Remote point

In general topology, a remote point is a point ${\displaystyle p}$ that belongs to the Stone–Čech compactification ${\displaystyle \beta X}$ of a Tychonoff space ${\displaystyle X}$ but that does not belong to the topological closure within ${\displaystyle \beta X}$ of any nowhere dense subset of ${\displaystyle X}$.[1]

Let ${\displaystyle \mathbb {R} }$ be the real line with the standard topology. In 1962, Nathan Fine and Leonard Gillman proved that, assuming the continuum hypothesis:

There exists a point ${\displaystyle p}$ in ${\displaystyle \beta \mathbb {R} }$ that is not in the closure of any discrete subset of ${\displaystyle \mathbb {R} }$ ...[2]

Their proof works for any Tychonoff space that is separable and not pseudocompact.[1]

Chae and Smith proved that the existence of remote points is independent, in terms of Zermelo–Fraenkel set theory, of the continuum hypothesis for a class of topological spaces that includes metric spaces.[3] Several other mathematical theorems have been proved concerning remote points.[4][5]