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A bitopological space is a set $X$ with two fixed Hausdorff topologies $\tau_1, \tau_2$. In my case I am interested in the case where $\tau_1 \subseteq \tau_2$. Say that a bitopological space $(X, \tau_1, \tau_2)$ is compactly almost metric, when

  • $\tau_1 \subseteq \tau_2$,
  • $\tau_2$ is metrisable
  • $\tau_1, \tau_2$ have the same compact sets.

Are there any non-trivial sufficient conditions so that if $(X, \tau_1, \tau_2)$ and $(Y, \sigma_1, \sigma_2)$ are compactly almost metric and $(X,\tau_2)$ is homeomorphic to $(Y,\sigma_2)$, then $(X, \tau_1)$ is homeomorphic to $(Y, \sigma_1)$?

Since the topic is rather obscure, an idea where to look up such things would be appreciated too.

Tomasz Kania
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    I don't have a useful answer to your question, but let me make a perhaps useless remark instead: this seems like a very "unbitopological" notion and an even more "unbitopological" question to me. (For example, bitopological Hausdorffness does not mean having a pair of Hausdorff topologies.) Therefore I suspect that you might have more success finding related results in the literature if you look for papers on topological instead of bitopological spaces, even if there are indeed two topologies on the same space involved in your question. The bitopological angle may be a red herring here. – Pilcrow Mar 22 '22 at 21:23
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    You could recast this problem as "Let $(X, d)$ be a metric space and $\tau, \sigma$ two Hausdorff topologies on $X$ coarser than the metric topology. If both $\tau$ and $\sigma$ have the same compact sets as the metric topology, is $(X, \tau)$ homeomorphic to $(X, \sigma)$?" A somewhat stronger conclusion would be that "$\tau = \sigma$". And it may be prudent to investigate that statement first. Even if it is false, how it fails may be useful in deducing where some homeomorphism can be made to bypass the problem. – Paul Sinclair Mar 23 '22 at 18:32
  • @PaulSinclair, thank you. In the cases that are of interest to me, I am almost certain that the topologies are different but want to distill cases where they are homeomorphic. – Tomasz Kania Mar 23 '22 at 19:22

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