Every locally compact Hausdorff space is Čech-complete

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

A Tychonoff space $X$ is Čech-complete when there is a Hausdorff compactification $(K,i)$ of $X$ (def-compactification-of-a-tychonoff-space) for which $i[X]$ is a $G_{\delta }$ subset of $K$ (def-g-delta-and-f-sigma-in-a-topological-space). The definition asks for one compactification; thm-cech-completeness-is-independent-of-compactification proves the equivalent every-compactification form. (Čech-complete spaces as $G_{\delta }$ subspaces of Hausdorff compactifications).

Let $(X,\mathcal{T})$ be a topological space (def-topological-space) and let $(X^{\ast },{\mathcal{T}}^{\ast })$ be its one-point compactification, with added point $\mathrm{\infty }$ (def-one-point-compactification). Then: 1. $X^{\ast }$ is compact (def-compact-space). 2. $X$ is an open subspace of $X^{\ast }$: $X\in {\mathcal{T}}^{\ast }$, and the subspace topology that $X$ inherits from $X^{\ast }$ (def-subspace-topology-top) is $\mathcal{T}$ itself. 3. $X$ is dense in $X^{\ast }$ (def-dense-top) if and only if $X$ is not compact. 4. $X^{\ast }$ is Hausdorff (def-hausdorff-space) if and only if $X$ is locally compact (def-locally-compact-space) and Hausdorff. In particular, a locally compact Hausdorff space is an open subspace of a compact Hausdorff space, which is the reason the construction is made. No choice principle is used: the only cover thinned below is thinned by the indexed form of lem-compactness-of-a-subspace-is-ambient, which returns its own indices. ($X^{\ast }$ is compact and contains $X$ as an open subspace; $X$ is dense in $X^{\ast }$ exactly when $X$ is not compact; and $X^{\ast }$ is Hausdorff exactly when $X$ is locally compact and Hausdorff).

Let $(X,\mathcal{T})$ be a topological space (def-topological-space) and let $A\subseteq X$. $A$ is a $G_{\delta }$ set of $X$ when there is a sequence $(V_n{)}_{n\in \mathbb{N}}$ of open subsets of $X$ with $A={\bigcap }_{n\in \mathbb{N}}{V}_n$, and an $F_{\sigma }$ set of $X$ when there is a sequence $(F_n{)}_{n\in \mathbb{N}}$ of closed subsets of $X$ with $A={\bigcup }_{n\in \mathbb{N}}{F}_n$. ($G_{\delta }$ and $F_{\sigma }$ subsets of a topological space, agreeing with the real-line notion).

Assume the Axiom of Dependent Choice. If $(X,\mathcal{T})$ is locally compact and Hausdorff, then $X$ is completely regular, and hence, being Hausdorff, Tychonoff (Under dependent choice a locally compact Hausdorff space is completely regular, hence Tychonoff).