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We construct a space having the properties in the title, and with the same technique, a countably compact $T_2$ topological group which is not absolutely countably compact.

We consider the spaces called $Seq\left(u_t\right)$, 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 $S\left(u_t\right)$ 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 $Seq\left(p\right)$ (i.e., $u_t=p$ for all $t\in Seq$). It follows that for a Ramsey ultrafilter $p$, $Seq\left(p\right)$ is a topological group....

Let ω denote the set of natural numbers. We prove: for every mod-finite ascending chain $T_{\alpha }:\alpha <\lambda$ of infinite subsets of ω, there exists $ℳ\subset {\left[\omega \right]}^{\omega }$, an infinite maximal almost disjoint family (MADF) of infinite subsets of the natural numbers, such that the Stone-Čech remainder βψ∖ψ of the associated ψ-space, ψ = ψ(ω,ℳ ), is homeomorphic to λ + 1 with the order topology. We also prove that for every λ < ⁺, where is the tower number, there exists a mod-finite ascending chain $T_{\alpha }:\alpha <\lambda$, hence a ψ-space with Stone-Čech remainder...