Lifting Theorems for Continuous Order-Preserving Functions and Continuous Multi-Utility

Abstract

We present some lifting theorems for continuous order-preserving functions on locally and $\sigma$-compact Hausdorff preordered topological spaces. In particular, we show that a preorder on a locally and $\sigma$-compact Hausdorff topological space has a continuous multi-utility representation if, and only if, for every compact subspace, every continuous order-preserving function can be lifted to the entire space. Such a characterization is also presented by introducing a lifting property of ≾-C-compatible continuous order-preserving functions on closed subspaces. The assumption of paracompactness is also used in connection to lifting conditions.

1. Introduction

The problem concerning the continuous representability of not necessarily total preorders is very interesting not only from a purely mathematical viewpoint, but also for its possible applications to economics and social sciences. General achievements concerning the existence of continuous utility representations were very recently presented by Bosi [1] in the case of nontotal preorders, and by Bosi and Zuanon [2] in the case of total preorders.

Several authors have presented contributions to this topic by following a direct approach, which is mainly based on the existence of a separable system or decreasing scale in a topological preordered space. Such a notion generalizes the concept of a scale in a topological space (see for example, Burgess and Fitzpatrick [3] and Gillman and Jerison [4]). In particular, Herden [5,6] proved very general results in this direction, but also other authors contributed to this field (see, for example, Herden and Pallack [7], Levin [8,9], Mehta [10,11] and Minguzzi [12]).

On the other hand, another possible approach to the existence of continuous representations of preorders by means of one real-valued function is based on lifting theorems, which concern the possibility of lifting a continuous (strictly) isotone function from a generic (typically closed or compact) subspace of the topological preordered space to the entire space.

Nachbin [13] generalized to normally preordered topological spaces the well-known Tietze–Urysohn extension theorem in normal spaces (see, for example, Engelking [14]), according to which it is always possible to lift a continuous real-valued function from a closed subspace of a normal space over the entire space.

Herden [15] was concerned with the possibility of lifting continuous order-preserving functions from (closed) subsets of preordered topological spaces. Further, Herden [16] generalized to arbitrary topological preordered spaces the aforementioned extension result proved by Nachbin and, as a consequence of his main result, characterized the possibility of lifting every bounded increasing continuous real-valued function from a closed subset of a preordered topological space to the whole space.

Mehta [17] studied a variant of Nachbin’s lifting theorem. Recently, some results concerning the extensions of continuous increasing or order-preserving functions were presented by Evren and Hüsseinov [18].

In this paper, we present a lifting theorem for a closed preorder (i.e., a preorder which is closed with respect to the product topology), which guarantees the possibility of lifting a continuous real-valued order-preserving function defined on compact or closed subsets of a locally compact topological preordered space. The interest for closed preorders primarily arises in connection to the fact that the condition of being closed is necessary for the existence of a continuous multi-utility representation.

We recall that a preorder ≾ is defined to admit a continuous multi-utility representation on a topological space $(X,t)$ if there is a collection $\mathcal{F}$ of continuous increasing functions such that, for all points $x,y\in X$, we have that $x\precsim y$ if, and only if, $f\left(x\right)\le f\left(y\right)$ for every $f\in \mathcal{F}$. It is worth noticing that continuous multi-utility representations, which were first introduced and investigated by Evren and Ok [19] (see also Bosi and Herden [20]), fully characterize the given closed preorder, while order-preserving functions only provide very particular continuous extensions by means of continuously representable total preorders (see, for example, Bosi and Herden [21]).

We concentrate our attention on locally and $\sigma$-compact Hausdorff spaces. We prove that a preorder on a locally and $\sigma$-compact Hausdorff space is closed (or equivalently, representable by a continuous multi-utility) if, and only if, for every compact subspace, every continuous order-preserving function can be lifted to the entire space. We further inaugurate the notion of a ≾-C-compatible real-valued function on a topological preordered space $(X,\precsim ,t)$, and we prove that a preorder on a locally and $\sigma$-compact Hausdorff space is closed if, and only if, for every closed subspace, every bounded, continuous and ≾-C-compatible order-preserving function can be lifted to the entire space. Finally, we show that the assumption of $\sigma$-compactness cannot be avoided in such a characterization since the aforementioned lifting property from compact subspaces is equivalent to $\sigma$-compactness when the topological space is locally compact and paracompact.

2. Basic Concepts

In the sequel, $N=\{0,1,2,...,,n,n+1,...\}$ is the set of natural numbers, R the set of real numbers, and $[a,b]$ a non-degenerate (non-trivial) real interval. As usual, we denote by $[a,b[$, $]a,b]$ and $]a,b[$, respectively, the corresponding half-closed half-open, half-open half-closed and open real intervals, respectively. $[0,1]$ is the real unit interval. For every set A, we abbreviate by $\mid A\mid$ the cardinality of A. ${\mathsf{\Delta }}_A:=\{(a,a)\mid a\in A\}$ is the diagonal of A.

Definition 1.

A preorder ≾ on a set X is a binary relation on X satisfying reflexivity and transitivity. The pair $(X,\precsim )$ is referred to as a preordered set.

The strict part ≺ of a preorder ≾ on a set X is defined to be, for all $x,y\in X$,

$$x\prec y\iff (x\precsim y)andnot (y\precsim x).$$

In the sequel, $(x,y)\in \prec$ will occasionally replace $x\prec y$.

Definition 2.

A real-valued function f on a preordered set $(X,\precsim )$ is defined to be

(i)

Increasing, if, for all $x,y\in X$,

$$x\precsim y\Rightarrow f\left(x\right)\le f\left(y\right);$$

(ii)

Order-preserving, if f is increasing and, for all $x,y\in X$,

$$x\prec y\Rightarrow f\left(x\right)<f\left(y\right).$$

An increasing (order-preserving) function is sometimes called an isotone (respectively, strictly isotone) function.

If $(X,\precsim ,t)$ is a preordered topological space, C is a subset of X and h is a real-valued function on X, then $t_{\mid C}$, ${\precsim }_{\mid C}$ and $h_{\mid C}$ are the topology on C, which is induced by t, the restriction of ≾ to C, and, respectively, the restriction of h to C.

$\overline{A}$ stands for the topological closure (with respect to t) of any subset A of X. In addition, $t_{nat}$ will stand for the natural topology on $\mathbb{R}$.

Definition 3.

Let ≾ be a preorder on a topological space $(X,t)$. Then ≾ is defined to be closed if ≾ is a closed subset of $X\times X$ with the product topology $t\times t$.

A closed preorder is referred to as a continuous preorder by some authors (see, for example, Evren and Ok [19]). For every closed preorder ≾ on X and every point $x\in X$, we have that the sets

$$d\left(x\right):=\{y\in x\mid y\precsim x\},i\left(x\right):=\{z\in x\mid x\precsim z\}$$

are both closed subsets of X. It is well known that a preorder ≾ on X that has the property that, for every point $x\in X$ both sets $d\left(x\right)$ and $i\left(x\right)$ are closed, is not necessarily closed. However, if a preorder ≾ is total, then the closedness of ≾ is equivalent to the requirement according to which both sets $d\left(x\right)$ and $i\left(x\right)$ are closed. This latter property is sometimes referred to as semiclosedness (see, for example, Bosi and Herden [22]).

Definition 4.

Let $(X,\precsim )$ be a preordered set, and let $C\ne \varnothing$ be a subset of X. Then a real-valued function f on C is defined to be ≾-C-compatible if, for every pair $(x,y)\in \prec$, the sets $\overline{f(C\cap d(x\left)\right)}$ and $\overline{f(C\cap i(y\left)\right)}$ are disjoint.

Let $E_{\prec }^C$ be the family of all pairs $(x,y)\in \prec$ for which neither $C\cap d\left(x\right)$ nor $C\cap i\left(y\right)$ is empty. Let f be a real-valued function on X. For every pair $(x,y)\in E_{\prec }^C$, we set

$$s_x^C\left(f\right):=supf(C\cap d\left(x\right)),i_y^C\left(f\right):=inff(C\cap i\left(y\right)).$$

Then the following proposition, the simple proof of which may be omitted for the sake of brevity, somewhat characterizes real-valued ≾-C-compatible functions.

Proposition 1.

Let f be a real-valued function on a preordered set $(X,\precsim )$, and let C be a subset of X. The following conditions, concerning a real-valued function f on C, are equivalent:

(i)

f is ≾-C-compatible;

(ii)

$s_x^C\left(f\right)<i_y^C\left(f\right)$ for every pair $(x,y)\in E_{\prec }^C$;

(iii)

For every pair $(x,y)\in E_{\prec }^C$, the following implication holds:

$$f^{-1}\left(\right]-\infty ,s_x^C\left(f\right)\left]\right)\cup f^{-1}\left(\right[i_y^C\left(f\right),\infty \left[\right)=C\Rightarrow s_x^C\left(f\right)<i_y^C\left(f\right).$$

Definition 5.

Let $(P,\precsim )$ be a preordered set. Then a subset A of P is said to be decreasing if $x\in A$ and $y\precsim x$ imply that $y\in A$. Dually, the notion of an increasing subset B of P is expressed.

Definition 6.

A preorder ≾ on $(X,t)$ is said to be representable by continuous multi-utility if, for some family $\mathcal{F}$ of continuous increasing real-valued functions f on $(X,\precsim ,t)$, the following equivalence is valid for all $x\in X$ and all $y\in Y$:

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

Definition 7.

A preordered topological space $(X,\precsim ,t)$ is defined to be normally preordered if, for every pair $(F_0,F_1)$ of disjoint closed sets, $F_0$ being decreasing and $F_1$ being increasing, there exists a pair of disjoint open sets $(A_0,A_1)$, where $A_0$ is decreasing and contains $F_0$, and $A_1$ is increasing and contains $F_1$.

Nachbin [13], Theorem 2 on page 36, proved the following generalization to continuous increasing functions of the Tietze–Urysohn extension theorem (see, for example, Engelking [14], Theorem 2.1.8).

Theorem 1

(Nachbin [13]). Consider a normally preordered topological space $(X,\precsim ,t)$, and let f be a bounded, continuous and increasing real-valued function defined on some closed subset C of X. The function f can be extended to X in such a way that the resulting extension is bounded, continuous and increasing on $(X,\precsim ,t)$ if, and only if, for every pair of real numbers $r<r^{\prime }$, the smallest closed decreasing subset $A\left(r\right)$ of X that contains the set $A_r$ of all points $x\in C$ such that $f\left(x\right)\le r$, and the smallest closed increasing subset $B\left(r^{\prime }\right)$ of X that contains the set $B_{r^{\prime }}$ of all points $y\in C$ such that $r^{\prime }\le f\left(y\right)$, are disjoint.

Bosi and Herden [22] used the following definition.

Definition 8.

A preordered topological space $(X,\precsim ,t)$ is defined to be strongly normally preordered if, for every pair $(A,B)$ of disjoint closed subsets of X with $not(x\succsim y)$ for every pair $(x,y)\in A\times B$, there exists a pair $(U,V)$ of disjoint open subsets of X, where U is decreasing and contains A, and V is increasing and contains V.

It is immediate to check that a strongly normally preordered topological space is normally preordered.

3. The Lifting Theorems

We are going to show the validity of a general lifting theorem concerning continuous order-preserving functions on compact and, respectively, closed subspaces of a topological space satisfying particular conditions of compactness. In order to prove our theorem, characterizing closed preorders in terms of a lifting property on locally and $\sigma$-compact (a topological space $(X,t)$ is said to be locally compact if every point in X has an open neighborhood whose closure is compact, and $(X,t)$ is said to be σ-compact if it is a union of countably many compact subsets) Hausdorff topological spaces, we need to use Theorem 1 presented by Evren and Ok [19] and to prove two lemmas, together with a resulting proposition.

Theorem 2

(Evren and Ok [19]). Every closed preorder ≾ on a locally and σ-compact Hausdorff space $(X,t)$ is representable by a continuous multi-utility.

Lemma 1.

A preordered locally and σ-compact Hausdorff space $(X,\precsim ,t)$ is strongly normally preordered provided that the preorder ≾ is closed.

Proof. 

Consider two disjoint closed subsets A and B of X, with the property that, for every pair $(x,y)\in A\times B$, $not(x\succsim y)$. By Theorem 2, ≾ is representable by a continuous multi-utility $\mathcal{F}$. Therefore, for every pair $(x,y)\in A\times B$, there is some continuous increasing function $f_{xy}\in \mathcal{F}$, $f_{xy}:(X\precsim ,t)\to ([0,1],{\le }_{\mid [0,1]},t_{nat\mid [0,1]})$, with the property that $f_{xy}\left(x\right)=0$ and $f_{xy}\left(y\right)=1$ (see Evren and Ok [18], Remark 3). It follows that ${\displaystyle A\times B\subset \bigcup _{(x,y)\in A\times B}{f}_{xy}^{-1}\left(\left[0,\frac{1}{2}\right[\right)\times f_{xy}^{-1}\left(\left[\frac{1}{2},1\right[\right)}$. Since a $\sigma$-compact (Hausdorff) space is Lindelöf (a topological space $(X,t)$ is said to be Lindelöf if every open cover of X has a countable subcover), and since, in addition, finite products of locally and $\sigma$-compact Hausdorff spaces are locally and $\sigma$-compact Hausdorff spaces, it follows that there is a countable collection ${\left\{(x_n,y_n)\right\}}_{n\in N}$ of pairs $(x_n,y_n)\in A\times B$ such that ${\displaystyle A\times B\subset \bigcup _{(x_n,y_n)}{f}_{x_n{y}_n}^{-1}\left(\left[0,\frac{1}{2}\right[\right)\times f_{x_n{y}_n}^{-1}\left(\left[\frac{1}{2},1\right[\right)}$. Let ${\displaystyle F:=\sum _{n\in N}\frac{1}{2^{n+1}}{f}_{x_n{y}_n}}$. Then $U:=F^{-1}\left(\left[0,\frac{1}{2}\right[\right)$ and $V:=F^{-1}\left(\left]\frac{1}{2},1\right]\right)$ are two disjoint open subsets of X, with the additional property that U is decreasing and contains A, and V is increasing and contains B. □

Let us now show that a continuous increasing function on a closed subset of a strongly normally preordered topological space can be lifted to the entire space in order for it to remain continuous and increasing.

Lemma 2.

Let $(X\precsim ,t)$ be a strongly normally preordered space. Then the following property holds:

“If C is any closed subset of X, and $f:(C,{\precsim }_{\mid C},t_{\mid C})\to (R,\le ,t_{nat})$ is any continuous increasing function, then $F_{\mid C}=f$ for some continuous increasing function $F:(X,\precsim ,t)\to (R,\le ,t_{nat})$".

Proof. 

By the above Theorem 1, it suffices to show that, for every pair $r<r^{\prime }$ of real numbers, the smallest closed decreasing subset $A\left(r\right)$ of X that includes the set $A_r$ of all points $x\in C$ such that $f\left(x\right)\le r$, and the smallest closed increasing subset $B\left(r^{\prime }\right)$ of X that includes the set $B_{r^{\prime }}$ of all points $y\in C$ such that $r^{\prime }\le f\left(y\right)$ are disjoint. Let, therefore, real numbers $r<r^{\prime }$ be arbitrarily chosen. Since $r<r^{\prime }$ and since f is continuous and increasing, $A_r$ and $B_{r^{\prime }}$ are disjoint closed subsets of X such that $not(x\succsim y)$ for every pair $(x,y)\in A_r\times B_{r^{\prime }}$. Hence, the assumption that $(X\precsim ,t)$ is a strongly normally preordered space implies that there exist disjoint open decreasing and increasing subsets U, respectively, V of X, such that $A_r\subset U$ and $B_{r^{\prime }}\subset V$. It follows that $X\backslash V$ is a closed decreasing subset of X such that $A_r\subset X\backslash V$. This means, in particular, that $A\left(r\right)\subset X\backslash V$. With help of the observations that $A\left(r\right)$ is decreasing, V is increasing, $A\left(r\right)\cap V=\varnothing$ and $B_{r^{\prime }}\subset V$, we may conclude that $A\left(r\right)\cap B_{r^{\prime }}=\varnothing$ and that $not(x\succsim y)$ for all pairs $(x,y)\in A\left(r\right)\times B_{r^{\prime }}$. Therefore, there exist disjoint open decreasing and increasing subsets H, respectively, W of X, such that $A\left(r\right)\subset H$ and $B\left(r^{\prime }\right)\subset W$. These inclusions imply that $X\backslash H$ is a closed increasing subset of X that includes $B_{r^{\prime }}$. It, thus, follows that $B\left(r^{\prime }\right)\subset X\backslash H$. Therefore, we have that $A\left(r\right)\cap B\left(r^{\prime }\right)=\varnothing$. □

Needless to say, we can put together Lemma 1 and Lemma 2 so that the following proposition holds true.

Proposition 2.

Let $(X,\precsim ,t)$ be a preordered locally and σ-compact Hausdorff space, the preorder ≾ of which is closed. Then the following property holds:

“If C is any closed subset of X, and $f:(C,{\precsim }_{\mid C},t_{\mid C})\to (R,\le ,t_{nat})$ is any continuous increasing function, then $F_{\mid C}=f$ for some continuous increasing function $F:(X,\precsim ,t)\to (R,\le ,t_{nat})$".

The following theorem characterizes the existence of a continuous multi-utility representation for a preorder on a locally and $\sigma$-compact Hausdorff space in terms of lifting properties from closed and, respectively, compact subspaces.

Theorem 3.

Consider a preordered locally and σ-compact Hausdorff space $(X,\precsim ,t)$. Then the following conditions are equivalent:

(i)

≾ is representable by a continuous multi-utility;

(ii)

If ≾ is any closed preorder on $(X,t)$, then the following property is verified:

“If C is any closed subset of X, and $f:(C,{\precsim }_{\mid C},t_{\mid C})\to (R,\le ,t_{nat})$ is any bounded, continuous, order-preserving and ≾-C-compatible function, then $F_{\mid C}=f$ for some continuous order-preserving function $F:(X,\precsim ,t)\to (R,\le ,t_{nat})$";

(iii)

If ≾ is any closed preorder on $(X,t)$, then the following property is verified:

“If C is any compact subset of X, and $f:(C,{\precsim }_{\mid C},t_{\mid C})\to (R,\le ,t_{nat})$ is any continuous order-preserving function, then $F_{\mid C}=f$ for some continuous order-preserving function $F:(X,\precsim ,t)\to (R,\le ,t_{nat})$”.

Proof. 

(i) ⇒ (ii) and (i) ⇒ (iii): We prove jointly the two implications for the sake of convenience. Since ≾ is representable by a continuous multi-utility, we have that ≾ is closed by Bosi and Herden [22], Proposition 2.1. We consider a subset C of X and a continuous order-preserving function $f:(C,{\precsim }_{\mid C},t_{\mid C})\to (R,\le ,t_{nat})$. In order to verify that f can be lifted to a continuous order-preserving function $F:(X,\precsim ,t_{nat})\to (R,\le ,t_{nat})$, according to other suitable assumptions suggested by the consideration of the implication we want to prove, we arbitrarily choose a pair $(x,y)\in \prec$, and we define $C_{xy}:=C\cup \{x,y\}$. We proceed by showing that f can be extended to a continuous order-preserving function $f_{xy}:(C_{xy},{\precsim }_{\mid C_{xy}},t_{\mid C_{xy}})\to (R,\le ,t_{nat})$. Therefore, we distinguish between the following four cases:

Case 1:

$C\cap d\left(x\right)=\varnothing$ and $C\cap i\left(y\right)=\varnothing$.

Case 2:

$C\cap d\left(x\right)\ne \varnothing$ and $C\cap i\left(y\right)=\varnothing$.

Case 3:

$C\cap d\left(x\right)=\varnothing$ and $C\cap i\left(y\right)\ne \varnothing$.

Case 4:

$C\cap d\left(x\right)\ne \varnothing$ and $C\cap i\left(y\right)\ne \varnothing$.

The only case that needs particular reflection is the case that both sets $C\cap d\left(x\right)$ and $C\cap i\left(y\right)$ are not empty (i.e., the pair $(x,y)\in E_{\prec }^C$). In this case it, clearly, suffices to prove that $s_x^C\left(f\right)$ and $i_y^C\left(f\right)$ exist and that $s_x^C\left(f\right)<i_y^C\left(f\right)$. Indeed, having proved the existence of $s_x^C\left(f\right)$ and $i_y^C\left(f\right)$ as well as the strong inequality $s_x^C\left(f\right)<i_y^C\left(f\right)$, we may assume that $x\notin C$ or $y\notin C$. In this situation, the inequality $s_x^C\left(f\right)<i_y^C\left(f\right)$ allows us to set $f_{xy}\left(x\right):=s_x^C\left(f\right)+\frac{i_y^C\left(f\right)-s_x^C\left(f\right)}{4}$ if $x\notin C$ and $y\in C$ or $f_{xy}\left(y\right):=i_y^C\left(f\right)-\frac{i_y^C\left(f\right)-s_x^C\left(f\right)}{4}$ if $x\in C$ and $y\notin C$ or $f_{xy}\left(x\right):=s_x^C\left(f\right)+\frac{i_y^C\left(f\right)-s_x^C\left(f\right)}{4}$ and $f_{xy}\left(y\right):=i_y^C\left(f\right)-\frac{i_y^C\left(f\right)-s_x^C\left(f\right)}{4}$ if $x\notin C$ and $y\notin C$. It, thus, remains to verify that $s_x^C\left(f\right)$ and $i_y^C\left(f\right)$ exist and that the strong inequality $s_x^C\left(f\right)<i_y^C\left(f\right)$ holds.

Let us now concentrate on the implication (i) ⇒ (ii). In this case, since C is a closed subset of X and $(X,t)$ is a Hausdorff space, we may conclude that $C_{xy}:=C\cup \{x,y\}$ is a closed subset of X. In addition, besides the assumption that f is continuous and order-preserving, we have that f is bounded and ≾-C-compatible. Using the fact that f is bounded and continuous on C closed, and that $d\left(x\right)$ and $i\left(y\right)$ are closed due to the closedness of ≾, we have that, actually, $s_x^C\left(f\right):=maxf(C\cap d\left(x\right))$ and $i_y^C\left(f\right):=minf(C\cap i\left(y\right))$, and $s_x^C\left(f\right)<i_y^C\left(f\right)$ for every pair $(x,y)\in E_{\prec }^C$ by condition (ii) of Proposition 1.

Let us now consider the implication (i) ⇒ (iii). Since C is a compact subset of X and $(X,t)$ is a Hausdorff space, we may conclude that $C_{xy}:=C\cup \{x,y\}$ is a compact subset of X. Well, the compactness of C implies that there exist points $v\in C\cap d\left(x\right)$ and $z\in C\cap i\left(y\right)$ such that $f\left(v\right)=s_x^C\left(f\right)$ and $f\left(z\right)=i_y^C\left(f\right)$. Since f is order-preserving, we thus may conclude that $f\left(v\right)=s_x^C\left(f\right)<f\left(z\right)=i_y^C\left(f\right)$. Let us abbreviate the above considerations by ${(}^{*})$. Since $C_{xy}$ is compact, there exist real numbers $a<b$ such that $f_{xy}\left(C_{xy}\right)\subset [a,b]$. Applying ${(}^{*})$, it follows that the real numbers $a<b$ can be chosen in such a way that $f_{xy}\subset [a,b]$ for all pairs $(x,y)\in \prec$.

For both implications, Proposition 2 now implies that every function $f_{xy}$ can be lifted to a continuous increasing function $F_{xy}:(X,\precsim ,t)\to ([a,b],{\le }_{\mid [a,b]},t_{\mid [a,b]})$. In particular, we may conclude that, for every pair $(x,y)\in \prec$, there exists a real number ${ϵ}_{xy}\in ]a,b[$ such that $(x,y)\in F_{xy}^{-1}\left([a,{ϵ}_{xy}\left[\right)\times F_{xy}^{-1}\left(\right]{ϵ}_{xy},b]\right)$. Hence, ${\displaystyle \prec \subset \bigcup _{(x,y)\in \prec }{F}_{xy}^{-1}\left([a,{ϵ}_{xy}\left[\right)\times F_{xy}^{-1}\left(\right]{ϵ}_{xy},b]\right)}$. Now, we may apply the results on the Lindelöf property of $(X,t)$ that already have been quoted in the proof of Lemma 1 in order to conclude that there exists a countable family ${\left\{(x_n,y_n)\right\}}_{n\in N}$ of pairs $(x_n,y_n)\in \prec$ such that the inclusion ${\displaystyle \prec \subset \bigcup _{(x_n,y_n)\in \prec }{F}_{x_n{y}_n}^{-1}\left([a,{ϵ}_{x_n{y}_n}\left[\right)\times F_{x_n{y}_n}^{-1}\left(\right]{ϵ}_{x_n{y}_n},b]\right)}$ holds. Hence, we set ${\displaystyle F:=\sum _{n\in N}\frac{1}{2^{n+1}}{F}_{x_n{y}_n}}$. The definition of F allows to conclude that F is a continuous order-preserving real-valued function such that $F_{\mid C}=f$. In this way, we have proven both implications (i) ⇒ (ii) and (i) ⇒ (iii).

(ii) ⇒ (i). Consider a preorder ≾ on $(X,t)$ satisfying property (ii). In order to show that ≾ is representable by a continuous multi-utility, consider any pair $(x,y)\in X\times X$ with $not(x\succsim y)$. It suffices to verify that $f_{xy}\left(x\right)<f_{xy}\left(y\right)$ for some continuous increasing real-valued function $f_{xy}$ on $(X\precsim ,t)$. Therefore, we set $C:=\{x,y\}$. Clearly, C is a closed subset of X. Furthermore, the function $g_{xy}:(C,{\precsim }_{\mid C},t_{\mid C})\to (R,\le ,t_{nat})$ defined by $g_{xy}\left(x\right):=0$ and $g_{xy}\left(y\right):=1$ is a bounded, continuous, order-preserving and ≾-C-compatible function on $(C,{\precsim }_{\mid C},t_{\mid C})$. Hence, there exists a continuous order-preserving function $f_{xy}:(X,\precsim ,t)\to (R,\le ,t_{nat})$ such that $f_{xy}\left(x\right)=0$ and $f_{xy}\left(y\right)=1$, which implies that ≾ is representable by a continuous multi-utility.

(iii) ⇒ (i). Consider a preorder ≾ on $(X,t)$ satisfying property (iii). We proceed as in the proof of the previous implication, by considering any pair $(x,y)\in X\times X$ with $not(x\succsim y)$. Then, we set $C:=\{x,y\}$. Clearly, C is a compact subset of X. Furthermore, the function $g_{xy}:(C,{\precsim }_{\mid C},t_{\mid C})\to (R,\le ,t_{nat})$ defined by $g_{xy}\left(x\right):=0$ and $g_{xy}\left(y\right):=1$ is a continuous order-preserving real-valued function on $(C,{\precsim }_{\mid C},t_{\mid C})$. Hence, there exists a continuous order-preserving function $f_{xy}:(X,\precsim ,t)\to (R,\le ,t_{nat})$ such that $f_{xy}\left(x\right)=0$ and $f_{xy}\left(y\right)=1$, which implies that ≾ is representable by a continuous multi-utility. This observation finishes the proof of the implication and, thus, of the theorem. □

It seems that the postulate of $(X,t)$ being $\sigma$-compact cannot be avoided in Theorem 3. Indeed, the following restrictive theorem that is based upon the additional assumption $(X,t)$ to be paracompact holds (a topological space $(X,t)$ is said to be paracompact if it is Hausdorff and every open cover of X has a locally finite open refinement).

Theorem 4.

Let $(X,t)$ be a locally compact paracompact topological space. Then the following assertions are equivalent:

(i)

$(X,t)$ is σ-compact;

(ii)

If ≾ is any closed preorder on $(X,t)$, then the following property is verified:

“If C is any compact subset of X, and $f:(C,{\precsim }_{\mid C},t_{\mid C})\to (R,\le ,t_{nat})$ is any continuous order-preserving function, then $F_{\mid C}=f$ for some continuous order-preserving function $F:(X,\precsim ,t)\to (R,\le ,t_{nat})$".

Proof. 

(i) ⇒ (ii). This implication follows from the implication “(i) ⇒ (ii)" of Theorem 3.

(ii) ⇒ (i). This implication is based upon the well-known result that a locally compact topological space is paracompact if and only if it is the direct sum of locally and $\sigma$-compact topological spaces (cf., for instance, Grotemeyer [23], Satz 97). In order to prove the validity of assertion (i) it, therefore, suffices to show that $(X,t)$ must be the direct sum of countably many locally and $\sigma$-compact topological spaces. Indeed, let us assume, in contrast, that $(X,t)$ is the direct sum of uncountably many locally and $\sigma$-compact topological spaces. Then we may assume these summands to be indexed by the ordinal numbers $\alpha$ that are strictly smaller than some uncountable cardinal number $\kappa$, i.e., we may assume X to be given in the form ${\displaystyle X=\underset{\alpha <\kappa }{⨁}{X}_{\alpha }}$, where each summand $X_{\alpha }$ ($\alpha <\kappa$) is a locally and $\sigma$-compact topological space. Therefore, we consider the binary relation ≾ on X that is defined by setting

$$\begin{array}{ccc}\hfill \precsim :=& & \left\{\right(x,y)\in X\times X\mid \text{there} \text{exist} \text{ordinal} \text{numbers} \alpha \le \beta <\kappa \text{such} \text{that}\hfill \\ & & x\in X_{\alpha } and y\in X_{\beta }\}.\hfill \end{array}$$

Obviously, ≾ is a closed (continuous) total preorder on $(X,t)$. In addition, since ≾ contains uncountable well-ordered sub-chains, there cannot exist any compact subset C of X and any continuous order-preserving function $f:(C,{\precsim }_{\mid C},t_{\mid C})\to (R,\le ,t_{nat})$ for which there exists a continuous order-preserving function $F:(X,\precsim ,t)\to (R,\le ,t_{nat})$ such that $F_{\mid C}=f$. □

Consider that the proof of Theorem 4 demonstrates that the assumption that $(X,t)$ is paracompact does not mean a great loss of generality.

4. Conclusions

A lifting theorem was presented for continuous order-preserving functions on locally and $\sigma$-compact Hausdorff preordered topological spaces. In particular, we showed that, on such spaces, a preorder is closed (or equivalently, representable by a continuous multi-utility) if, and only if, for every compact subspace, every continuous order-preserving function can be lifted to the entire space. A lifting property from closed sets was also introduced in such spaces for a bounded, continuous, order-preserving and ≾-C-compatible function. We showed that the assumption of $\sigma$-compactness cannot be avoided in such a characterization since the aforementioned lifting property is equivalent to $\sigma$-compactness when the topological space is locally compact and paracompact.

These theorems are helpful in order to provide necessary and sufficient conditions on a topology on a preordered set, according to which every closed preorder is representable by a continuous multi-utility. The corresponding more general results will be presented in a future paper.