Type: Article

Spaces of remote points

Rodrigo Hernández-Gutiérrez; Michael Hrušák; Angel Tamariz-Mascarúa;

Abstract:

Given a Tychonoff space X, let rho(X) be the set of remote points of X. We view rho(X) as a topological space. In this paper we assume that X is metrizable and ask for conditions on Y so that rho(X) is homeomorphic to rho(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 P and rho(X) is homeomorphic to rho(Y), then Y also has P. We also characterize the remote points of the rationals and irrationals up to some restrictions. Further, we show that rho(X) and rho(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.