Comfort types of ultrafilters

Abstract:

Let $\alpha$ be an infinite cardinal. Comfort pre-order on $\beta (\alpha )\backslash \alpha$ is defined as follows: for $p,q \in \beta (\alpha )\backslash \alpha ,\;p{ \leqslant _{\text {c}}}q$ if every $q$-compact space is $p$-compact. For $p \in U(\alpha )$, we let ${T_{\operatorname {RK} }}(p)$ be the type of $p$ and ${T_{\text {c}}}(p) = \{ q \in U(\alpha ):q{ \leqslant _{\text {c}}}p{ \leqslant _{\text {c}}}q\}$. Since ${T_{\text {c}}}(p)$ is a union of types, it is natural to define ${c_p} = |\{ {T_{\operatorname {RK} }}(q):q \in {T_{\text {c}}}(p)\} |$. It is evident that $\omega \leqslant {c_p} \leqslant {2^\alpha }$ for $p \in U(\alpha )$. We show that if $p \in U(\alpha )$ then $|{T_{\operatorname {RK} }}(p)| = {|^\alpha }\alpha /p|$, and we use this equality to prove that ${c_p} = {2^\alpha }$ whenever $p$ is decomposable. We also note that if $p$ is countably incomplete then ${2^\omega } \leqslant {c_p} \leqslant {2^\alpha }$; if $p$ is RK-minimal (selective) and $\omega < \alpha$ then ${c_p} = \omega$ and ${c_p} = {2^{{\aleph _n}}}$ for each $p \in U({\aleph _n})$ and for $n < \omega$. Finally, we prove that, if $\alpha$ is a strong limit and $p \in U(\alpha )$ is indecomposable, then ${\beta _p}(\alpha )$ is a $p$-compact, noninitially $\alpha$-compact space, where ${\beta _p}(\alpha )$ is the $p$-compactification of $\alpha$.