Corollary 1

Let G be a topological group operating continuously on the right in a topological space E. The following conditions are equivalent:

(i) The orbit space $E/G$ is Hausdorff and the space E, endowed with the canonical mapping $p: E\rightarrow E/G$, is a principal fibre space with group G;

(ii) The group G operates properly (TG, III, p. 27) and freely in E, and every point of $E/G$ possesses a neighbourhood over which there exists a continuous section of the mapping $p$.

Put $B = E/G$. Under each of the hypotheses (i) and (ii), the group G operates continuously and freely in E, so that the mapping $\theta : (x, g)\mapsto (x, x\cdot g)$ is a continuous bijection of $E\times G$ onto $E\times_BE$. Let us place ourselves under this hypothesis and denote by $\varphi : E\times_BE\rightarrow G$ the mapping pr$_2\circ \theta^{-1}$. Having regard to formula (1), in order that $\theta$ be a homeomorphism, it is necessary and sufficient that the mapping $\varphi$ be continuous. On the other hand, since the equivalence relation defined by G is open (TG, III, p. 10, lemma 2), in order that the space $E/G$ be Hausdorff, it is necessary and sufficient that the graph $E\times_BE$ of this equivalence relation be closed in $E\times E$ (TG, I, p. 55, prop. 8). Finally (TG, III, p. 31, prop. 6), in order that G operate properly in E, it is necessary and sufficient that $E\times_BE$ be closed in $E\times E$ and that the mapping $\varphi$ be continuous. The equivalence of conditions (i) and (ii) then follows from prop. 3.