Ultrafilters, monotone functions and pseudocompactness

Abstract

that is, {n<&omega;:V&AND;U-n&NOTEQUAL;&phi;} &ISIN

q for every neighborhood V of x. The P-RK(p)-pseudocompact spaces were studied in [ST]. In this article we analyze M-pseudocompactness when M is one of the classes S(p), R(p), T(p), I(p), P-RB(p) and P-RK(p). We prove that every Frolik space is S(p)-pseudocompact for every p &ISIN

&beta;&omega

with &omega

&SUB

X is M-pseudocompact.

In this article we, given a free ultrafilter p on omega, consider the following classes of ultrafilters: (1) T(p)-the set of ultrafilters Rudin-Keisler equivalent to p, (2) S(p)={q is an element of omega:There Exists f is an element of omega(omega), strictly increasing, such that q=f(beta)(p)}, (3) I(p)-the set of strong Rudin-Blass predecessors of p, (4) R(p)-the set of ultrafilters equivalent to p in the strong Rudin-Blass order, (5) P-RB(p)-the set of Rudin-Blass predecessors of p, and (6) P-RK(p)-the set of Rudin-Keisler predecessors of p, and analyze relationships between them. We introduce the semi-P-points as those ultrafilters p is an element of omega for which P-RB(p)=P-RK(p), and investigate their relations with P-points, weak-P-points and Q-points. In particular, we prove that for every semi-P-point p its alpha-th left power (alpha)p is a semi-P-point, and we prove that non-semi-P-points exist in ZFC. Further, we define an order < in T(p) by r < q if and only if r is an element of S(q). We prove that (S(p),<) is always downwards directed, (R(p), <) is always downwards and upwards directed, and (T(p), <) is linear if and only if p is selective. We also characterize rapid ultrafilters as those ultrafilters p is an element of omega for which R(p)\S(p) is a dense subset of omega. A space X is M-pseudocompact (for M subset of omega) if for every sequence (U-n)(n<omega) of disjoint open subsets of X, there are q is an element of M and x is an element of X such that x=q-lim (U-n)

&omega;, and determine when a subspace X &SUB