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Feb 7, 2019 at 17:48 comment added YCor Side remark: it's used implicitly that on a compact Hausdorff topological space $X$, a distance induces the topology iff it's continuous. Indeed $\Rightarrow$ is clear, and conversely, $d$ continuous implies that the identity map $X\to (X,d)$ is continuous bijective; since $X$ is compact Hausdorff this implies that it is a homeomorphism.
Feb 7, 2019 at 17:38 history edited YCor CC BY-SA 4.0
fixed typo
Feb 7, 2019 at 17:23 history answered YCor CC BY-SA 4.0