Cantor and Baire sequence spaces and coordinate codings

Symmetry and separation of $d$ follow from the first differing coordinate. If $x,y$ and $y,z$ both agree through the first $n$ coordinates, so do $x,z$. Consequently $d(x,z)\le \max (d(x,y),d(y,z))\le d(x,y)+d(y,z)$, proving the metric law. For $n\ge 1$, $N_{x\upharpoonright n}=\{y:d(x,y)<2^{-n}\}$, so the metric induces precisely the cylinders. A Cauchy sequence has each coordinate eventually constant: use the Cauchy bound $2^{-k-1}$ for coordinate $k$. Define $x(k)$ to be that unique eventual value. The Cauchy bound at $2^{-n}$ shows all sufficiently late terms agree with $x$ on the first $n$ coordinates, hence converge to $x$. In the binary case each value remains binary.

Let $\mathcal{U}$ be an open cover of $\mathcal{C}$. If no finite subfamily covers it, the root cylinder is not finitely covered. Whenever $N_s$ is not finitely covered, at least one of $N_{s^{⌢}0},N_{s^{⌢}1}$ is not finitely covered: otherwise combine their two finite covers. Recursively take the least such bit. The resulting $x$ lies in some $U\in \mathcal{U}$, and openness gives $N_{x\upharpoonright n}\subseteq U$ for some $n$, contradicting the construction. Hence every cover has a finite subcover, as F3 requires. Only least choices from two bits were used.

The displayed pairing is a bijection ${\mathbb{N}}^2\to \mathbb{N}$: on diagonal $i+j=r$ its values are the consecutive integers from $r(r+1)\mathrm{/}2$ to $(r+1)(r+2)\mathrm{/}2-1$, and the diagonals partition $\mathbb{N}$. Define $H(x{)}_i(j)=x(⟨i,j⟩)$. Its inverse assigns $x(⟨i,j⟩)=z_i(j)$, so both compositions are identities coordinate by coordinate. A finite restriction on either side constrains finitely many coordinates on the other; at each point a long enough initial cylinder fixes all those coordinates. Thus both directions are continuous by F4, in the binary case as well.

Each block in $h$ ends in $1$ and has positive length, so $h(x)\in D$. Conversely for $z\in D$, let $p_n$ be the position of its $n$th $1$, indexed starting at zero, obtained by successive least search. Put $x(0)=p_0$ and $x(n+1)=p_{n+1}-p_n-1$. These are natural numbers and the block concatenation reconstructs $z$. Reading block lengths from $h(x)$ returns $x$, giving a two-sided inverse. Fixing enough input coordinates to finish the first $k$ output bits proves continuity of $h$; fixing through the $k$th separator proves continuity of its inverse on $D$.

Finite words admit an explicit natural-number coding: encode a finite word by its length and recursively pair its entries, using $⟨i,j⟩=(i+j)(i+j+1)\mathrm{/}2+j$. Appending infinitely many zeros gives a countable family meeting every cylinder in either space. Thus they are separable; combined with step 1.1 this proves Polishness. Given any binary cylinder containing $x$, change the next unrestricted bit of $x$ and keep all other bits. The resulting different point is in the same cylinder, so no point is isolated.

If $z\mathrm{\notin }D$, it has only finitely many $1$s. Associate the integer $c(z)={\sum }_{j:z(j)=1}{2}^j$. Distinct finite binary supports give distinct sums: at their largest differing index $r$, the term $2^r$ exceeds the sum ${\sum }_{j<r}{2}^j=2^r-1$. Thus $c$ is an injection into $\mathbb{N}$, with the zero sequence mapped to zero. This proves the countability assertion and completes all constructions. QED.