Some Generalizations of Pseudocompactness

Abstract

In this paper, we introduce the concepts of p‐boundedness for pɛω*, (α, M)‐pseudocompactness and (α, M)‐compactness, for a cardinal number α and Ø≠M⊆β(ω)\ω. We prove that Xα is pseudocompact (respectively, countably compact) iff X is (α, M)‐pseudocompact (respectively, (α, M)‐compact), for some Ø≠M⊆β(ω)\ω; the Rudin‐Keisler order on β(ω)\ω can be defined in terms of p‐boundedness and p‐pseudocompactness; and if pɛβ(ω)\ω then p is RK‐minimal (selective) iff the space ω∪T(p) is p‐pseudocompact, where T(p) is the type of p in β(ω)\ω.

Ultrafilter Convergence in Ordered Topological Spaces

“…This is Remark 30 in [16], relying on an example by García-Ferreira [9], which in turn builds on a construction by Kanamori [12]. However, (5) implies (6) (7) implies (7) reg trivially, hence, for GO spaces, they are all equivalent, since we have already proved that for GO spaces (5) and (7) reg are equivalent.…”

“…Let A and B be defined as in the statement of Proposition 3.1. By condition (9) in the same proposition, (A, B) is either a gap or a pseudo-gap of X, in particular,…”

“…(9) We use the characterization given in (7). If one of (a), (b) or (c) in (7) holds, then trivially (A, B) is neither a gap nor a pseudo-gap.…”

“…This is the case for conditions (1), (3)-(5), (7) reg , (8) and (9), by the remarks before the statement of the corollary, and where by (7) reg we denote condition (7) restricted to regular ν's.…”

“…The particular case when D is an ultrafilter over ω is due to Sanchis and Tamariz-Mascarúa [25], whose paper contains many other related theorems. More generally, we characterize D-converging sequences in GO spaces (Proposition 3.1 (7)- (9)). We also show that the D-compactness of some GO space X depends exclusively on the "decomposability spectrum" of D and on the cardinal types of (pseudo-)gaps in X, equivalently, on the existence of non converging strictly monotone sequences of certain cardinal order types (Theorem 4.1).…”