Topological characterization of the closed interval $[0, 1]$.

It is the unique second countable continuum with precisely two non-cutpoints. This is due to Veblen, according to this overview.

A continuum is a connected and compact Hausdorff space, and a cutpoint (in a connected space) is a point that when removed leaves the remainder disconnected. The second countable is a (non-reals using) way of saying metrisable. One can prove the compactness and connectedness purely from order completeness of the order.

This is a duplicate of my answer on MO

Consider the class of all Hausdorff compacts with distinct points (i.e. which have more than $1$ point) that are absolute retracts in the class of Hausdorff compacts. Then $[0,1]$ is up to homeomorphism the only member of this class that embeds into every other.