A separable space with no remote points

Abstract

In the model obtained by adding ω 2 {\omega _2} side-by-side Sacks reals to a model of C H {\mathbf {CH}} , there is a separable nonpseudocompact space with no remote points. To prove this it is also shown that in this model the countable box product of Cantor sets contains a subspace of size ω 2 {\omega _2} such that every uncountable subset has density ω 1 {\omega _1} . Furthermore assuming the existence of a measurable cardinal κ \kappa with 2 κ = κ + {2^\kappa } = {\kappa ^ + } , a space X X is produced with no isolated points but with remote points in υ X − X \upsilon X - X . It is also shown that a pseudocompact space does not have remote points.