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Continuum theory, point-set topology, spaces with algebraic structure, foundations, dimension theory, local and global properties.

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On a result by Rubin on elementary equivalence of homeomorphism groups and homeomorphisms of...

There's a new paper that proves something even stronger for compact manifolds: https://arxiv.org/abs/2302.01481. I don't know about noncompact manifolds, though, and it seems neither do the authors.
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