Theorem 2.1 Any connected, locally path connected, and locally simply connected topological space admits a universal cover.

Proof. Indeed, let $ A $ be such a space. Choose a basepoint $ a \in A $, and set $ E $ to be the homotopy classes of paths starting at $ a $. For each path connected open subset $ U \subseteq A $ and $ [\gamma] \in E $ with \([\gamma](1) \in U\), denote $ U_{[\gamma]} $ to be the collection of curves $ [\omega \circ \gamma] $ with $ \omega \colon [0, 1] \to U $. Then the sets $ U_{[\gamma]} $ form a basis for a topology on $ E $, and the endpoint map \(\pi([\gamma]) = [\gamma](1)\) is a universal cover. See May, chapter 3, §8 for the remainder of the proof. Q.E.D.