9. embedding 𝒩 into 𝒞

problem

Show that $𝒩$ is homeomorphic to the set of all sequences $x∈𝒞$ with infinitely many $1$s.

proof

Define $ψ:𝒩→𝒞$ by

$$ψ(x) ≔ (0^x(0), 1, 0^x(1), 1, 0^x(2), 1, …).$$

since there are infinitely many digits in $x$, there are infinitely many $1$s in $ψ(x)$. also this is a bijection because this encoding is invertible: given any binary string $b$ obtain $[ψ⁻¹(b)](i)$ as the number of $1$s between the $i$-th and $(i+1)$-th occurrence of $1$ in $b$.

it remains to show that both $ψ$ and $ψ⁻¹$ are continuous. for this it suffices to show that subbasis elements get mapped to open sets.

• fix any $i∈ℕ$ and $V⊆{0,1}$ and let $S=ϕᵢ⁻¹(V)⊆𝒞$. we wish to show $ψ⁻¹(S)$ open in $𝒩$.

  fix any $x∈ψ⁻¹(S)$, and let $U=N_{[i]}(x)$ be the set of all strings sharing digits up to index $i$ with $x$. let $y∈U$. by construction then $ψ(x)$ and $ψ(y)$ share (at least) a common length-$i$ prefix as well, so in particular $ψ(y)(i)=ψ(x)(i)$ so $ψ(y)∈S$. so $U⊆ψ⁻¹(S)$.

• fix any $i∈ℕ$ and $V⊆ℕ$ and let $S=ϕᵢ⁻¹(V)⊆𝒩$. we wish to show $ψ(S)$ open in $𝒞$.

  fix any $x∈ψ(S)$, and let $y=ψ⁻¹(x)$. let $k = y(0) + 1 + y(1) + 1 + ⋯ + y(i) + 1.$ let $U=N_{[k]}(x)$ be the set of all strings sharing first prefix up to index $k$ with $x$. then observe that for any $x'∈U$ we have $ψ⁻¹(x')$ sharing at least prefix up to $k$ with $y$, so in particular $ψ⁻¹(x')(i)=y(i)∈V$, so $ψ⁻¹(x')∈S$, so $x'∈ψ(S)$. since this holds for arbitrary $x'∈U$ we have $U⊆ψ(S)$.