Corollary 2

Let B be a separated topological space and let E be a B-space. Suppose that E is compact and that its projection $p: E\rightarrow B$ is étale. Then, E is a finite covering of B.

The mapping $p$ is separated (I, p. 26, remark 2) and proper (TG, I, p. 76, cor. 2). By corollary 1, E is therefore a locally finite covering of B. Since E is compact, $p(E)$ is a quasi-compact subset of B; the mapping of $p(E)$ into $\mathbf{N}$ given by $b\mapsto$ Card $\overset{-1}{p}(b)$ being locally constant, it is bounded above (TG, IV, p. 30, corollary).