On the relation between continuous functions in two 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 $\phi :{\mathbb{R}}^m\to {\mathbb{R}}^n\setminus \sim$ 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 $\phi$. This theorem is wrong in the complex case. A counterexample is given in [1].

4 pages; the result had not achieved full generality yet; small mistake on page 4; some corrections regarding to the english language as kindly suggested by a friend of mine