A short proof of Rubin's theorem

TL;DR

This paper provides a concise, self-contained proof of Rubin's theorem, demonstrating the uniqueness of group actions on certain topological spaces using ultrafilters, simplifying previous complex proofs.

Contribution

It introduces a new, streamlined proof of Rubin's theorem employing ultrafilters, enhancing understanding and accessibility of the original result.

Findings

The proof confirms the uniqueness of the space and group action up to homeomorphism.

Ultrafilters on a poset effectively reconstruct the space points.

The approach simplifies the original proof of Rubin's theorem.

Abstract

In a remarkable theorem, M. Rubin proved that if a group $G$ acts in a locally dense way on a locally compact Hausdorff space $X$ without isolated points, then the space $X$ and the action of $G$ on $X$ are unique up to $G$-equivariant homeomorphism. Here we give a short, self-contained proof of Rubin's theorem, using equivalence classes of ultrafilters on a poset to reconstruct the points of the space $X$.