Corollary 5

Let B be a connected topological space and let $(E, p)$ be a finite covering of B. The space E has only a finite number of connected components, and each of them is a covering of B.

The result is true if B is empty; henceforth we assume it nonempty.

If X is an open and closed subset of E, the restriction of $p$ to X is a separated mapping (I, p. 25, prop. 2, a) and I, p. 26, remark 1), etale and proper (TG, I, p. 74, corollary 1); by theorem 1, $(X, p)$ is a locally finite covering of B. Since B is connected, this covering has finite degree. If X is distinct from E, there exists a point $b\in B$ such that $X_b(E_b$, whence $[X : B]<[E : B]$ because B is connected. It follows that every decreasing sequence of open and closed sets in E is stationary.

Let $x\in E$; then there exists a smallest open and closed subset X of E containing $x$ (E, III, p. 51, prop. 6). Such a set X is connected; it is therefore the connected component of $x$ in E. This shows that the connected components of E are open and closed and that each of them is a covering of B. Since B is connected, each connected component of E meets each fibre of $p$; the connected components of E are therefore finite in number.