Spaces of Remote Points

TL;DR

This paper investigates the topological structure of remote points in metrizable spaces, characterizing when their spaces are homeomorphic and exploring conditions under which remote point spaces share topological properties.

Contribution

It provides new characterizations of remote points for certain spaces and introduces the concept of cellular type to compare remote point spaces.

Findings

Remote points of rationals and irrationals characterized

Homeomorphism conditions based on topological properties established

Open dense homeomorphic subspaces identified for specific spaces

Abstract

Given a Tychonoff space $X$, let $\varrho(X)$ be the set of remote points of $X$. We view $\varrho(X)$ as a topological space. In this paper we assume that $X$ is metrizable and ask for conditions on $Y$ so that $\varrho(X)$ is homeomorphic to $\varrho(Y)$. This question has been studied before by R. G. Woods and C. Gates. We give some results of the following type: if $X$ has topological property $\mathbf{P}$ and $\varrho(X)$ is homeomorphic to $\varrho(Y)$, then $Y$ also has $\mathbf{P}$. We also characterize the remote points of the rationals and irrationals up to some restrictions. Further, we show that $\varrho(X)$ and $\varrho(Y)$ have open dense homeomorphic subspaces if $X$ and $Y$ are both nowhere locally compact, completely metrizable and share the same cellular type, a cardinal invariant we define.