Under the Axiom of Choice, topological sums of Čech-complete spaces are 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).

The underlying set. Let $I$ be a set and let $X_i$ be a set for each $i\in I$. The disjoint union is ${⨆}_{i\in I}{X}_i:={\bigcup }_{i\in I}(X_i\times \{i\}),$ whose elements are the pairs $(x,i)$ with $i\in I$ and $x\in X_i$. For $j\in I$ the $j$-th canonical injection is ${\kappa }_j:X_j\to {⨆}_{i\in I}{X}_i,{\kappa }_j(x):=(x,j)\mathrm{.}$ The construction is what makes the word "disjoint" honest. Each ${\kappa }_j$ is injective (def-injection-surjection-bijection), since $(x,j)=(x^{\mathrm{\prime }},j)$ forces $x=x^{\mathrm{\prime }}$; the images ${\kappa }_j[X_j]=X_j\times \{j\}$ are pairwise disjoint, since the second coordinate determines $j$; and their union is the whole set. So no assumption that the $X_i$ are disjoint as sets is needed, and none is made: the tag $i$ separates the copies even when $X_i=X_{i^{\mathrm{\prime }}}$ for $i\mathrm{\ne }{i}^{\mathrm{\prime }}$. (The disjoint union (coproduct) ${⨆}_i{X}_i$ with the final topology of the canonical injections: a set is open exactly when each of its traces is).

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).

The Axiom of Choice (AC) is the following statement. The statement is: every family of nonempty sets has a choice function; that is, for every set $\mathcal{F}$ all of whose members are nonempty there is a function $g$ with domain $\mathcal{F}$ satisfying $g(S)\in S$ for every $S\in \mathcal{F}$. (The Axiom of Choice).