Constructing arcs from paths using Zorn's Lemma

TL;DR

This paper provides a straightforward proof that path-connected Hausdorff spaces are arcwise connected using Zorn's Lemma, and improves understanding of certain spaces in algebraic topology.

Contribution

It offers a direct proof leveraging Zorn's Lemma and refines conditions for arcwise connectivity in specific topological spaces.

Findings

Simplified proof of arcwise connectivity from path-connectedness

Explicit use of Zorn's Lemma in topological proofs

Improved conditions for arcwise connectivity in algebraic topology

Abstract

It is a well-known fact that every path-connected Hausdorff space is arcwise connected. Typically, this result is viewed as a consequence of a sequence of fairly technical results from continuum theory. In this note, we exhibit a direct and simple proof of this statement, which makes explicit use of Zorn's Lemma. Additionally, by carefully breaking down the proof, we identify a modest improvement to a class of spaces relevant to algebraic topology.