Isometric embeddings and continuous maps onto the irrationals

TL;DR

This paper explores the properties of continuous maps from complete separable metric spaces onto the irrationals, focusing on isometric embeddings and the complexity of zero-dimensional sets within the hyperspace.

Contribution

It establishes conditions under which a metric space contains isometric copies of fibers of such maps and analyzes the complexity of zero-dimensional sets in the hyperspace.

Findings

Complete spaces containing all closed relatively discrete sets also contain isometric fibers of the map.

If all fibers have positive dimension, the collection of zero-dimensional sets is non-analytic.

Provides new insights into the structure of fibers and zero-dimensional sets in metric spaces.

Abstract

Let f be a continuous map of a complete separable metric space E onto the irrationals. We show that if a complete separable metric space M contains isometric copies of every closed relatively discrete set in E, then M contains also an isometric copy of some fiber of f. We shall show also that if all fibers of f have positive dimension, then the collection of closed zero-dimensional sets in E is non-analytic in the Wijsman hyperspace of E.