Cantor space is an example of a compact and uncountable Polish space. In fact, every uncountable Polish space contains Cantor space, as stated by the following theorem.

For example, the set $\mathbb{R}$ of real numbers contains the Cantor middle thirds set. Note that, being homeomorphic to Cantor space, $S$ must be a compact and hence closed subset of $X$ . The result is trivial in the case of Baire space $\mathcal{N}$ , in which case we may take $S$ to be the set of all $s\in\mathcal{N}$ satisfying $s_n\in\{1,2\}$ for all $n$ . Then, for any uncountable Polish space $X$ there exists a continuous and one-to-one function $f\colon\mathcal{N}\to X$ (see here). Then $f$ gives a continuous bijection from $S$ to $f(S)$ . The inverse function theorem implies that $f$ is a homeomorphism between $S$ and $f(S)$ and, therefore, $f(S)$ is homeomorphic to Cantor space.