The Rudin-Kiesler pre-order and the Pixley-Roy spaces over ultrafilters

Abstract

For a $T_1$-space X, we denote by $PR\left(X\right)$ the Pixley-Roy space over X. For $p\in {\omega }^{⁎}$, let $X_p=\left\{p\right\}\cup \omega$ be the subspace of the Stone-Čech compactification βω of the discrete space ω. Motivated by Gul'ko's theorem (Theorem 1.1), we show: (1) $PR\left(X_p\right)$ and $PR\left(X_q\right)$ are homeomorphic if and only if $X_p$ and $X_q$ are homeomorphic (i.e., p and q are type-equivalent), (2) if q is selective and $X_q$ can be embedded into $PR\left(X_p\right)$, then $X_p$ and $X_q$ are homeomorphic, (3) if p is selective, then $PR\left(X_p\right)$ contains copies of some $X_{q_n}(n\in N)$ which are pairwise non-homeomorphic, and (4) $PR\left(X_p\right),PR(X_p\oplus X_p)$ and $PR(X_p⁎X_p)$ are pairwise non-homeomorphic, where $X_p⁎X_p$ is the quotient space obtained by identifying the limit points of the topological sum $X_p\oplus X_p$.