Universal covering spaces, a footnote

TL;DR

This paper provides a concise proof that, for well-behaved spaces, the topologically enriched fundamental groupoid serves as the universal covering space, connecting algebraic and topological structures.

Contribution

It offers a simplified proof establishing the fundamental groupoid as the universal cover for certain spaces, clarifying the relationship between topology and covering spaces.

Findings

The fundamental groupoid with compact open topology is the universal cover for nice spaces.

The proof simplifies understanding of covering space theory.

Connects topological and algebraic perspectives on coverings.

Abstract

We give a short proof that, for nice $X$, the based fundamental groupoid of $X$ with topology induced by the compact open topology on the space of paths, is indeed the universal covering space of $X$.