Proposition 4

Let B be a topological space, $(E, p)$ a B-space, and G a discrete topological group operating on the right on E. The following assertions are equivalent:

(i) The B-space E is a principal covering with group G;

(ii) For every $g\in G$ and every $x\in E$, one has $p(x\cdot g) =p(x)$, the mapping $p$ induces a homeomorphism of $E/G$ onto B, and the mapping $\theta : (x, g)\mapsto (x, x\cdot g)$ is a homeomorphism of $E\times G$ onto $E\times_BE$;

(iii) The group G operates continuously and freely in E, the mapping $p$ is etale, and its fibres are the orbits of G.

(i)$\Rightarrow$(ii) : This follows from I, p. 91 and from proposition 3 of I, p. 94.

(ii)$\Rightarrow$(iii): Under assumption (ii), the action law of G is continuous, and the mapping $p$ is surjective and open (TG, III, p. 10, lemma 2). Let $x\in E$; the mapping $\theta$ induces a bijection of $\{x\} \times G$ onto $\{x\} \times$ $\overset{-1}{p}(p(x))$, hence the group G operates freely and its orbits are the fibres of $p$. If $e$ denotes the identity element of G, the diagonal of $E\times_BE$ is the image of $E\times \{e\}$ under the homeomorphism $\theta$. Since the group G is discrete, the set $\{e\}$ is open in G, therefore the diagonal $\Delta_E$ is open in $E\times_BE$. By proposition 7 of I, p. 31, the mapping $p$ is étale.

(iii)$\Rightarrow$(i): Every fibre of $p$ being an orbit of the operation of G in E, the mapping $p$ is surjective. Since the mapping $p$ is étale, for every point $b$ of B, there exists an open neighbourhood U of $b$ and a continuous section $s$ of $p$ over U. The set $s(U)$ is open in E and $s$ induces a homeomorphism of U onto $s(U)$ (I, p. 30, cor. 3). For every $g\in G$, the set $s(U)\cdot g$ is open in E and the union of the sets $s(U)\cdot g$, for all $g\in G$, is equal to $\overset{-1}{p}(U)$. If $g$ and $g'$ are two distinct elements of G, the sets $s(U)\cdot g$ and $s(U)\cdot g'$ are disjoint, since G operates freely in E. The mapping $f: U\times G\rightarrow$ $\overset{-1}{p}(U)$ defined by $f(u, g) =s(u)\cdot g$ for $(u, g)\in U\times G$ is therefore a homeomorphism of $U\times G$ onto $\overset{-1}{p}(U)$, compatible with the right operations of G. By definition, E is therefore a principal covering of B with group G.