Measuring the complexity of characterizing $[0, 1]$, $S^1$, and $\mathbb{R}$ up to homeomorphism

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.