𝒞 is compact

claim

$𝒞=\{0,1\}^ℕ$ is compact.

For every $p∈\{0,1\}^*$ let $Uₚ⊆𝒞$ denote the set of all strings with prefix $p$.

proof

Let $X=(xₙ)_{n∈ℕ}$ be an arbitrary sequence in $𝒞$. For all $p∈\{0,1\}^*$ let $Xₚ$ denote the set of $x∈X$ with prefix $p$.

We will define an infinite string $y$ inductively. For each $k∈ℕ$ let $pₖ$ denote the prefix $y(0)\,y(1)\,\dotsm\,y(k-1)$. We will define $y$ so that for each $k$, $X_{pₖ}$ is infinite.

• We start with $p₀$ the empty string, and $X_{p₀}=X$ empty.
• Fix $k∈ℕ$ such that $pₖ$ has already been specified, and $X_{pₖ}$ is infinite. Note that $X_{pₖ}=X_{pₖ0}∪X_{pₖ1}$, so between the two “halves” at least one set is infinite; define $y(k)∈\{0,1\}$ to be a/the value so that $X_{pₖ₊₁}=X_{pₖ \, y(k)}$ is infinite.

Finally for each $k∈ℕ$ let $yₖ∈X_{pₖ}$ be an element of the sequence $X$ occurring after $yₖ₋₁$. Then $(yₖ)_{k∈ℕ}$ is a subsequence of $X$ converging to $y$.