Far points and discretely generated spaces

TL;DR

This paper investigates properties of discretely generated spaces, especially under PFA, and explores the existence and characteristics of far points in various topological spaces.

Contribution

It provides a partial solution to a question about discretely generated spaces under PFA and studies the cardinality and existence of far points in separable metrizable spaces.

Findings

Under PFA, the one-point compactification of certain spaces is discretely generated.

The smallest character of remote and far sets in separable metrizable spaces is analyzed.

Conditions are identified under which countable spaces have far points.

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.