A tale of three hedgehogs

TL;DR

This paper explores three topologies on the hedgehog space, proves the Kowalsky hedgehog theorem relating metrizable spaces to the metric hedgehog, and characterizes collectionwise normality via extension theorems.

Contribution

It provides new proofs of the Kowalsky hedgehog theorem and characterizes collectionwise normality using Tietze extension theorems.

Findings

Every metrizable space embeds into a countable power of the metric hedgehog.

Collectionwise normality is hereditary with respect to $F_{\sigma}$ sets.

Extension theorems characterize collectionwise normality.

Abstract

In this work we study three topologies defined over the same set: the hedgehog. As the name suggests, the hedgehog can be described as a set of spines identified at a single point. Among others, we give a proof of the Kowalsky hedgehog theorem, which asserts that every metrizable space is embeddable into a countable cartesian power of the metric hedgehog. Besides, the concept of collectionwise normality will characterized by means of a Tietze type extension theorem. In particular we give a proof of the fact that collectionwise normality is hereditary with respect to $F_{\sigma}$ sets.