Every filter is homeomorphic to its square

TL;DR

This paper proves that any filter on the natural numbers, when viewed as a topological space, is homeomorphic to its square, extending previous results known for Borel filters.

Contribution

It generalizes van Engelen's theorem by showing all filters, not just Borel filters, are homeomorphic to their squares.

Findings

All filters on ω are homeomorphic to their squares.

Extension of van Engelen's theorem to non-Borel filters.

Topological characterization of filters as self-homeomorphic spaces.

Abstract

We show that every filter $\mathcal{F}$ on $\omega$, viewed as a subspace of $2^\omega$, is homeomorphic to $\mathcal{F}^2$. This generalizes a theorem of van Engelen, who proved that this holds for Borel filters.