A local homeomorphism from a nonempty compact Hausdorff space to a connected Hausdorff space is a finite-sheeted covering

List the finite fibre as $f^{-1}(y)=\{x_1,\dots ,x_m\}$ with $m\ge 1$ by [L1]. Using [F1] finitely many times and [F2] for the finite selections, choose pairwise disjoint open neighbourhoods $N_i$ of the $x_i$. Intersect each $N_i$ with a local-homeomorphism chart at $x_i$; its image is still an open neighbourhood of $y$. Let $W$ be the finite intersection of these images, and replace each chart by its inverse image of $W$. We obtain pairwise disjoint open sets $U_i$ with $f{\mathrm{\mid }}_{U_i}:U_i\to W$ a homeomorphism.

The set $K=X\setminus {\bigcup }_i{U}_i$ is closed and therefore compact by [F3]. Its image $f[K]$ is compact by [F4] and closed in $Y$ by [F5]. No point of the fibre over $y$ lies in $K$, so $y\mathrm{\notin }f[K]$. Hence $V:=W\setminus f[K]$ is an open neighbourhood of $y$.

Put $V_i=U_i\cap f^{-1}(V)$. Each $V_i$ is open and $f{\mathrm{\mid }}_{V_i}:V_i\to V$ is a homeomorphism. If $x\in f^{-1}(V)$ then $x\mathrm{\notin }K$, so $x$ lies in exactly one $U_i$ and hence in exactly one $V_i$. Thus $f^{-1}(V)$ is the disjoint union of the finitely many $V_i$. Since $y$ was arbitrary and $f$ is surjective by [L1], [F6] makes $f$ a finite-sheeted covering.