Under the Axiom of Choice, countable products 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).

Assume the Axiom of Choice (def-axiom-of-choice). Let $I$ be a set and let $(X_i,{\mathcal{T}}_i{)}_{i\in I}$ be a family of compact topological spaces (def-compact-space, def-topological-space). Then the product $P:={\prod }_{i\in I}{X}_i$ with the product topology (def-product-topology) is compact. The Axiom of Choice is spent twice, and both uses are flagged below. Once inside thm-alexander-subbase-lemma, through Zorn's lemma (thm-zorn), and once directly at step 2.1, to produce a point of a product of nonempty sets. (Tychonoff's theorem: an arbitrary product of compact spaces is compact in the product topology, assuming the Axiom of Choice).

The Axiom of Countable Choice, written ${\mathrm{A}\mathrm{C}}_{\omega }$, is the following statement. The statement is: for every family $(X_n{)}_{n\in \mathbb{N}}$ of nonempty sets indexed by $\mathbb{N}$ there is a function $f$ with domain $\mathbb{N}$ such that $f(n)\in X_n$ for every $n\in \mathbb{N}$. Equivalently, every at most countable family of nonempty sets has a choice function. (The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$)).

$\mathbb{N}\times \mathbb{N}\approx \mathbb{N}$ (def-equinumerous): the plane of pairs of naturals is countably infinite (def-countable). The bijection is exhibited, not merely asserted to exist. Define $2^m$ by recursion on $m$ (thm-recursion) by $2^0=1$ and $2^{\sigma (m)}=2^m+2^m$, and set $J(m,n)=2^m\cdot \sigma (n+n),\text{that is}J(m,n)=2^m(2n+1)\mathrm{.}$ Then $J$ is a bijection from $\mathbb{N}\times \mathbb{N}$ onto $\mathbb{N}\setminus \{0\}$, and $\sigma$ is a bijection from $\mathbb{N}$ onto $\mathbb{N}\setminus \{0\}$, so ${\sigma }^{-1}\circ J$ is a bijection $\mathbb{N}\times \mathbb{N}\to \mathbb{N}$. What makes $J$ bijective is the decomposition of a nonzero natural into a power of two times an odd number, existence and uniqueness both. ($\mathbb{N}\times \mathbb{N}\approx \mathbb{N}$).

The product set. Let $I$ be a set and let $X_i$ be a set for each $i\in I$. The product is ${\prod }_{i\in I}{X}_i:=\{x:x\text{ is a function with domain }I\text{ and }x(i)\in X_i\text{ for every }i\in I\},$ and we write $x_i:=x(i)$, the $i$-th coordinate of $x$. Two elements of the product are equal exactly when they agree at every index, functions being equal when they have the same domain and the same values. For $j\in I$ the $j$-th projection is ${\pi }_j:{\prod }_{i\in I}{X}_i\to X_j,{\pi }_j(x):=x_j\mathrm{.}$. The product topology ${\mathcal{T}}^{\mathrm{\Pi }}$ on ${\prod }_i{X}_i$ is the initial topology of the projections: the topology generated by the subbasis $\{{\pi }_i^{-1}[U]:i\in I,U\in {\mathcal{T}}_i\}$. Finite intersections of subbasic sets form a basis for it, and they are exactly the boxes ${\prod }_{i\in I}{U}_i$ with every $U_i$ open in $X_i$ and $U_i=X_i$ for all but finitely many $i$. (The product set ${\prod }_{i\in I}{X}_i$ of functions choosing a point in each factor, the projections, the box topology, and the product topology as the initial topology of the projections; the empty product is a one-point space).