Hilbert cube

Remarks

The map $(t_n)_{n \ge 1} \mapsto (t_n / n)_{n \ge 1}$ is a homeomorphism from $[0, 1]^{\N}$, with the product topology, onto $H$. So $H$ is compact, for example by Tychonoff's theorem. It is also convex in $\ell^2$.

Every separable metrizable space is homeomorphic to a subspace of $H$. This is the embedding behind the Urysohn metrization theorem.

The Hilbert cube has the fixed-point property: every continuous map $H \to H$ has a fixed point. This follows from the Schauder fixed-point theorem, since $H$ is a compact convex subset of a Banach space.