For a finite-sheeted covering, the total space is compact exactly when the base is compact

Given: The objects, hypotheses, and choice principles stated above.

A covering map is a continuous surjection $p:E\to B$ such that every $b\in B$ has an open neighbourhood $U$ for which $p^{-1}(U)$ is a disjoint union of open sets $V_j$, called sheets, and each restriction $p{\mathrm{\mid }}_{V_j}:V_j\to U$ is a homeomorphism (def-continuous-map-top, def-homeomorphism-and-open-maps, def-disjoint-union-topology). Such a $U$ is evenly covered, and $p^{-1}(b)$ is the fibre over $b$. A covering is trivial when it is isomorphic over $B$ to a product projection $B\times F\to B$ with $F$ discrete. (Covering maps, evenly covered neighbourhoods, fibres, sheets, and trivial coverings).

Let $(X,\mathcal{T})$ be a topological space (def-topological-space). An open cover of $(X,\mathcal{T})$ is a family $\mathcal{U}\subseteq \mathcal{T}$ of open sets with $X=\bigcup \mathcal{U}$; a subcover of $\mathcal{U}$ is a subfamily that is itself an open cover; and $(X,\mathcal{T})$ is compact when every open cover of it has a finite subcover. (Open cover, subcover, and compact topological space; a compact subset is a subspace that is compact in its own right).

Let $(X,{\mathcal{T}}_X)$ and $(Y,{\mathcal{T}}_Y)$ be topological spaces (def-topological-space), and let $\mathbb{R}$ carry its usual topology, the metric topology of $d_{\mathbb{R}}(s,t)=\mathrm{\mid }s-t\mathrm{\mid }$ (lem-real-line-is-a-metric-space, def-metric-topology, def-metrizable-space). Then: 1. Continuous images. If $f:X\to Y$ is continuous (def-continuous-map-top) and $(X,{\mathcal{T}}_X)$ is compact (def-compact-space), then $f[X]$ is a compact subset of $Y$. More generally, if $K\subseteq X$ is a compact subset of $X$ then $f[K]$ is a compact subset of $Y$. 2. Extreme values. If $(X,{\mathcal{T}}_X)$ is compact and nonempty and $g:X\to \mathbb{R}$ is continuous, then $g[X]$ has a maximum and a minimum (def-max-min): there are $x_{\max },x_{\min }\in X$ with $g(x_{\min })\le g(x)\le g(x_{\max })\text{for every }x\in X\mathrm{.}$ 3. Compact to Hausdorff. If $(X,{\mathcal{T}}_X)$ is compact, $(Y,{\mathcal{T}}_Y)$ is Hausdorff (def-hausdorff-space) and $f:X\to Y$ is a continuous bijection, then $f$ is a homeomorphism (def-homeomorphism-and-open-maps). Nonemptiness in claim 2 is a hypothesis and not an oversight: for $X=\mathrm{\varnothing }$ the image is empty and has neither a maximum nor a minimum. No choice principle is used: the one selection made below is over a finite index set, where lem-finite-choice is a theorem of ZF. (A continuous image of a compact space is compact; a continuous real-valued map on a nonempty compact space attains a maximum and a minimum; and a continuous bijection from a compact space to a Hausdorff space is a homeomorphism).