Lemma 3

Let $\varepsilon$ be a real number $>0$. There exists an integer $m\geqslant 1$ such that, for every integer $n\geqslant m$, X admits a partition formed of $n$ nonempty open and closed sets of diameter $\leqslant \varepsilon$.

Every point of X admits an open and closed neighbourhood of diameter $\leqslant \varepsilon$ (TG, II, p. 32, corollary of prop. 6). Since the space X is compact, it has a finite covering by such sets. Choose one, $(U_i)_{1\leqslant i\leqslant m}$, for which $m$ is minimal. We have $m\geqslant 1$ since X is nonempty. For $1\leqslant i\leqslant m$, denote by $V_i$ the intersection of $U_i$ and the $X-U_k$ for $k < i$. Then $(V_i)_{1\leqslant i\leqslant m}$ is a partition of X, formed of $m$ nonempty open and closed sets of diameter $\leqslant \varepsilon$.

Since X has no isolated point, every nonempty open and closed subset V of X contains at least two points. Since moreover X is compact and totally disconnected, V is the union of two disjoint nonempty open and closed subsets (loc. cit.). It follows, by induction, that for every integer $n\geqslant m$, X admits a partition formed of $n$ nonempty open and closed sets of diameter $\leqslant \varepsilon$.

Let us now finish the proof of the prop. 14. Every nonempty open and closed subspace of X is a compact, totally disconnected metric space and has no isolated point. Lemma 3 therefore allows us to construct by induction a sequence $(J_n)_{n\geqslant 0}$ of finite sets and, for every $n\geqslant 0$, a mapping $\varphi_n$ from the set $C_n= J_0\times \cdots \times J_n$ into the set of nonempty open and closed subsets of X of diameter $\leqslant 2^{-n}$, in such a way that :

(i) For every $n\geqslant 0$, there exists an integer $m_n\geqslant 1$ such that Card(J$_n$) $=$ $2^{m_n}$;

(ii) The family $(\varphi_0(c))_{c\in C_0}$ is a partition of X ;

(iii) For every $n\geqslant 0$ and every $c\in C_n$, the family $(\varphi_{n+1}(c, j))_{j\in J_{n+1}}$ is a partition of $\varphi_n(c)$.

Let us denote by $p_n$ the canonical projection of $C_{n+1}$ onto $C_n$. The sequence C = $(C_n, p_n)_{n\geqslant 0}$ is a sieve (TG, IX, p. 63, def. 8). The topological space associated with this sieve (TG, IX, p. 63) is identified with the topological space J, product of the discrete topological spaces $J_n$. The sieve C and the sequence of mappings $(\varphi_n)_{n\geqslant 0}$ define a strict sieving of the metric space X (TG, IX, p. 63 and p. 64). The mapping $f: J\rightarrow X$ deduced from this sieving is continuous and bijective (TG, IX, p. 65). Since the topological space J is compact (TG, I, p. 63, th. 3) and X is separated, $f$ is a homeomorphism (TG, I, p. 63, cor. 2). Since $J_n$ is homeomorphic to $\{0,1\}^{m_n}$ for every $n\geqslant 0$, J is homeomorphic to $\{0,1\}^{\mathbf{N}}$ (TG, I, p. 25, prop. 2).