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Can someone come up with two topological spaces $X$ and $Y$ satisfying

  1. Both are Hausdorff,
  2. $X$ and $Y$ are not homeomorphic; however
  3. $X$ and $Y$ are “compactomorphic” in the sense that there is a proper map between them with a proper inverse.

In other words, the spaces themselves are different, but the compact sets are the same.

This would be trivial in the non-Hausdorff case, using the same finite set with the discrete and trivial topologies. However, the discrete topology is the only Hausdorff topology on a finite set.

Gaussler
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    See here http://math.stackexchange.com/questions/1651142/are-two-metrics-with-same-compact-sets-topologically-equivalent for an answer in the metric case. – Nitrogen Mar 10 '16 at 18:09
  • @Nitrogen I had actually noticed it, and it is very much related. In particular, at least one of the topologies must be non-metric. – Gaussler Mar 10 '16 at 18:10

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