Proposition 6

Let B be a topological space and G a group. Let $(E, p)$ be a principal covering of B with group G. Suppose the space B connected and locally connected. Let $E_0$ be a connected component of E and let $G_0$ be the stabilizer subgroup of G of $E_0$ (A, I, p. 51). The B-space $(E_0, p|E_0)$ is a principal covering for the right operation of $G_0$ in $E_0$. This covering is Galois.

Since the space B is locally connected, the same is true of E, so that $E_0$ is an open subspace of E (TG, I, p. 85, prop. 11). Since it is also closed (TG, I, p. 83), the space $E_0$ is therefore a covering of B (I, p. 80, cor. 1). Since B is connected, this covering has a degree; since $E_0$ is not empty, all the fibres of $p|E_0$ are therefore non-empty. Let $x$ be a point of $E_0$, let $x'\in E_0$ be such that $p(x') =p(x)$; there exists an element $g\in G$ such that $x\cdot g=x'$. Since the connected components $E_0$ and $E_0\cdot g$ then have a common point, they are equal, whence $g\in G_0$. Thus, the group $G_0$ operates transitively in the fibre of the B-space $E_0$ at $p(x)$. Proposition 6 follows.