Under Dependent Choice, every nonempty compact metric space is a continuous image of Cantor space

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

If $K$ is a nonempty compact metric space, there is a rooted levelled tree $T={\bigcup }_{n\in \mathbb{N}}{T}_n$, finitely branching with every level $T_n$ finite and nonempty — so $T$ itself is infinite — and nonempty compact sets $(K_s{)}_{s\in T}$ such that $T_0$ has one root $r$ with $K_r=K$, every node has a finite nonempty set of children whose sets cover its set, every child set is contained in its parent set, and $\mathrm{diam}(K_s)\le 2^{-n}$ for $s\in T_n$ after a harmless rescaling of the metric. (Under Dependent Choice, a compact metric space carries a finitely branching refining tree of covers of arbitrarily small diameter).

Let $(X,d)$ be a compact metric space (def-metric-compactness, def-metric-space). Then $(X,d)$ is totally bounded (def-totally-bounded) and complete (def-complete-metric-space). Both implications are theorems of ZF. Completeness is obtained here from the finite intersection characterisation (thm-compact-iff-finite-intersection-property) applied to the closures of the tails of a Cauchy sequence, and not from the extraction of a convergent subsequence, which would route the argument through sequential compactness. What matters for the ledger is that the route taken below selects nothing at all; the first remark below says why the other route was not taken. (A compact metric space is complete and totally bounded, and neither implication uses any choice principle).

Let $(X,d)$ be a metric space (def-metric-space). Call a sequence $(F_k{)}_{k\in \mathbb{N}}$ of subsets of $X$ a Cantor chain if every $F_k$ is nonempty, closed (def-metric-topology) and bounded, $F_{k+1}\subseteq F_k$ for every $k$, and $\mathrm{diam}(F_k)\to 0$ in $\mathbb{R}$ (def-metric-bounded-diameter, def-real-limit). Then: 1. If $(X,d)$ is complete (def-complete-metric-space), every Cantor chain in $X$ has an intersection ${\bigcap }_{k\in \mathbb{N}}{F}_k$ with exactly one element. 2. Conversely, if every Cantor chain in $X$ has nonempty intersection, then $(X,d)$ is complete. Boundedness of each $F_k$ is part of the definition of a Cantor chain because $\mathrm{diam}$ is defined for nonempty bounded sets only in this library (def-metric-bounded-diameter); it is not an extra hypothesis but the precondition for writing the diameter condition down. (In a complete metric space nested nonempty closed sets whose diameters tend to $0$ meet in exactly one point, and this property characterises completeness).

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).

Throughout, a topology is as in def-topological-space, and finite, at most countable and uncountable are as in def-countable, so that "countable" always means "at most countable" and every finite set is countable. Let $X$ be a set. The six families below are topologies on $X$; that each really satisfies (T1), (T2) and (T3) is discharged in full after the list. Among those six is the discrete topology ${\mathcal{T}}_{\mathrm{d}\mathrm{i}\mathrm{s}\mathrm{c}}:=\mathcal{P}(X)$, in which every subset is open and hence every subset is also closed. (The discrete, indiscrete, cofinite, cocountable, particular-point and Sierpinski topologies).

Let $X$ be a set and let $R\subseteq X\times X$ be a binary relation on $X$. Call $R$ entire on $X$ when $\text{for every }x\in X\text{ there is }y\in X\text{ with }xRy\mathrm{.}$ The Axiom of Dependent Choice, written $\mathrm{D}\mathrm{C}$, is the following statement. The statement is: for every nonempty set $X$, every relation $R$ entire on $X$, and every $a\in X$, there is a sequence $x:\mathbb{N}\to X$ with $x_0=a$ and $x_{n}R{x}_{n+1}$ for every $n\in \mathbb{N}$. (The axiom of dependent choice: a relation in which every element is related to something admits an $\mathbb{N}$-indexed chain).