On isometric embeddings and continuous maps onto the irrationals

Roman Pol,Elżbieta Pol

doi:10.1007/s00605-018-1157-z

Abstract

Let f,{:},E rightarrow F be a continuous map of a complete separable metric space E onto the irrationals. We shall 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 f^{-1}(y). 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. These results, based on a classical Hurewicz’s theorem, refine some results from Pol and Pol (Isr J Math 209:187–197, 2015) and answer a question in Banakh et al. (in: Pearl (ed) Open problems in topology II. Elsevier, Amsterdam, 2007).

Highlights

In [13] we proved that each complete separable metric space containing isometric copies of every countable complete metric space contains isometric copies of every separable metric space
Theorem 1.1 Let f : E → F be a continuous map of a complete separable metric space onto a non-σ -compact metric space
Sin E such that, for any complete separable metric space M containing isometric copies of every subset of S closed in E, some fiber f −1(y) embeds isometrically in M

Summary

Introduction

In [13] we proved that each complete separable metric space containing isometric copies of every countable complete metric space contains isometric copies of every separable metric space.

B Elzbieta Pol

The Effros Borel spaces

A remark on continuous maps onto the irrationals

An application of the Hurewicz theorem

The collections of zero-dimensional sets in Effros Borel spaces