On the relation between continuous functions in two different metric spaces

TL;DR

This paper explores the relationship between continuous functions mapping between Euclidean spaces and a specific metric space of unordered tuples, establishing a key theorem in the real case and providing a counterexample in the complex case.

Contribution

It proves a new theorem linking continuous functions into a space of unordered tuples with continuous functions into Euclidean space, and shows this does not hold in the complex case.

Findings

Existence of a continuous lifting of functions into unordered tuple spaces in real case

Counterexample demonstrating the theorem fails in the complex case

Clarification of the relationship between functions in different metric spaces

Abstract

Let the metric space $\mathbb R^n \setminus \sim$ be the metric space of $n$-sized unordered tuples of real numbers. In the following, it will be shown that if a function $\varphi: \mathbb R^m \to \mathbb R^n \setminus \sim$ is continuous, then there is a continuous function $f: \mathbb R^m \to \mathbb R^n$ such that a natural embedding of $f$ into $\mathbb R^n \setminus \sim$ is equal to $\varphi$. This theorem is wrong in the complex case. A counterexample is given in [1].