  In analogy to the study of Scott rank/complexity of countable structures, we
initiate the study of the Wadge degrees of the set of homeomorphic copies of
topological spaces. One can view our results as saying that the classical
characterizations of $[0,1]$ (e.g., as the unique continuum with exactly two
non-cut points, and other similar characterizations), appropriated expressed,
are the simplest possible characterizations of $[0,1]$. Formally, we show that
the set of homeomorphic copies of $[0,1]$ is $\mathbf{\Pi}^0_4$-Wadge-complete.
We also show that the set of homeomorphic copies of $S^1$ is
$\mathbf{\Pi}^0_4$-Wadge-complete. On the other hand, we show that the set of
homeomorphic copies of $\mathbb{R}$ is $\mathbf{\Pi}^1_1$-Wadge-complete. It is
the local compactness that cannot be expressed in a Borel way; the set of
homeomorphic copies of $\mathbb{R}$ is $\mathbf{\Pi}^0_4$-Wadge-complete within
the locally compact spaces.
