Topologies for the Continuous Representability of All Continuous Total Preorders

In this paper, we present a new simple axiomatization of useful topologies, i.e., topologies on an arbitrary set, with respect to which every continuous total preorder admits a continuous utility representation. In particular, we show that, for completely regular spaces, a topology is useful, if and only if it is separable, and every isolated chain of open and closed sets is countable. As a specific application to optimization theory, we characterize the continuous representability of all continuous total preorders, which admit both a maximal and a minimal element.


Introduction
From Herden [1], a topology on any fixed chosen set is said to be useful if every continuous total preorder has a continuous utility representation, i.e., it can be represented by a continuous real-valued order-preserving function (utility function) (see, e.g., Herden [2,3], and Herden and Pallack [4]). Continuity of the preorder means that the given topology is finer than the order topology induced by the preorder.
Other authors call continuously representable the topologies satisfying the aforementioned property (see, e.g., Candeal et al. [5], and Campión et al. [6][7][8][9]). In this paper, we prefer the original terminology of a useful topology, inherited from the seminal paper Herden [1], who first explicitly inaugurated and started a systematic study of this concept.
In some sense, the problem of characterizing all useful topologies on a set can be thought as the most fundamental problem in utility theory. Indeed, the classical theorems by Eilenberg [10] (ET) and Debreu [11,12] (DT), according to which every continuous total preorder on a connected and separable, and, respectively, on a second countable topological space, actually present sufficient conditions in order that a topology is useful.
Another very important result, in the spirit of the previous considerations, was proven by Estévez and Hervés [13], who showed that in every non-separable metric space there is a continuous total preorder, which does not admit a (continuous) utility function. This result, in combination with DT, can be used to state that a metrizable topology is useful if and only if it is second countable, or equivalently separable. By the way, we recall that the well-known Debreu Open Gap Lemma (see Debreu [11,12]) guarantees the equivalence, for a continuous total preorder, of admitting a utility function on one hand, and of admitting a continuous utility function on the other hand. We will refer to this latter result as Estévez-Hervés' theorem (EHT; see also Candeal et al. [5,Theorem 1]).
With help of the concept of a useful topology, the fundamental theorems above can be restated as follows (see the introduction in Bosi and Herden [14]): ET: Every connected and separable topology is useful. DT: Every second countable topology is useful. EHT: A metrizable topology is useful if and only if it is separable, or equivalently second countable On the other hand, it is well known that second countability or separability, in general, is not necessary for t to being useful. For instance, the Niemitzki plane, that is extensively discussed in Steen and Seebach [15], is useful, since it is second countable, even if it is not separable.
Different characterizations of useful (or representable) topologies appear in the literature. In particular, Campión et al. [8,Theorem 5.1] proved that a topology is useful if and only if all its preorderable subtopologies are second countable, where a topology is preorderable if it is the order topology of some continuous total preorder. Bosi and Herden [14,Theorem 3.1] showed that a topology is useful if and only if the topology generated by every complete separable system is second countable. In particular, this latter result can be regarded as the simplest axiomatization of useful topologies, even if it, nevertheless, invokes the concept of a complete separable system (see Bosi and Zuanon [16,Definition 2.20]).
To the best of our knowledge, there is not in the literature any axiomatization of a useful topology, which directly refers to appropriate conditions relative of the considered topology, without referring to properties of some subtopologies. This paper is aimed at filling this gap. Indeed, we show that, for completely regular spaces, a topology is useful, if and only if it is separable, and every isolated chain of open and closed sets is countable. The concept of an isolated chain (i.e., a chain containing only isolated sets) of open and closed subsets is already found in Herden [4], and it is denoted by OCCC in the present paper.
The paper is structured as follows. Section 2 presents the main definitions and the preliminary results. Section 3 contains the main new results concerning the formulation of, so to say, purely topological characterizations of useful topologies. In particular, we prove a fundamental result, generalizing the aforementioned Estévez and Hervés' [13] theorem, according to which a completely regular useful topology is separable (see Theorem 3.1, point 3). We show that the Eilenberg and, respectively, the Debreu utility representation theorem are immediate corollaries of our result. Finally, Sect. 4 is devoted to applications to optimization theory. In particular, Theorem 4.2 below shows that a topology is useful, and every continuous total preorder has both a minimal and a maximal element, if and only if the weak topology of continuous functions is compact, separable, and satisfies OCCC.

Notation and Preliminary Results
A preorder on a nonempty set X is a reflexive and transitive binary relation on X . A preorder on X is said to be total if, for all x, y ∈ X , either x y or y x. The strict part (or asymmetric part) of a preorder on X is defined as follows, for all x, y ∈ X : x ≺ y if and only if (x y) and not(y x). Further, the symmetric part ∼ of a preorder on X is defined as follows, for all x, y ∈ X : x ∼ y if and only if (x y) and (y x). We have that ∼ is an equivalence on X , and we denote by X | ∼ the quotient set, made up by the equivalence classes [x] = {z ∈ X : z ∼ x} (x ∈ X ).
An order on X is a preorder which in addition is antisymmetric (i.e., for all x, y ∈ X , (x y) and (y x) implies that x = y). An order on a set X is said to be linear (or complete) if, for all x = y (x, y ∈ X ), either x y or y x. If is a preorder on X , then the quotient order |∼ on the quotient set X |∼ is defined as follows, for all x, y ∈ X : If is a total preorder on X , then := |∼ is a linear order on X |∼ . A subset D of a preordered set (X ) is said to be decreasing if (x ∈ D) and (z x) imply z ∈ D, for all z ∈ X .
If t is a topology on X , then a family R ⊂ t is said to be a subbasis of t, if the family R consisting of all possible intersections of finitely many elements of R is a basis of t (i.e., every set O ∈ t is the union of some sets of R).
A topology t on X is said to be second countable if there is a countable basis R = {B n : n ∈ N + } for t. Further, a topology t on X is said to be separable, if there exists a countable subset D of X such that D ∩ O = ∅ for every O ∈ t.
We recall that, for two topologies t , t on set X , t is said to be coarser (respectively, finer) than t if it happens that t ⊂ t (t ⊂ t ). If t is coarser than t, we also say that t is a subtopology of t.
Further, a topological space (X , t) is said to be (i) T 1 , if all singleton sets {x} are closed; (ii) Hausdorff, if, given any two points x, y ∈ X with x = y, there exist two open disjoint sets U , V such that x ∈ U and y ∈ V ; (iii) completely regular, if it is T 1 and, for every x ∈ X , and every closed set F ⊆ X not containing x, there exists a continuous function f : Let (X , ) be an arbitrarily chosen preordered set. We define, for every point x ∈ X , the following subsets of X : All the previous definitions can be found, for example, in Herden [2]. A total preorder on the topological space (X , t) is said to be continuous if the sets l (x) = {z ∈ X : z ≺ x} and r (x) = {z ∈ X : x ≺ z} are open subsets of X for every x ∈ X . Equivalently, this is the case when t is finer than the order topology t on X associated to , which is precisely the topology generated by the family In other words, t is the coarsest topology on X such that the sets l (x) and r (x) are open for every x ∈ X .
If t is a topology on X , and X is any nonempty subset of X , then the relativized topology t |X on X is defined as follows: A real-valued function u on a preordered set (X , ) is said to be 2. order-preserving, if u is increasing and, for all x, y ∈ X , A topology t on X is said to be useful (see Herden [1]), if every continuous total preorder on the topological space (X , t) has a continuous utility representation (orderpreserving function) u, i.e., there exists a continuous real-valued function u on the We shall denote by t nat the natural (interval) topology on the real line R.
The following definition is found in Herden and Pallack [17].
Definition 2.1 A preorder on a topological space (X , t) is said to be weakly continuous if, for every pair (x, y) ∈≺, there exists a continuous and increasing real-valued function u xy on X , such that u xy (x) < u xy (y).
We now recall the definition of a complete separable system on a topological space (X , t).
The following definition is found in Bosi and Herden [14].
In the case when is a preorder on X , a complete separable system E on (X , t) is said to be a complete decreasing separable system on (X , , t) as soon as every set E ∈ E is required to be decreasing.
The following theorem holds, presenting different conditions all equivalent to the continuity of a total preorder on a topological space (see Bosi is weakly continuous; (vi) For every pair (x, y) ∈≺, a complete decreasing separable system E xy on X can be chosen, in such a way that there exist sets E ⊂ E ⊂ E in E xy such that x ∈ E and y / ∈ E .
The following theorem holds true, which provides a simple characterization of useful topologies (see Bosi and Herden [14, Theorem 3.1]).

Theorem 2.2
The following assertions are equivalent on a topology t on set X : (ii) For every complete separable system E on (X , t), the topology t E generated by E is second countable.

Proposition 2.1
Let be a total preorder on a set X . Then, the following conditions are equivalent: (i) There exists a utility function u on (X , ); (ii) The following conditions are verified: (a) The order topology t on X is separable; (b) There are only countable many jumps in (X |∼ , |∼ ).
The easy proof of the following Proposition is left to the reader.

Proposition 2.2
Let (X , ) be a totally preordered set. Then, the following conditions are equivalent for all x, y ∈ X:

Continuous Representability of All Continuous Total Preorders
We are going to furnish a very simple and direct characterization of useful topologies. To this aim, starting from a topological space (X , t), denote by OC(t) the set of all open and closed non-empty subsets of X , and let, in addition, C be set of all chains in OC(t) (i.e., an element O of C is a subset of OC(t), which is totally ordered by set inclusion). Two more definitions are needed. We now present the main definition of open and closed countable chain condition ( OCCC), which is originally due to Herden [4]. The simple proof of the following proposition, which is based on Proposition 2.2, is left to the reader.

Proposition 3.1 Let a continuous total preorder on a topological space (X , t), and assume that there exists some jump ]x, y[ in (X , ). Then, the family O := {l (y) : y is the right endpoint of some jump ]x, y[in (X , )} is an isolated family of open and closed subsets of X .
We are now ready to state and prove a very general result, characterizing useful topologies, at least in the completely regular case. Theorem 3.1 Let t be a topology on a nonempty set X . Then, the following statements hold true:

If t is separable and satisfies OCCC, then t is useful; 2. If t is useful, then t satisfies OCCC; 3. If t is a completely regular useful topology, then t is separable.
Proof 1. In order to show that t is useful, if t is separable, and satisfies OCCC, consider any continuous total preorder in on (X , t). Since t is separable, we have that the order topology t (⊂ t) is also separable. Further, by Proposition 3.1, there are only countably many jumps in (X |∼ , |∼ ), and the continuous total preorder on (X , t) has a continuous utility representation by Proposition 2.1. Indeed, the aforementioned Debreu Open Gap Lemma (see Debreu [11,12]) guarantees that a continuous total preorder, which admits a utility function, also admits a continuous utility function. This consideration completes this part of the proof. 2. In order to show that a useful topology t satisfies OCCC, assume, by contraposition, that there exists an uncountable isolated chain O of open and closed sets. Consider the continuous total preorder O on (X , t), defined to be, for all x, y ∈ X , Since there are uncountably many jumps in (X |∼ O , |∼ O ), the continuous total preorder O does not admit a (continuous) utility representation by Proposition 2.1. Hence, t is not useful. 3. The proof of the fact that a completely regular and useful topology is separable is based upon some straightforward transfinite induction argument on all countable ordinal numbers 0 ≤ α < , where is the first uncountable ordinal number. In particular, at the generic step α − 1, we are going to guarantee the existence of a countable set D α−1 ⊂ X , thought of as endowed with the relativized topology t |D α−1 , and of a continuous total preorder α−1 on D α−1 , in such a way that either D α−1 = X , or D α−1 D α , and (D |∼ α , α |∼ α ) has more jumps than (D |∼ α−1 , α−1 |∼ α−1 ).
0 < α < is not a limit ordinal: now the induction hypothesis allows us to assume that α−1 is a continuous total preorder on the set D α−1 , endowed with the relativized topology t |D α−1 . Then, we distinguish between the following two cases.
Case 1: D α−1 = X . In this case we finish the induction procedure. Case 2: D α−1 X . In this situation we arbitrarily choose some point x α ∈ X \D α−1 in order to set D α := D α−1 ∪ {x α }. Since t is assumed to be completely regular, there exists some function f α ∈ C(X , [0, 1]) such that f α (D α−1 ) = {0} and f α (x α ) = 1. Then, the preorder is a continuous total preorder on D α , since f α is a continuous increasing function with respect to α , and α−1 is a continuous total preorder [see Theorem 2.1,(v)]. Now we may continue the transfinite induction procedure. 0 < α < is a limit ordinal: In this situation we set D α = β<α D β . Then, we go the same way as in the case α < to not being a limit ordinal. This means that we finish the induction procedure if D α = X . Otherwise, we consider the totally preordered set α is a continuous total preorder on D α , which is obtained by "extending" to D α the continuous total preorder β on D β by using complete regularity of t, as follows. Define, for every β < α, the total preorder (β) α on D α by setting: In order to show that (β) α is a actually a continuous total preordered on D α for every β < α, consider that, for every z ∈ D β , and for every x We notice that D α is a countable subset of X for every ordinal number 0 ≤ α < . Indeed, either we set D α := D α−1 ∪ {x α }, or D α = β<α D β , and clearly there are countably many ordinal numbers α ∈ [0, α] for every 0 ≤ α < .
It still remains to verify that the transfinite induction procedure described above stops at some countable ordinal number α < . Let us, therefore, assume, in contrast, that the procedure does not stop. Then, we consider the preorder * on X that is defined by setting * : The transfinite induction process implies that * is a continuous total preorder on X that cannot be representable by a continuous utility function. Indeed, there are uncountable many jumps in (X |∼ * , * |∼ * ). This contradiction completes the proof of the last statement of the theorem.

Remark 3.1
Since every metrizable topology is completely regular, condition 3 of Theorem 3.1 generalizes Estévez and Hervés' theorem [13], according to which a non-separable metric topology is not useful.

Remark 3.2
The fact that a useful topology t on X satisfies OCCC, without requiring complete regularity of t, can be seen in another way by using Theorem 2.2. Indeed, it is immediate to check that the topology generated by an uncountable isolated chain O of open and closed subsets of X is not second countable.
We recall that, when we consider the space C(X , t, R) of all continuous real-valued function on some topological space (X , t), the weak topology on X , σ (X , C(X , t, R)), is the coarsest topology on X satisfying the property that every continuous real-valued function on (X , t) remains being continuous. Two points x, y ∈ X are considered as being equivalent if f (x) = f (y) for all functions f ∈ C(X , t, R). For two equivalent points x, y ∈ X , we write x ∼ C(X ,t,R) y (x ∼ C y for the sake of brevity).
It is well known that (X , σ (X , C(X , t, R))) is a completely regular space (cf., for instance, Cigler The following lemma holds true.

Lemma 3.1
The coarsest topology on X satisfying the property that all continuous total preorders on (X , t) remain being continuous is σ (X , C(X , t, R)). (Of course, this assertion is equivalent to the statement that a total preorder on (X , t) is continuous if and only if it is continuous with respect to σ (X , C(X , t, R))).
Proof Although the validity of this assertion appears somewhat surprisingly, its trueness is trivial. Indeed, since continuity of a preorder on (X , t) is described by continuous (increasing) real-valued functions, its validity is immediate (see the equivalence "(i) ⇔ (v)" in Theorem 2.1).
The previous considerations guarantee that complete regularity is a necessary condition for a topology t on X to be useful, when we further assume that t is the coarsest topology on X such that every continuous preorder remains continuous. Corollary 3.1 Let t be a topology on a nonempty set X . The the following conditions are equivalent: C(X , t, R)) is separable and satisfies OCCC.

Theorem 3.2 (Eilenberg theorem) Every connected and separable topology is useful.
Proof Just consider that t is separable, and it obviously satisfies OCCC.

Lemma 3.2 Let t be a second countable topology on a set X , Then, every linearly ordered subtopology t l of t is also second countable.
Proof Let, therefore, t be a second countable topology on X , and consider a subtopology t l of t which is linearly ordered by set inclusion. Then, we choose a countable base R of t and consider the countable subset of t l , in order to easily verify that R l is a base of t l . Indeed, consider any element x ∈ X such that x ∈ O ∈ t l for some set O ∈ t l . Then, there exists B ∈ R such that

Theorem 3.3 (Debreu theorem) Every second countable topology is useful.
Proof Clearly, every second countable topology is separable. Further, from Lemma 3.2, every second countable topology must satisfy OCCC, since a chain of open and closed sets which contains uncountably many isolated sets generates a linearly ordered subtopology, which is not second countable. Therefore, the thesis follows from Theorem 3.1, point 1.
The following corollary concerns the case of connected topologies. The immediate proof is omitted.

Application to Optimization of All Continuous Total Preorders
We apply the results of the previous section to optimization of all continuous total preorders on a topological space.
A total preorder on a topological space (X , t) is said to be upper semicontinuous (respectively, lower semicontinuous) if l (x) := {z ∈ X : z ≺ x} (respectively r (x) := {z ∈ X : x ≺ z}) is an open subset of X for every x ∈ X (see, e.g., Bosi and Zuanon [16]).
We recall that, if (X , ) is any preordered set, an element x 0 ∈ X is said to be maximal (respectively, minimal), if for no x ∈ X it occurs that x 0 ≺ x (respectively, for no x ∈ X it occurs that x ≺ x 0 ). Clearly, an element x 0 ∈ X is maximal (minimal) for a totally preordered set (X , ) if and only if x x 0 for every x ∈ X (respectively, x 0 x for every x ∈ X ).
Compact spaces are obviously of particular interest when searching maximal/minimal elements for total preorders representable by continuous utilities. We recall that a topology t on a set X is said to be compact if every open cover of X admits a finite subcover (see e.g., Gutiérrez [20]).
The following proposition may be viewed as a corollary of Theorem 3.1, given the fact that every compact Hausdorff space is normal, and therefore completely regular (see, e.g., Aliprantis and Border [19, Theorem 2.48]).

Proposition 4.1 The following conditions are equivalent on a compact Hausdorff topology t on a nonempty set X .
(i) t is useful; (ii) t is separable and satisfies OCCC.
Gutiérrez [20, Theorem 2.1] proved the following interesting result concerning the existence of maximal elements for all continuous total preorders on a topological space.

Theorem 4.1 A topology t on a nonempty set X is compact, provided that every upper semicontinuous total preorder on (X , t) admits a maximal element.
We are now able to characterize all the topologies such that every continuous total preorder is continuously representable, and, in addition, admits both a maximal and minimal element. (i) Every continuous total preorder on (X , t) is representable by a continuous utility function, and has both a maximal and a minimal element; (ii) Every continuous total preorder on (X , σ (X , C(X , t, R)) is representable by a continuous utility function, and has both a maximal and a minimal element; (iii) σ (X , C(X , t, R) is compact, separable, and satisfies OCCC.
(iii) ⇒ (i). This implication is an immediate consequence of Corollary 3.1, and of the consideration according to which the maximization/minimization of a continuous utility function for a total preorder on a compact space (see the famous Weierstrass Theorem) leads to maximal/minimal elements for the total preorder.