Normal Spaces via Urysohn's Lemma as a Lifting Property

TL;DR

This paper reinterprets the classical characterization of normal and hereditarily normal spaces using the language of lifting properties in topology, correcting previous inaccuracies in the translation.

Contribution

It provides a precise translation of Urysohn's lemma into lifting properties and clarifies the concept for hereditarily normal spaces within this framework.

Findings

Corrects previous translation errors of Urysohn's lemma

Provides a new perspective on normal spaces via lifting properties

Clarifies the structure of hereditarily normal spaces

Abstract

We present a translation of Urysohn's description of normal spaces (as those where disjoint closed subsets are separated by a continuous function) into the language of lifting properties in $\mathbf{Top}$, correcting a frequently-cited previous erroneous translation. We also present a translation of the definition of hereditarily normal spaces as those in which every open subspace is normal, by directly 'mapping' the translation of the usual description of normal spaces.