Existence of continuous euclidean embeddings for a weak class of orders

TL;DR

This paper proves that certain weak order relations on topological spaces with utility representations can be continuously embedded into Euclidean spaces, resolving a conjecture for compact metric spaces and exploring limitations under Pareto order.

Contribution

It establishes conditions under which weak orders on topological spaces can be embedded into Euclidean spaces, including a resolution of Nishimura & Ok's conjecture for compact metric spaces.

Findings

Order relations with milder completeness can be embedded in 1^I.

The conjecture is confirmed for compact metric spaces.

Embedding fails under Pareto order with non-standard partial orders.

Abstract

We prove that if $X$ is a topological space that admits Debreu's classical utility theorem (eg.\ $X$ is separable and connected, second countable, etc.), then order relations on $X$ satisfying milder completeness conditions can be continuously embedded in $\mathbb R^I$ for $I$ some index set. In the particular case where $X$ is a compact metric space, this closes a conjecture of Nishimura \& Ok (2015). We also show that when $\mathbb R^I$ is given a non-standard partial order coinciding with Pareto improvement, the analogous embedding theorem fails to hold in the continuous case.