Can someone please explain this example? It’s not clear why it doesn’t induce homeomorphism. Does anyone have any simpler example?
2026-03-27 04:56:35.1774587395
Quasi-isomorphic doesn't induce homeomorphic example
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Quasi-isometry is really a tool for studying the large-scale structure of a metric space. It's completely insensitive to fine details. In particular, any two bounded metric spaces are quasi-isometric, via any map from one to the other (this is a good exercise).
So pick two bounded complete CAT(0) spaces with different $\partial$s. For example, the interval $[0, 1]$ and the square $[0, 1]^2$.