to x. In other terms, there exists a subnet $(J,\le )$ of $(I,\le )$ that converges to x.

The following characterization of the compactness of Hausdorff topological spaces using nets is found, for example, in Engelking ([19] [Theorem 3.1.23]).

Theorem 3.

A Hausdorff topological space $(X,t)$ is compact if and only if every net in X has at least one accumulation point (or equivalently, every net in X has a converging subnet).

Let us now present the main definition of this paper.

Definition 12.

Let $(X,t)$ be a topological space, and denote by $\mathbb{D}$ the collection of all directed sets $(I,\le )$ for which there exists in X some net ${\left(x_i\right)}_{i\in I}$ converging to a point $x\in X$ such that $x\notin \left\{x_i\right|i\in I\}$. Then, $(X,t)$ is said to be net-compact if, for every directed set $(I,\le )\in \mathbb{D}$, all nets ${\left(y_i\right)}_{i\in I}$ of points $y_i\in X$ have an accumulation point.

We now recall two popular topological definitions.

Definition 13.

A topological space $(X,t)$ is said to be

• (i) 

  first countable if every point $x\in X$ possesses a countable local basis;

• (ii) 

  sequentially compact if every sequence in X admits a convergent subsequence.

The following proposition presents a class of topological spaces that are net-compact.

Proposition 1.

Every first countable and sequentially compact topological space is net-compact.

Proof. 

Let $(X,t)$ be a first countable and sequentially compact topological space. Consider any directed set $(I,\le )\in \mathbb{D}$ and a net ${\left(x_i\right)}_{i\in I}$, as defined in Definition 12. The definition of $\mathbb{D}$ implies that there exists a subsequence ${\left\{x_n\right\}}_{n\in \mathbb{N}}$ of ${\left(x_i\right)}_{i\in I}$. Since $(X,t)$ is sequentially compact, there exists a subsequence ${\left\{x_{n_k}\right\}}_{k\in \mathbb{N}}$ of ${\left\{x_n\right\}}_{n\in \mathbb{N}}$ that converges to a point $x\in X$. Such a point is an accumulation point of ${\left(x_i\right)}_{i\in I}$ because $(X,t)$ is first countable. □

Remark 4.

Theorem 3 guarantees that the compactness implies the net-compactness. Contrarily, according to Proposition 1, net-compact spaces are not necessarily compact. The net-compactness, in some sense, may be considered a natural generalization of the concept of sequential compactness.

We conclude this section by presenting an example of a net-compact space that is not compact.

Example 1.

Let ${\aleph }_1$ be the first uncountable ordinal, and consider the interval of ordinals $[0,{\aleph }_1[$ endowed with the order topology $t:=t^{\le }$. Then, t is first countable and sequentially compact, but it is not compact (see Example 43 in Steen and Seebach [20]).

3. Net-Compactness and Continuous Multi-Utility

The concept of net-compactness is motivated by the following results. However, before stating the announced outcomes, the following ad hoc definition is needed.

Definition 14.

Let ${\left(x_i\right)}_{i\in I}$ and ${\left(y_j\right)}_{i\in J}$ be nets in X, and the corresponding directed sets are $(I,⪯)$ and $(J,\le )$, respectively. Then, ${\left(y_j\right)}_{j\in J}$ is said to be a quasi-subnet of ${\left(x_i\right)}_{i\in I}$ if $\{y_j\mid j\in J\}\subset \{x_i\mid i\in I\}$ and $\le \subset {⪯}_{\mid J}$.

In order to motivate this ad hoc definition, let $([0,{\aleph }_1],\le ,t^{\le })$ be the well-ordered set of all ordinal numbers $0\le \alpha \le {\aleph }_1$ endowed with its order topology. Then, we choose the subset $\Lambda$ of $[0,{\aleph }_1]$ that consists of all limit ordinals $\lambda \in [0,{\aleph }_1]$ in order to set $\Gamma :=[0,{\aleph }_1]\setminus \Lambda$. Of course, ${\left(\gamma \right)}_{\gamma \in \Gamma }$ is a net in $[0,{\aleph }_1]$ that converges to ${\aleph }_1$. In addition, ${\aleph }_1$ is the only one accumulation point of ${\left(\gamma \right)}_{\gamma \in \Gamma }$. However, for every $\lambda \in \Lambda$, there exists some quasi-subnet ${\left(\chi \right)}_{\chi \in X}$ of ${\left(\gamma \right)}_{\gamma \in \Gamma }$ that converges to $\lambda$. It is worth stressing that this situation appears as a crucial part in the proof of the following Proposition 2. For those reasons, it is necessary to introduce the ad hoc concept of a quasi-subnet.

Before stating the next proposition, we recall that ${\Delta }_X:=\{(x,x)\mid x\in X\}$ denotes, as usual, the diagonal of X.

Proposition 2.

Let $(X,t)$ be a Hausdorff space. Then, the following assertions are equivalent:

• (i) 

  Every closed preorder ≾ on $(X,t)$ is d-i-closed.

• (ii) 

  Every closed preorder ≾ on $(X,t)$ is D-I-closed.

• (iii) 

  $(X,t)$ is net-compact.

Proof. 

In order to verify the equivalence of the assertions (i), (ii), and (iii), it is enough to prove the following assertions.

• (j). In order for every closed preorder ≾ on $(X,t)$ to be d-i-closed, it is necessary and sufficient that $(X,t)$ is net-compact.

• (jj). In order for every closed preorder ≾ on $(X,t)$ to be D-I-closed, it is necessary and sufficient that $(X,t)$ is net-compact.

Necessity: We prove the necessity parts of both assertions (j) and (jj) in one step. Hence, we assume, by contradiction, that there exists some directed set $(I,\le )\in \mathbb{D}$ and some net ${\left(x_i\right)}_{i\in I}$ that converges with respect to $(X,t)$ to some point $x\in X$ such that $x\notin \left\{x_i\right|i\in I\}$, but for which there exists a net ${\left(y_i\right)}_{i\in I}$ having no accumulation points. Let us abbreviate this assumption by ${(}^{\ast })$. Then, we have to construct a closed preorder ≾ on $(X,t)$ that neither can be d-i-closed nor D-I-closed. We proceed by assuming that $(I,\le )$ has been divided into two disjoint directed sets $(K,\le )$ and $(L,\le )$ in such a way that $\left|K\right|=\left|L\right|=\left|I\right|$ and both nets ${\left(x_k\right)}_{k\in K}$ and ${\left(x_l\right)}_{l\in L}$ converge to x. Since a routine induction procedure that, in each step, adds new points $x_k$ to ${\left(x_k\right)}_{k\in K}$ and $x_l$ to ${\left(x_l\right)}_{l\in L}$, respectively, allows us to construct directed subsets $(K,\le )$ and $(L,\le )$ such that $\left|K\right|=\left|L\right|$ and both nets ${\left(x_k\right)}_{k\in K}$ and ${\left(x_l\right)}_{l\in L}$ converge to x, the above assumption does not entail any loss of generality. Now, we consider the subsets $\left\{(x_k,y_k)|k\in K\right\}$ and $\left\{(y_l,x_l)|l\in L\right\}$ of $X\times X$ in order to choose the set U of all pairs $(u,v)\in X\times X$ for which there exist quasi-subnets ${\left(u_p\right)}_{p\in P}$ of ${\left(x_k\right)}_{k\in K}$ and ${\left(v_p\right)}_{p\in P}$ of ${\left(y_k\right)}_{k\in K}$ that converge to u and v, respectively (cf. the afore given comments on quasi-subnets). In addition, we consider the set T of all pairs $(s,t)\in X\times X$ for which there exist quasi-subnets ${\left(s_q\right)}_{q\in Q}$ of ${\left(y_l\right)}_{l\in L}$ and ${\left(t_q\right)}_{q\in Q}$ of ${\left(x_l\right)}_{l\in L}$ that converge to s and t, respectively. Then, we set $S_k:=\left\{(x_k,y_k)|k\in K\right\}\cup U$ and $S_L:=\left\{(y_l,x_l)|l\in L\right\}\cup T$ in order to finally consider the preorder

$\precsim :={\Delta }_X\cup S_K\cup S_L$

on $(X,t)$. Because of ${(}^{\ast })$, we can suppose without loss of generality that the sets $S_K^1:=\{x_k\mid k\in K\}$ and $S_K^2:=\{y_k\mid k\in K\}$ as well as the sets $S_L^1:=\{y_l\mid l\in L\}$ and $S_L^2:=\{x_l\mid l\in L\}$ are disjoint. Let us abbreviate this assumption by ${(}^{\ast \ast })$. Since $(X,t)$ is a Hausdorff space and since the net ${\left(y_i\right)}_{i\in I}$ is assumed to have no accumulation point, the definition of ≾ allows us to conclude, with the help of ${(}^{\ast \ast })$, that ≾ is a closed preorder on $(X,t)$. Now, we set $A:=\overline{\{y_k\mid k\in K\}}$ and $B:=\overline{\{y_l\mid l\in L\}}$. Because of the definition of ≾, it follows that x is neither contained in $d\left(A\right)$ nor in $i\left(B\right)$. This observation implies that ≾ is not d-i-closed. In addition, the relation $x\in \overline{d\left(A\right)}\cap \overline{i\left(B\right)}$ implies that ≾ cannot be D-I-closed. The previous contradictions prove the necessity parts of both assertions.

Sufficiency: In order to simultaneously show the sufficiency part of both assertions (j) and (jj), we use the fact that a d-i-closed preorder ≾ on $(X,t)$ is D-I-closed. Indeed, if ≾ is d-i-closed and A is a closed subset of X, then we have that $D\left(A\right)=d\left(A\right)$ and $I\left(A\right)=i\left(A\right)$. Therefore, it is enough to verify the sufficiency part of assertion (j). In order to verify this part of assertion (j), let A be a closed subset of $(X,t)$ and ≾ a closed preorder on $(X,t)$. Then, we show that $d\left(A\right)$ is a closed subset of $(X,t)$. By duality, we may conclude that $i\left(A\right)$ is also a closed subset of $(X,t)$. Let us suppose, by contraposition, that $d\left(A\right)$ is not closed. Therefore, there exists some point $x\in X\setminus d\left(A\right)$ that has the property that $O\cap d\left(A\right)\ne \varnothing$ for every open subset of $(X,t)$ that contains x. This means, in particular, that there exists some directed set $(I\le )\in D$ for which there exists some net ${\left(x_i\right)}_{i\in I}$ of points $x_i\in d\left(A\right)$ that converges to x. Nevertheless, for every point $x_i\in d\left(A\right)$, there exists some point $y_i\in A$ such that $x_i\precsim y_i$. Since $(X,t)$ is net-compact and A is a closed subset of $(X,t)$, one, thus, can assume (without loss of generality) that the net ${\left(y_i\right)}_{i\in I}$ converges to some point $y\in A$. Using now our assumption that ≾ is a closed preorder on $(X,t)$ and the validity of the inequality $x_i\precsim y_i$ for all $i\in I$, it follows that $x\precsim y$. Hence, $x\in d\left(A\right)$. The previous contrast finishes the sufficiency parts of assertion (i) and, thus, assertion (ii). □

The next proposition links the notion of net-compactness with the one of strong normality. For the reader’s convenience, we recall two theorems, which can be found in Bosi and Herden [5] since they are exploited in the proof of Proposition 3 below.

Theorem 4.

Let ≾ be a preorder on a topological space $(X,t)$. Then, the following conditions are equivalent:

• (i) 

  Every d-i-closed preorder ≾ on $(X,t)$ has a continuous multi-utility representation;

• (ii) 

  Every D-I-closed preorder ≾ on $(X,t)$ admits a continuous multi-utility representation;

• (iii) 

  $(X,t)$ is a normal topological space.

Theorem 5.

Let $(X,t)$ be a Hausdorff topological space $(X,t)$. Then, the following conditions are equivalent:

• (i) 

  Every closed preorder ≾ on $(X,t)$ has a continuous multi-utility representation;

• (ii) 

  Every closed preorder ≾ on $(X,t)$ is normal.

By using the two previous theorems, we are now able to prove the following proposition.

Proposition 3.

Let $(X,t)$ be a Hausdorff space. Then, the following assertions are equivalent:

• (i) 

  Every closed preorder ≾ on $(X,t)$ is strongly normal;

• (ii) 

  $(X,t)$ is normal and net-compact.

Proof. 

(i) ⇒ (ii): Assertion (i) implies that ${\Delta }_X$ is a strongly normal preorder on $(X,t)$. Hence, $(X,t)$ is a normal space, and it is sufficient to verify that $(X,t)$ is net-compact. Therefore, we apply the definition of a strongly normal preorder ≾ on $(X,t)$ in order to conclude that every closed preorder ≾ on $(X,t)$ is D-I-closed. Proposition 2, thus, implies that $(X,t)$ is net-compact.

(ii) ⇒ (i): Let ≾ be a closed preorder on $(X,t)$. Then, assertion (ii) of Proposition 2 implies that ≾ is D-I-closed. Since $(X,t)$ is a normal space, from Theorem 4, we have that ≾ has a continuous multi-utility representation. Hence, Theorem 5 allows us to conclude that ≾ is normal. Now, the validity of assertion (i) is a consequence of Proposition 2. □

The following fundamental theorem appears as a consequence of Proposition 2.

Theorem 6.

Let $(X,t)$ be a net-compact Hausdorff space. Then, the following conditions are equivalent:

• (i) 

  Every closed preorder ≾ on $(X,t)$ has a continuous multi-utility representation;

• (ii) 

  $(X,t)$ is normal.

Proof. 

(i) ⇒ (ii). The assumption that every closed preorder ≾ on $(X,t)$ possesses a continuous multi-utility representation implies, with the help of Theorem 5, that every closed preorder ≾ on $(X,t)$ is normal. Since $(X,t)$ is a Hausdorff space, it follows, in addition, that ${\Delta }_X$ is a closed preorder on $(X,t)$. Hence, $(X,t)$ is a normal space.

(ii) ⇒ (i). Let $(X,t)$ be a normal and net-compact space, and let ≾ be a closed preorder on $(X,t)$. Then, Proposition 2 implies that ≾ is d-i-closed (D-I-closed). It suffices to verify that ≾ is a normal preorder on $(X,t)$. Therefore, we can arbitrarily choose two disjoint closed subsets A and B of X, with A decreasing and B increasing. Since $(X,t)$ is a normal space, there exist two disjoint open subsets U and V of X such that $A\subset U$ and $B\subset V$. Now, we show that U and V can be refined to open decreasing and open increasing subsets O and P of X, respectively, in such a way that $A\subset O\subset U$ and $B\subset P\subset V$. Now, it is sufficient to verify that there exists some open decreasing subset O of X such that $A\subset O\subset U$. The existence of P then follows by duality. In order to verify the existence of O, we consider the subset H of U that consists of all points $x\in U$ for which there exists some point $y\in X\setminus U$ such that $y\precsim x$. Then, we show that H is a closed subset of U endowed with the relativized topology $t_{\mid U}$. This result allows us to set $O:=U\setminus H$. Because of the definition of H, we, therefore, may infer that O is an open decreasing subset of U such that $A\subset O\subset U$, and this concludes the construction. In order to prove that H is a closed subset of X endowed with the relativized topology $t_{\mid U}$, we assume, in contrast, that there exists some point $z\in U\setminus H$ such that for every open neighborhood E of z, there exists some point $u\in U$ for which there exists some point $v\in X\setminus U$ such that $v\precsim u$. Thus, we, finally, may construct nets ${\left(u_i\right)}_{i\in I}$ and ${\left(v_i\right)}_{i\in I}$ of points $u_i\in X$ and $v_i\in X$ such that ${\left(u_i\right)}_{i\in I}$ converges to z. Moreover, since $(X,t)$ is net-compact, it follows that one can suppose (without loss of generality) that there exists some point $t\in X$ such that ${\left(v_i\right)}_{i\in I}$ converges to t. At this point, the validity of the relation $v_i\precsim u_i$ for all $i\in I$ implies, with the help of assumption ≾ to be a closed preorder on $(X,t)$, that $t\precsim z$. Hence, $t\in d\left(z\right)$ and $t\in X\setminus U$. This observation allows us to conclude that $z\in H$, which contradicts our assumption that $z\in U\setminus H$. □

As an immediate consequence of Proposition 1 and Theorem 6, we have that the following corollary holds.

Corollary 1.

Let $(X,t)$ be a first countable and sequentially compact Hausdorff space. Then, the following conditions are equivalent:

• (i) 

  Every closed preorder ≾ on $(X,t)$ admits a continuous multi-utility representation;

• (ii) 

  $(X,t)$ is a normal space.

4. Conclusions

In this paper, we deal with the problem of finding conditions on a topological space that imply the existence of a continuous multi-utility representation for every closed preorder. To this aim, the notion of net-compactness of a topology is introduced in order to prove an interesting theorem, according to which every closed preorder on a net-compact and normal Hausdorff topological space possesses a continuous multi-utility representation. Therefore, net-compactness, which generalizes sequential compactness, appears as an appropriate condition in order to tackle the general problem of characterizing topological spaces with the property that every closed preorder has a continuous multi-utility representation. Such a general and very intriguing problem will be hopefully analyzed in a future paper.

Author Contributions

Conceptualization, G.B.; supervision, M.Z.; writing—review and editing, G.S. All authors have read and agreed to the published version of the manuscript.

Data Availability Statement

There are no data to report in this manuscript.

Conflicts of Interest

The authors have no conflicts of interest to declare. All co-authors have seen and agree with the contents of the paper, and there is no financial interest to report.

Footnotes

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