Abstract:We consider the spaces called $Seq(u_t)$, constructed on the set $Seq$ of all finite sequences of natural numbers using ultrafilters $u_t$ to define the topology. For such spaces, we discuss continuity, homogeneity, and rigidity. We prove that $S(u_t)$ is homogeneous if and only if all the ultrafilters $u_t$ have the same Rudin-Keisler type. We proved that a space of Louveau, and in certain cases, a space of Sirota, are homeomorphic to $Seq(p)$ (i.e., $u_t = p$ for all $t\in Seq$). It follows that for a Ramsey ultrafilter $p$, $Seq(p)$ is a topological group.
Keywords: ultrafilters, continuity, homeomorphisms, homogeneous, rigid, topological group, Ramsey ultrafilters, selective ultrafilters
AMS Subject Classification: Primary 54D80, 54C05, 54G05, 54A35, 54H11