Properties of Topologies for the Continuous Representability of All Weakly Continuous Preorders

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Properties of Topologies for the Continuous Representability of All Weakly Continuous Preorders

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A peer-reviewed article of this preprint also exists.

Gianni Bosi^*,‡

,Laura Franzoi^‡,

Gabriele Sbaiz^‡

Gianni Bosi^*,‡

,Laura Franzoi^‡,

Gabriele Sbaiz^‡

^‡ These authors have contributed equally to this work.

This version is not peer-reviewed

Submitted:

18 September 2023

Posted:

19 September 2023

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Abstract

We investigate properties of strongly useful topologies, i.e. topologies with respect to which every weakly continuous preorder admits a continuous order-preserving function. In particular, we prove that a topology is strongly useful provided that the topology generated by every family of separable systems is countable. Focusing on normal Hausdorff topologies, whose consideration is fully justified and not restrictive at all, we show that strongly useful topologies are hereditarily separable on closed sets, and we identify a simple condition under which the Lindelöf property holds.

Keywords:

Subject: Computer Science and Mathematics - Geometry and Topology

1. Introduction

Gerhard Herden inaugurated a very general approach to Mathematical Utility Theory and in particular to Real Representation of Preferences (see, e.g., Herden [1,2,3] and Herden and Pallack [4]). In particular, Herden was able to provide characterizations of the existence of continuous order-preserving functions for a not necessarily total preorder on a generic topological space, by using the fundamental notion of a linear decreasing separable system, which generalizes the concept of a decreasing scale introduced by Burgess and Fitzpatrick [5].

The popular theorems by Eilenberg [6] and Debreu [7,8], according to which there exists a continuous order-preserving function (utility function) for every continuous total preorder on a connected and separable, and respectively on a second countable, topological space are particular cases of Herden’s axiomatization. We recall that a total preorder on a topological space is said to be continuous if all the weak lower and upper sections are closed (or, equivalently, if all the strict lower and upper sections are open).

The very important concept of a useful topology (i.e., a topology on a given set such that every continuous total preorder is representable by a continuous utility function) was inaugurated by Herden [3], who based his analysis on linear separable systems on a topological space. Indeed, to each linear separable system it is possible to associate a continuous total preorder in a very natural way. Useful topologies were recently studied and fully characterized in three subsequent papers by Bosi and Zuanon [9,10,11], after the seminal paper by Bosi and Herden [12], who introduced and applied the concept of a complete separable system on a topological space. This is the case of a particular linear separable system. According to Bosi and Zuanon [11, Theorem 2], a completely regular topology is useful if and only if it is separable and every chain of clopen sets is countable. It is very important to notice that the assumption of complete regularity is not restrictive, since a (total) preorder is continuous if and only if it is continuous with respect to the weak topology of continuous functions, which is completely regular (see Bosi and Zuanon [9, Lemma 3.1]). Other deep contributions to the study of useful topologies were presented by Herden and Pallack [13], Campión et al. [14,15,16] and Candeal et al. [17].

A very natural and general extension of the concept of a continuous total preorder to the nontotal case is the notion of a weakly continuous preorder introduced by Herden and Pallack [4]. This is the case of a preorder ≾ on a topological space

(X, t)

such that, for every

(x, y) \in ≺

, there exists a continuous increasing real-valued function

u_{x y}

such that

u_{x y} (x) < u_{x y} (y)

. In other words, points

(x, y) \in ≺

are separated by continuous increasing functions. In the case of a total preorder, the notion of weak continuity is equivalent to that of continuity.

In order to generalize the concept of a useful topology, Bosi and Zuanon [18] introduced the notion of a strongly useful topology on a set. This is the case of a topology such that every weakly continuous and not necessarily total preorder admits a continuous order-preserving function. It is easy to see that a strongly useful topology is useful. The concept of a strongly useful topology was further examined by Bosi [19], who studied the bijective correspondence between weakly continuous preorders on one hand, and (equivalence classes of) families of complete separable systems on the other hand. Therefore, some characterizations of strongly useful topology are already available. Actually, the situation is, needless to say, much more complicated when the preorder may fail to be total. However, there is some analogy with useful topologies, in the sense that, while in the case of useful topologies it is not restrictive to deal with completely regular topologies, in the case of strongly useful topologies it is not restrictive to deal with normal and Hausdorff spaces.

This paper is aimed to take advantage of this latter very important consideration, in order to prove interesting properties of strongly useful topologies which are also normal and Hausdorff. Indeed, we show that strongly usefulness is an hereditary property on open sets when the topology is normal Hausdorff. This is proven by using the fact that strongly usefulness depends, loosely speaking, on the properties of families of complete separable systems, and on the weakly continuous preorders which are naturally associated to them. We further prove that strongly useful normal Hausdorff topologies are hereditarily separable on closed sets, and we identify a simple condition concerning well-ordered families of open sets, which implies the Lindelöf property.

Incidentally, under fairly general conditions, we establish that a topology is strongly useful provided that the topology generated by every family of open sets is second countable, and we show that such condition is not necessary for strongly usefulness. We further furnish a very general characterization of the existence of continuous order-preserving functions, which is strictly related to the consideration of linear separable systems which are actually complete.

2. Notation and Preliminaries

The reader is assumed to be familiar with the definition of a preorder ≾ on a set X, and the related classical notation, according to which the associated strict part and symmetric part are denoted by ≺ and ∼, respectively. An order≲ is an antisymmetric preorder, and a chain is a total order. However, all the detailed definitions can be found for example, in Herden [1].

Definition 1.

A real-valued function u on a preordered set

(X, ≾)

is said to be

1.: increasing, if, for all $x, y \in X$ ,

$x ≾ y \Rightarrow u (x) \leq u (y);$
2.: order-preserving, if u is increasing and, for all $x, y \in X$ ,

$x ≺ y \Rightarrow u (x) < u (y);$
3.: a utility function, if $x ≾ y$ is equivalent to $u (x) \leq u (y)$ , for all $x, y \in X$ .

In the sequel,

t_{n a t}

will stand for the natural topology (i.e., interval topology) on

R

. Without loss of generality, we shall consider continuous increasing functions or continuous order-preserving functions u taking values in

[0, 1]

, i.e. we shall be authorized to use the notation

u : (X, ≾, t) \to ([0, 1], \leq, t_{n a t})

instead of

u : (X, ≾, t) \to (R, \leq, t_{n a t})

Definition 2

(Herden and Pallack [4]). A preorder ≾ on a topological space

(X, t)

is said to be weakly continuous if, for every pair

(x, y) \in ≺

, there exists a continuous and increasing real-valued function

u_{x y} : (X, ≾, t) \to ([0, 1], \leq, t_{n a t})

such that

u_{x y} (x) < u_{x y} (y)

It is immediate to check that weak continuity is a necessary condition for the existence of a continuous order-preserving function. Let us introduce the basic definition of a strongly useful topology.

Definition 3

(Bosi and Zuanon [18]). A topology t on a nonempty set X is said to be strongly useful if every weakly continuous preorder on the topological space

(X, t)

admits a continuous order-preserving function

u : (X, ≾, t) \to ([0, 1], \leq, t_{n a t})

Characterizations of a strongly useful topology are presented in Bosi and Zuanon [18, Theorem 3.4] and Bosi [19, Theorem 4.2].

A subset D of a preordered set

(X ≾)

is said to be decreasing if

(x \in D)

and

(z ≾ x)

imply

z \in D

, for all

z \in X

We now recall the definition of a complete separable system on a topological space

(X, t)

, which is a particularly relevant case of a linear separable system, as introduced by Bosi and Herden [12].

Definition 4

(Bosi and Herden [12]). Let a topology t on X be given. A family

E

of open subsets of the topological space

(X, t)

, such that

⋃_{E \in E} E = X

, is said to be a complete separable system on

(X, t)

if it satisfies the following conditions:

S1:: There exist sets $E_{1} \in E$ and $E_{2} \in E$ such that $\bar{E_{1}} \subset E_{2}$ .
S2:: For all sets $E_{1} \in E$ and $E_{2} \in E$ such that $\bar{E_{1}} \subset E_{2}$ , there exists some set $E_{3} \in E$ such that $\bar{E_{1}} \subset E_{3} \subset \bar{E_{3}} \subset E_{2}$ .
S3:: For all sets $E \in E$ and $E^{'} \in E$ , at least one of the following conditions $E = E^{'}$ , or $\bar{E} \subset E^{'}$ , or $\bar{E^{'}} \subset E$ holds.

In the case when ≾ is a preorder on X, a complete separable system

E

(X, t)

is said to be a complete decreasing separable system on

(X, ≾, t)

as soon as every set

E \in E

is required to be decreasing.

Denote by

S_{C} (X, t)

and

F_{C} (X, t)

the set of all complete separable systems and the set of all families of complete separable systems on

(X, t)

, respectively.

Let us now present the definition of the weakly continuous preorder associated to a family of complete separable systems.

Definition 5.

If we consider any family

E : = {E_{i}}_{i \in I} \in F_{C} (X, t)

, then the weakly continuous preorder

≾_{E}

naturally associated to

E

is defined as follows, for all points

x, y \in X

x ≾_{E} y \Leftrightarrow \forall i \in I, \forall E \in E_{i} : (y \in E \Rightarrow x \in E) .

We now present a characterization of weak continuity of a preorder on a topological space.

Proposition 1.

Let ≾ be a preorder on a topological space

(X, t)

. Then the following conditions are equivalent:

1.: ≾ is weakly continuous;
2.: For every $(x, y) \in ≺$ , there exists a complete decreasing separable system $E_{x y}$ on $(X, t)$ such that there exists $E \in E_{x y}$ with $x \in E$ , $y \notin E$ .

Proof.

1 ⇒ 2. Let ≾ be a weakly continuous preorder on a topological space

(X, t)

. Consider any pair

(x, y) \in ≺

, and a continuous and increasing real-valued function

u_{x y} : (X, ≾, t) \to ([0, 1], \leq, t_{n a t})

such that

u_{x y} (x) < u_{x y} (y)

. Then just define

E_{x y} : = {{u^{- 1} ({[0, q [)}}_{q \in Q \cap [0, 1 [} \cup X}

in order to immediately verify that condition 2 holds.

2 ⇒ 1. Assume that, given any pair

(x, y) \in ≺

, there exists a complete decreasing separable system

E_{x y}

(X, t)

with the indicated property. Then, from Bosi and Zuanon [10, Lemma 3.7]), there exists a complete decreasing separable subsystem

E_{x y}^{'} \subset E_{x y}

such that

x \in E^{'} \subset E

for every

E^{'} \in E_{x y}^{'}

. Therefore, we have that

E_{x y}^{'}

is a complete decreasing separable system on

(X, t)

such that

x \in E^{'}

y \notin E^{'}

for every

E^{'} \in E_{x y}^{'}

. Then the thesis follows from Bosi and Zuanon [20, Remark 2.21, 2], since to every complete decreasing separable system we can associate a continuous increasing function in a very natural way. □

Remark 1.

The weak continuity of the preorder

≾_{E}

presented in Definition 5 is an immediate consequence of the previous proposition.

Based on the above Proposition 1, it is easy to prove the following theorem, which presents a general characterization of the existence of a continuous order-preserving function. This theorem provides a slight generalization of the analogous characterizations presented by Herden [1, Theorem 4.1] and Herden [2, Theorem 3.1]. We omit the proof that is based on the proof of the aforementioned theorems and on the above Proposition 1.

Theorem 1.

Let ≾ be a preorder on a topological space

(X, t)

. Then the following condition are equivalent:

1.: There exists a continuous order-preserving function $u : (X, ≾, t) \to ([0, 1], \leq, t_{n a t})$ ;
2.: There exists a countable complete decreasing separable system $E = {E_{n}}_{n \in N}$ such that, for every $(x, y) \in ≺$ , there exists $n \in N$ with $x \in E_{n}$ , $y \notin E_{n}$ ;
3.: There exists a countable family ${E_{n}}_{n \in N}$ of complete decreasing separable systems on $(X, t)$ such that, for every $(x, y) \in ≺$ , there exist $n \in N$ and $E \in E_{n}$ with $x \in E$ , $y \notin E$ ;
4.: There exists a countable family ${u_{n} : (X, ≾, t) \to ([0, 1], \leq, t_{n a t})}_{n \in N ∖ {0}}$ of continuous increasing functions such that, for every $(x, y) \in ≺$ , there exists $n \in N$ with $u_{n} (x) < u_{n} (y)$ .

Denote by

F_{≾} (X, t)

the family of all continuous increasing functions for a preorder ≾ on a topological space

(X, t)

. Let us now define the family of complete separable systems associated to a (weakly continuous) preorder.

Definition 6.

Let ≾ be a weakly continuous preorder on a topological space

(X, t)

. Then the family

E_{≾} (X, t) \in F_{C} (X)

which is naturally associated to ≾ is defined as follows:

E_{≾} (X, t) : = {{u^{- 1} {({[0, q [)}}_{q \in Q \cap [0, 1 [} \cup X}}_{u \in F_{≾} (X, t)} .

Definition 7.

Let ≾ and

≾^{'}

be two preorders on a set X. Then

≾^{'}

is said to extend ≾ if

≾ \subset ≾^{'}

and

≺ \subset ≺^{'}

Remark 2.

Let ≾ be a weakly continuous preorder on a topological space

(X, t)

. Then the preorder

≾_{E_{≾} (X, t)}

is a weakly continuous preorder which extends ≾.

Definition 8.

Given a 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 continuous. Two points

x, y \in X

are considered equivalent if

f (x) = f (y)

for all continuous real-valued functions f. In this case, we write

x \sim_{C} y

The reader is assumed to be familiar with the fundamental concepts of general topology, like those of basis, subbasis, complete regularity and Hausdorff. However, for reader’s convenience we recall the definition of normal topological space since it is central in the present study.

Definition 9.

A topology t on a nonempty set X is said to be normal, if, for every pair

(A, B)

of disjoint closed subsets of X, there exist two open disjoint sets

U, V

such that

A \subseteq U

B \subseteq V

The famous Urysohn’s Lemma states that a topology t on a set X is normal if and only if for every pair of disjoint nonempty closed sets

A, B \subset X

there exists a continuous function

f : (X, t) \Rightarrow ([0, 1], t_{n a t})

such that

f (x) = 0

for all

x \in A

and

f (x) = 1

for all

x \in B

Bosi and Zuanon [18, Lemma 3.6 and Proposition 3.8] proved the following result.

Proposition 2.

Let

(X, t)

be a topological space. Then the following conditions hold:

1.: The coarsest topology on X satisfying the property that all weakly continuous preorders on $(X, t)$ remain being continuous is $σ (X, C (X, t, R))$ ;
2.: In order that t is strongly useful, it is necessary that $(X_{∣_{\sim_{C}}}, σ {(X, C (X, t, R))}_{∣_{\sim_{C}}})$ is a normal Hausdorff-space.

Notice that, according to the previous proposition, it is not restrictive to limit ourselves to take into account normal Hausdorff spaces when dealing with strongly useful topology. This consideration is of vital importance, and motivates the analysis performed in the next section.

A final proposition concludes this section, which is a direct consequence of normality. Its easy proof is therefore omitted.

Proposition 3.

Let

(X, t)

be a normal Hausdorff topological space. Then, for every open set

O \in t

such that

\bar{O} ⫋ X

, there exists a countable complete separable system

E (O) \in F_{C} (X, t)

such that O is the minimum element of the chain

(E (O), ⫋)

3. Properties of Strongly Useful Topologies

We now present a sufficient condition for the strong usefulness of a topology. This condition holds in the general case, without the aforementioned normality assumption. We recall that the topology

t_{E}

generated by any family

E \in F_{C} (X)

of complete separable systems on a topological space

(X, t)

is the topology on X which has

E

as a subbasis of open sets.

Proposition 4.

A topology t on a set X is strongly useful provided that the topology

t_{E}

generated by every family

E \in F_{C} (X)

is second countable.

Proof .

Let ≾ be a weakly continuous preorder on a topological space

(X, t)

, and consider the family

E_{≾} (X, t)

of complete separable systems associated to ≾ as in Definition 6. Since the topology

t_{E_{≾} (X, t)}

is second countable, we have that there exists a countable family

{u_{n}}_{n \in N ∖ {0}}

of continuous increasing functions such that the topology generated by

{{u_{n}^{- 1} {({[0, q [)}}_{q \in Q \cap [0, 1 [} \cup X}}_{n \in N ∖ {0}}

coincides with

t_{E_{≾} (X, t)}

. Therefore, based on this family

{u_{n}}_{n \in N ∖ {0}}

, we have that, for every

(x, y) \in ≺

, there exists

n \in N

with

u_{n} (x) < u_{n} (y)

. Hence, the function

u : = \sum_{n = 1}^{\infty} 2^{- n} u_{n}

is a continuous order-preserving function for ≾. This consideration completes the proof. □

The previous proposition is analogous to the sufficient part of the characterization of useful topologies presented by Bosi and Herden [12, Theorem 3.1], according to which a topology is useful if and only if the topology generated by every complete separable system is second countable. However, the converse of Proposition 4 does not hold in the case of a normal Hausdorff space, as the following example shows.

Example 1.

Consider any normal Hausdorff topology that is defined on a countable set X, but fails to be second countable. The reader may think, for instance, of Appert’s topology

t_{A}

that is defined on the positive integers

X : = N ∖ {0}

. For a detailed description of

t_{A}

the reader may consult Steen and Seebach [21]. The countability of X, of course, guarantees that

t_{A}

is strongly useful. But since

t_{A}

is not second countable, the countability of X implies the existence of at least one point

x \in X

having a basis

B (x) : = {O_{i} (x)}_{i \in I}

of (open) neighborhoods of x, the cardinality of which is not countable. Let now, for every

i \in I

, some (countable) complete separable system (chain)

E_{i} (O_{i} (x))

be chosen in such a way that

O_{i} (x)

is the minimum element of

(E (O_{i} (x)), ⫋)

. Since

(X, t_{A})

is a normal topological space, the arbitrary construction of these chains does not make any difficulties (see Proposition 3). But now we are already done. Indeed, the topology

t_{E}

that is generated by the family

E : = {E_{i}}_{i \in I}

cannot be second countable.

Another general property of strongly useful topologies is in order now. We shall denote by

C ((X, t), [0, 1])

the set of all continuous function on the topological space

(X, t)

taking values in the real interval

[0, 1]

Proposition 5.

Let

(X, t)

be a strongly useful normal Hausdorff topological space. Then for every open subset O of X the subspace

(O, t_{| O})

(X, t)

is strongly useful.

Proof.

Let some (non-empty) open subset O of X be arbitrarily chosen. First of all, we notice that our assumption

(X, t)

to be a normal Hausdorff-space allows us to choose a complete separable system

C

(O, t_{| O})

in such a way that there exists some countable strictly increasing chain

\emptyset ⫋ C_{0} ⫋ C_{1} ⫋ \dots ⫋ C_{n} ⫋ \dots ⫋ O

of closed subsets of O having the properties that

⋃_{n \in N} C_{n}

coincides with O and

X ∖ C_{n}

belongs to

C

for every

n \in N

. Let now

E_{O}

be a complete separable system on

(O, t_{| O})

, i.e. the open and closed sets with respect to

E_{O}

are open and closed with respect to the relativized topology

t_{| O}

on O. At this point, we consider for every

n \in N

the complete separable system

E_{O}^{n} : = {E \cap C_{n} | E \in E_{O}}

that is induced by

E_{O}

. Since O is open and every set

C_{n}

, where n runs through the set

N

of natural numbers, is closed we may apply our assumption

(X, t)

to being strongly useful in order to conclude that for every

n \in N

there exists a function

u_{n} \in C ((X, t), [0, 1])

that represents the total preorder

≾_{E_{O}^{n}}

(see Bosi and Zuanon [20]). Since

O = ⋃_{n \in N} C_{n}

the function

u_{O} : = \sum_{n \in N} \frac{1}{2^{n}} u_{n} \in C ((X, t), [0, 1])

is a function such that its restriction to O represents the total preorder

≾_{E_{O}}

. Summarizing our considerations, it follows that

E_{O}

is the restriction of some complete separable system that is defined on

(X, t)

, and, needless to say, such consideration naturally extends to the case of any family of complete separable systems on

(X, t)

. This means, however, that the case of an arbitrary open subset O of X to be given reduces to the situation that O coincides with X. Hence, we have completed the proof. □

Let us now present other properties relative to the assumption of normality of the topological space. The appropriateness of such hypothesis has been previously discussed.

The reader may recall that a topology t on a set X is said to be hereditarily separable on closed sets if every subspace

(A, t_{∣ A})

(X, t)

is separable whenever A is a closed subset of X.

Proposition 6.

Let

(X, t)

be a strongly useful normal Hausdorff topological space. Then t is hereditarily separable on closed sets.

Proof.

Consider a strongly useful normal Hausdorff topological space

(X, t)

. Assume, by contraposition, that t is not hereditarily separable on closed sets. Then, there exists some (non-empty) closed subset C of X such that the induced topology

t_{∣ C}

is not separable. Consider, however, that

t_{∣ C}

is a normal Hausdorff topology, since normality is an hereditary property on closed sets. Then, from Bosi and Zuanon [9, Theorem 3.1, statement 3], since

t_{∣ C}

is in particular a completely regular topology, there exists a continuous total preorder

≾_{∣ C}

which does not admit any continuous order-preserving function (i.e., utility function)

u^{C} : (X, ≾_{∣ C}, t_{∣ C}) \to ([0, 1], \leq, t_{n a t})

. Define the preorder ≾ on X as follows, for all

x, y \in X

x ≾ y \Leftrightarrow (x, y \in C) a n d (x ≾_{∣ C} y) .

Notice that, since

(X, t)

is a normal Hausdorff topological space, every increasing function

u^{C} : (X, ≾_{∣ C}, t_{∣ C}) \to ([0, 1], \leq, t_{n a t})

can be extended, by the Tietze-Urysohn Extension Theorem, to a continuous increasing function

u : (X, ≾, t) \to [0, 1], \leq, t_{n a t})

(see e.g. Engelking [22, Theorem 2.1.8]). Therefore ≾ is a weakly continuous preorder on

(X, t)

, and clearly it cannot admit any continuous order-preserving function

u : (X, ≾, t) \to [0, 1], \leq, t_{n a t})

(otherwise, its restriction to C would be a continuous order-preserving function for

≾_{∣ C}

). Hence, the topology t on X is not strongly useful, and the proof is complete. □

Since from Bosi [19, Theorem 4.2], a topology t is strongly useful if and only if for every family

E

of complete separable systems there exists a complete separable system

L

for which

t_{L}

is second countable and

t_{E} \subset t_{L}

, we have that the following proposition holds.

Proposition 7.

Let

(X, t)

be a strongly useful topological space. Then for every family

O = {O_{α}}_{α \in I}

of clopen subsets of X, there exists a complete separable system

E

(X, t)

such that

t_{O} \subset t_{E}

and

t_{E}

is second countable.

In order to proceed, we consider for every subset S of X the set

Σ (S)

of all well-ordered chains

O (S) : = S \subset O_{0} ⫋ O_{1} ⫋ . . . ⫋ O_{α} ⫋ . . . ⫋ X

of open subsets

O_{α}

of X such that

⋃_{α} O_{α} = X

. Then the length

λ (O (S))

O (S)

is the cardinality of

O (S)

, or equivalently the least ordinal which can be order-embeddable in

(O (S), ⫋)

The reader may recall that a topology t on X is said to be a Lindelöf-topology if for every open covering

C

of X there exists a countable covering

C^{'} \subset C

of X.

In order to prove the following theorem, which in some sense illustrates how close are strongly useful normal Hausdorff topological spaces to Lindelöf topological spaces, the following lemma is needed, which provides a sufficient condition for the non-strong usefulness of a topology.

Lemma 1.

Let t be a topology on X. If there exists a complete separable system

E

(X, t)

which contains an uncountable well-ordered sub-chain

E^{'}

, then t is not strongly useful.

Proof.

Consider the weakly continuous total preorder

≾_{E}

associated to the complete separable system

E

(see Definition 5). Since there exists an uncountable well-ordered sub-chain

E^{'}

E

, then

≾_{E}

does not admit a continuous order-preserving function (utility function) by the above Theorem 1. This consideration completes the proof. □

As usual,

ℵ_{1}

stands for the first uncountable ordinal number.

Theorem 2.

In order for a strongly useful normal Hausdorff topological space

(X, t)

to be Lindelöf, it is sufficient that every chain

O (\emptyset) : = \emptyset \subset O_{0} ⫋ O_{1} ⫋ . . . ⫋ O_{α} ⫋ . . . ⫋ X

, such that the length

λ (O (\emptyset))

exceeds

ℵ_{0}

, contains some open set

O_{α}

having the property that

\bar{O_{α}} \subset O_{λ}

for some

λ > α

Proof.

First consider that, if

(X, t)

is not Lindelöf, then there exists an open cover of X which does not admit a countable subcover of X, and therefore a standard transfinite induction argument guarantees the existence of a well-ordered chain

O (\emptyset) : = \emptyset \subset O_{0} ⫋ O_{1} ⫋ . . . ⫋ O_{α} ⫋ . . . ⫋ X

such that

⋃_{α < ℵ_{1}} O_{α} = X

. Therefore, it suffices to exclude this possibility given our hypotheses.

Let us consider any well-ordered chain

O (\emptyset)

which covers X. We have to show that the length

λ (O (\emptyset))

is not greater than

ℵ_{0}

. Therefore, we distinguish between the case

λ (O (\emptyset)) \leq ℵ_{0}

and the case

λ (O (\emptyset)) > ℵ_{0}

. In the first case we are done. In the second case there exists some set

O_{α} \in O (\emptyset)

having the property that

\bar{O_{α}} \subset O_{λ}

for some

λ > α

. Hence, we may consider the disjoint closed sets

\bar{O_{α}}

and

X ∖ O_{λ}

. Since

(X, t)

is a normal Hausdorff-space there exists some function

f_{0} \in C ((X, t), [0, 1])

such that

f_{0_{| \bar{O_{α}}}} = 0

and

f_{0_{| X ∖ O_{λ}}} = 1

. We, thus, may proceed by transfinite induction.

Let

0 < α < λ (O (\emptyset))

be not a limit ordinal and let some function

f_{α - 1} \in C ((X, t), [0, 1])

with corresponding sets

O_{γ} \in O (\emptyset)

and

O_{χ} \in O (\emptyset)

such that

\bar{O_{γ}} \subset O_{χ}

already have been defined. Then we consider the chain

O (O_{χ}) : = O_{χ} ⫋ O_{χ + 1} ⫋ O_{χ + 2} ⫋ . . . ⫋ O_{ξ} ⫋ . . . ⫋ X

and distinguish between the case

λ (O (O_{χ})) \leq ℵ_{0}

and the case

λ (O (O_{χ})) \geq ℵ_{1}

. In the first case nothing remains to be proved. In the second case there exists some set

O_{η} \in O (O_{χ})

having the property that

\bar{O_{η}} \subset O_{ρ}

for some

ρ > η

. Using the above normality argument it follows that in this case there exists some function

f_{α} \in C ((X, t), [0, 1])

satisfying the equations

f_{α_{| \bar{O_{η}}}} = 0

and

f_{α_{| X ∖ O_{ρ}}} = 1

Let

0 < α < λ (O (\emptyset))

be a limit ordinal. Now we may assume that for every ordinal number

β < α

already functions

f_{β} \in C ((X, t), [0, 1])

have been defined with respect to corresponding sets

\bar{O_{μ}} \subset O_{ν}

. Let

τ

be the minimum of all ordinal numbers that are not smaller than any of the ordinal numbers

ν

that already have been considered. Then we consider the chain

O (O_{τ}) : = O_{τ} ⫋ O_{τ + 1} ⫋ O_{τ + 2} ⫋ . . . ⫋ O_{θ} ⫋ . . . ⫋ X

and distinguish, in analogy to the above considerations, between the case

λ (O (O_{τ})) \leq ℵ_{0}

and the case

λ (O (O_{τ})) \geq ℵ_{1}

. In the first case the limit step is done. In the second case there again exist sets

O_{τ} \subset \bar{O_{σ}} \subset O_{δ}

and a corresponding function

f_{α} \in C ((X, t), [0, 1])

such that

f_{α_{| \bar{O_{σ}}}} = 0

and

f_{α_{| X ∖ O_{δ}}} = 1

Having finished the transfinite induction procedure we must distinguish between the case that there exists an ordinal number

θ

such that the transfinite induction process stops at

θ

and the case that there exists no such ordinal number. In the first case it follows that

λ (O (\emptyset)) < ℵ_{1}

and everything is shown. In the second case, however, the complete separable system

E : = {l (x)}_{x \in X} : = {{z \in X | \forall α (f_{α} (z) \leq f_{α} (x)) \land \exists β (f_{β} (z) < f_{β} (x))}}_{x \in X}

(X, t)

that is induced by the functions

f_{α} \in C ((X, t), [0, 1])

contains some well-ordered sub-chain, the length of which is at least

ℵ_{1}

. This conclusion contradicts our assumption

(X, t)

to be strongly useful since the above Lemma 1 applies. So, the proof is complete. □

4. Conclusions

We have presented some new properties of topologies such that every weakly continuous preorder admits a continuous order-preserving function, namely strongly useful topologies. Most part of the results concern the case of normal Hausdorff topologies, whose interest arises in connection with the consideration of weak topologies of continuous functions. Indeed, the concept of weak continuity of a preorder is based on the consideration of continuous (increasing) functions.

Hopefully, we shall be able to arrive at some characterization of strong usefulness simpler than existing ones in a future paper. Indeed, we formulate the conjecture according to which a normal Hausdorff topology is strongly useful if and only if it is hereditarily separable on closed sets (a condition whose necessity is proven in this paper) and in addition every family of clopen sets is (at most) countable. A result of this type would be the counterpart to the best existing characterization of completely regular useful topologies. We recall that a topology is useful if every continuous total preorder admits a continuous order-preserving function (utility function). According to this latter characterization, a completely regular topology is useful if and only if it is separable and every chain of clopen sets is (at most) countable (see the already mentioned Bosi and Zuanon [11, Theorem 2]).

Funding

This research received no external funding.

Acknowledgments

G. Sbaiz is member of the INdAM (Italian Institute for Advanced Mathematics) group.

Conflicts of Interest

The authors declare no conflict of interest.

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Properties of Topologies for the Continuous Representability of All Weakly Continuous Preorders

Abstract

1. Introduction

2. Notation and Preliminaries

3. Properties of Strongly Useful Topologies

4. Conclusions

Funding

Acknowledgments

Conflicts of Interest

References

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