Every filter is homeomorphic to its square
  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. 
Andrea Medini, Lyubomyr Zdomskyy
math.GN math.LO