Proposition 1

Let B be a connected and locally connected topological space and let $b$ be a point of B. Let $(E, x)$ be a universal covering of $(B, b)$. Then E is a Galois covering of B and every covering of B can be associated with E.

The space E is locally connected, since B is. Let $E_0$ be the connected component of $x$ in E, so that the space $(E_0, x)$ is a pointed covering of $(B, b)$ (I, p. 80, Cor. 1 of Prop. 6). There then exists a unique morphism of coverings $f: E\rightarrow E_0$ such that $f(x) =x$. If $i$ denotes the canonical injection of $E_0$ into E, the mapping $i\circ f: E\rightarrow E$ is a morphism of coverings which maps $x$ to $x$, just as does the mapping Id$_E$; since $(E, x)$ is a universal covering of $(B, b)$, we therefore have $i\circ f=$ Id$_E$. It follows that $i$ is surjective, hence $E_0= E$. Consequently, E is connected.

Let $y$ be a point of $E_b$ and let us consider the pointed covering $(E, y)$ of $(B, b)$; by hypothesis there exists a unique morphism of coverings $f: E\rightarrow E$ such that $f(x) =y$. The mapping $s: E\rightarrow E\times_BE$ defined by $t\mapsto (t, f(t))$ is a continuous section of the mapping pr$_1: E\times_BE\rightarrow E$. By Corollary 4 of I, p. 81, this proves that the covering $E\times_BE$ of E given by the mapping pr$_1$ is trivializable. The covering E is therefore Galois (Theorem 2 of I, p. 102). It then follows from I, p. 112, corollary to Proposition 10, that every covering of B can be associated with E.