Three orderings on \(\beta(\omega)\setminus\omega\) (Q690249)

The focus of this paper is the Comfort (pre)order on \(\omega^*\), with comparisons made to the Rudin-Keisler order and the Rudin-Frolík order. The paper starts by reviewing basic facts and definitions related to Bernstein's notions of \(p\)-limit and \(p\)-compactness. These are used to define the Comfort order on \(\omega^*\), namely \(p\leq_ C q\) if every \(q\)-compact space is \(p\)-compact. In general, this gives an order different from the Rudin-Keisler order. It is always true that if \(p\leq_{RK} q\) then \(p\leq_ C q\); however, there are points in \(\omega^*\) which are Rudin-Keisler incomparable, but Comfort equivalent. Interestingly, the two orders coincide on the set of weak \(P\)-points. The set \(T_ C(p)=\{q\in \omega^*\mid p\leq_ C q\) and \(q\leq_ C p\}\) contains exactly \(2^ \omega\) types of \(\omega^*\). If \(p\) is Rudin-Keisler minimal then \((T_ C(p), \leq_{RF})\) is a linearly ordered set, where \(\leq_{RF}\) is the Rudin-Frolík order. The following results are also shown: \((\forall p, q\in\omega^*)\) \((\exists r\in\omega^*)\) \([r\leq_{RK} q\land r\leq_{RK} q]\) if and only if \((\forall p, q\in\omega^*)\) \((\exists r\in\omega^*)\) \([r\leq_ C p\land r\leq_ C q]\); for \(p\in\omega^*\), \(T_ C(p)\) is countably compact; and if \(p\) is a \(P\)-point then \(T_ C(p)\) is \(p\)-compact. The paper lists a number of open questions related to the Comfort order.