# Characterization of Useful Topologies in Mathematical Utility Theory by Countable Chain Conditions

^{1}

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## Abstract

**:**

## 1. Introduction

**ET**) and Debreu [7,8] (

**DT**), which guarantee the continuous representability of every continuous total preorder on a connected and separable, and, respectively, on a second countable topological space. Therefore,

**ET**and

**DT**illustrate particular situations in order that a topology is useful.

**DT**, in order to guarantee that second countability (or, equivalently, separability) is equivalent to usefulness when dealing with a metrizable topology. This latter result will be referred to as Estévez–Hervés’ theorem (

**EHT**).

**ET:**- A topology is useful provided that it is connected and separable.
**DT:**- A topology is useful provided that it is second countable.
**EHT:**- For a metrizable topology, usefulness and separability (or equivalently second countability) are equivalent concepts.

- The weak topology of continuous functions is the coarsest topology with the property that all continuous total preorders are still continuous;
- Since the weak topology of continuous functions is completely regular, actually it is not restrictive to limit ourselves to the consideration of completely regular topologies, when dealing with useful topologies;
- A useful completely regular topology is necessarily separable.

## 2. Notation and Preliminaries

**ZFC**(Zermelo–Fraenkel + Axiom of Choice) set theory.

**Definition**

**1.**

- 1.
- reflexive, if $x\precsim x$, for every $x\in X$;
- 2.
- transitive, if$$(x\precsim y)and(y\precsim z)\Rightarrow (x\precsim z)\mathit{for}\mathit{all}x,y,z\in X;$$
- 3.
- antisymmetric, if$$(x\precsim y)and(y\precsim x)\Rightarrow (x=y)\mathit{for}\mathit{all}x,y\in X;$$
- 4.
- total, if $(x\precsim y)$ or $(y\precsim x)$, for all $x,y\in X$;
- 5.
- linear (or complete), if either ($x\precsim y$) or ($y\precsim x$), for all $x\ne y$ ($x,y\in X$);
- 6.
- a preorder, if ≾ is reflexive and transitive;
- 7.
- an order, if ≾ is an antisymmetric preorder;
- 8.
- a chain, if ≾ is a linear order.

**Definition**

**2.**

- (i)
- second countable, if t has a countable basis $\mathcal{B}=\{{B}_{n}:n\in {\mathbb{N}}^{+}\}$;
- (ii)
- separable, if there is a countable subset D of X such that $D\cap O\ne \varnothing $ for every nonempty $O\in t$;
- (iii)
- completely regular, if for every $x\in X$, and every closed set $F\subseteq X$ such that $x\notin F$, there exists a continuous function $f:(X,t)\to ([0,1],{t}_{nat})$ such that $f\left(x\right)=0$ and $f\left(y\right)=1$ for every $y\in F$.

**Definition**

**3.**

**Definition**

**4.**

**Definition**

**5.**

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

**Proposition**

**1.**

- (i)
- ${\mathcal{E}}^{c}:=\mathcal{E}\cup \{\overline{E}:E\in \mathcal{E}\}$ is linearly ordered by set inclusion;
- (ii)
- $E=\bigcup _{\overline{{E}^{\prime}}\subset E,{E}^{\prime}\in \mathcal{E}}{E}^{\prime}=\bigcup _{\overline{{E}^{\prime}}\subset E,{E}^{\prime}\in \mathcal{E}}\overline{{E}^{\prime}}$ for every $E\in \mathcal{E}$;
- (iii)
- $\overline{E}=\bigcap _{\overline{E}\subset {E}^{\prime}\in \mathcal{E}}{E}^{\prime}=\bigcap _{\overline{E}\subset {E}^{\prime}\in \mathcal{E}}\overline{{E}^{\prime}}$ for every $E\in \mathcal{E}$.

**Theorem**

**1.**

- (i)
- There is a continuous utility representation u for ≾;
- (ii)
- There is a countable decreasing complete separable system $\mathcal{E}={\left\{{E}_{n}\right\}}_{n\in \mathbb{N}}$ on $(X,t)$ with the property that, for all $x,y\in X$ with $x\prec y$, there exists $n\in \mathbb{N}$ with $x\in {E}_{n}$, $y\notin {E}_{n}$.

**Definition**

**6.**

- (i)
- ${O}_{1}=X$;
- (ii)
- $\overline{{O}_{{r}_{1}}}\subseteq {O}_{{r}_{2}}$ for every ${r}_{1},{r}_{2}\in \mathbb{S}$ such that ${r}_{1}<{r}_{2}$.

## 3. New Characterization of Useful Topologies

**Definition**

**7.**

**Definition**

**8.**

**Lemma**

**1.**

**Proposition**

**2.**

**Proof.**

**Theorem**

**2.**

- (i)
- t is useful;
- (ii)
- The following statements hold true:
- (a)
- t is separable;
- (b)
- t satisfies SOCCC.

**Proof.**

**Example**

**1.**

**Proposition**

**3.**

**Theorem**

**3.**

**Proof.**

**Theorem**

**4**

**.**If a topology t on a set X is connected and separable, then it is useful.

**Proof.**

**Theorem**

**5**

**.**If a topology t on a set X is second countable, then it is useful.

**Proof.**

**Theorem**

**6**

**.**Separability and usefulness are equivalent concepts on a metrizable topology t on a set X.

**Proof.**

## 4. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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Bosi, G.; Zuanon, M.
Characterization of Useful Topologies in Mathematical Utility Theory by Countable Chain Conditions. *Axioms* **2022**, *11*, 195.
https://doi.org/10.3390/axioms11050195

**AMA Style**

Bosi G, Zuanon M.
Characterization of Useful Topologies in Mathematical Utility Theory by Countable Chain Conditions. *Axioms*. 2022; 11(5):195.
https://doi.org/10.3390/axioms11050195

**Chicago/Turabian Style**

Bosi, Gianni, and Magalì Zuanon.
2022. "Characterization of Useful Topologies in Mathematical Utility Theory by Countable Chain Conditions" *Axioms* 11, no. 5: 195.
https://doi.org/10.3390/axioms11050195