# On a result by Rubin on elementary equivalence of homeomorphism groups and homeomorphisms of the underlying spaces

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?

• Damn, I really miss talking to Matti about math. Jul 29 at 19:06