New approaches to remote points

Abstract

For a given Tychonoff space $X$, a point $p\in {\hat{I}}^2(X)\setminus X$ is called {\em remote} if $p$ is not in the closure of any nowhere dense subset of $X$. In this paper, we characterize spaces with remote points in terms of certain topological ultrafilters, measures, and compact-like properties corresponding to the ideal consisting of nowhere dense sets. It is shown that the space of remote points is homeomorphic to a subspace of the Stone space taken over the smallest Boolean algebra containing all open and nowhere dense sets. Also, we show that the space of remote points of $\mathbb{R}$ is $\ddot{I}‰$-bounded.