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In the known paper On the reconstruction of topological spaces from their group of homeomorphisms by Matatyahu Rubin several deep reconstruction theorems of the form "if $X$ and $Y$ are topological spaces in a broad class of spaces $K$ and there is an isomorphism between $\mathrm{Homeo}(X)$ and $\mathrm{Homeo}(Y)$, then $X$ and $Y$ are homeomorphic" are proved. Moreover the following result is claimed

Assume $V=L$. If $X$ and $Y$ are second countable connected Euclidean manifolds and $\mathrm{Homeo}(X)$ is elementary equivalent to $\mathrm{Homeo}(Y)$, then $X$ and $Y$ are homeomorphic.

to appear in Second countable connected manifolds with elementarily equivalent homeomorphism groups are homeomorphic in the constructible universe. Unfortunately I cannot find any information on a paper with this title online. Has a proof of this theorem been published by Rubin? What is known about this result in $\mathsf{ZFC}$ without extra set theoretic assumptions?

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  • $\begingroup$ Damn, I really miss talking to Matti about math. $\endgroup$
    – Asaf Karagila
    Jul 29 at 19:06

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