Martin's axiom and pathological points in β X⧹ X

Abstract

P(c), a well-known consequence of Martin's Axiom, implies that if X is a nonpseudocompact space with countable π-weight, then there is a point p in β X⧹ X such that: (1) p is not in the closure of any nowhere dense subset of X (i.e., p is a remote point), (2) p is not simultaneously in the closure of any two disjoint open subsets of β X (i.e., β X is extremally disconnected at p), and (3) if X is locally compact, then each intersection of fewer than c neighborhoods of p in β X⧹ X is again a neighborhood of p (i.e., p is a P c-point). The construction of such points depends on the fact that P(c) is equivalent to the proposition that if X is any noncompact realcompact locally compact space with countable π-weight, then each nonempty intersection of fewer than c open subsets of β X⧹ X has a nonempty interior. The condition on the π-weight is essential: There is a noncompact σ-compact (hence realcompact) locally compact separable space M such that each point of β M⧹ M is contained in an intersection of ω 1 open sets, which has empty interior. M has the remarkable property that β M⧹ M and β N ⧹ N are homeomorphic under CH, the Continuum Hypothesis, but not under P(c) + ¬op;CH. P(c) also implies that if X is a noncompact realcompact locally compact space with countable π-weight, then each point of β X⧹ X is simultaneously in the closure of c pairwise disjoint open subsets of β X⧹ X (i.e., is a c-point of β X⧹ X). Finally, if ƒ: β Q →β R is the (unique) continuous map such that ƒ↾ Q = id Q , then P(c) implies the existence of a subset E of β R ⧹ R with cardinality 2 c, such that ƒ ←[{x}] is a one-point set for each x ϵ E(hence ƒ↾( Q∪ƒ ←[E]) is a homeomorphism).