Spaces of Remote Points
Rodrigo Hern\'andez-Guti\'errez, Michael Hru\v{s}\'ak, Angel, Tamariz-Mascar\'ua

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 , let be the set of remote points of . We view as a topological space. In this paper we assume that is metrizable and ask for conditions on so that is homeomorphic to . This question has been studied before by R. G. Woods and C. Gates. We give some results of the following type: if has topological property and is homeomorphic to , then also has . We also characterize the remote points of the rationals and irrationals up to some restrictions. Further, we show that and have open dense homeomorphic subspaces if and are both nowhere locally compact, completely metrizable and share the same cellular type, a cardinal invariant we define.
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Taxonomy
TopicsAdvanced Topology and Set Theory · Homotopy and Cohomology in Algebraic Topology · Rings, Modules, and Algebras
