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, homogenity, and rigity. 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 $t\in Seq$).