A Separable Space with no Remote Points

Abstract

In the model obtained by adding ${\omega _2}$ side-by-side Sacks reals to a model of ${\mathbf {CH}}$, 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 ${\omega _2}$ such that every uncountable subset has density ${\omega _1}$. Furthermore assuming the existence of a measurable cardinal $\kappa$ with ${2^\kappa } = {\kappa ^ + }$, a space $X$ is produced with no isolated points but with remote points in $\upsilon X - X$. It is also shown that a pseudocompact space does not have remote points.