Far points and discretely generated spaces

Abstract

We give a partial solution to a question by Alas, Junqueria and Wilson by proving that under PFA the one-point compactification of a locally compact, discretely generated and countably tight space is also discretely generated. After this, we study the cardinal number given by the smallest possible character of remote and far sets of separable metrizable spaces. Finally, we prove that in some cases a countable space has far points.0.1. Question [2] Let X be a locally compact and discretely generated space. Is the one-point compactification of X also discretely generated?So far, this question has been answered consistently in the negative. First countable counterexamples have been constructed using CH [2] or the existence of a Souslin line [1]. Later, it was shown in [17] that a similar construction could be carried out assuming the cardinal equation p = cof (M); the resulting space has an Date: October 15, 2018. 2010 Mathematics Subject Classification. 54D35, 54A25, 54G12, 54D80, 03E10, 03E75.