Lonely points revisited

TL;DR

This paper revisits the concept of lonely points in topological spaces, constructing examples within -star using a specific space, thereby answering an open question from previous research.

Contribution

It introduces new constructions of lonely points in -star, utilizing the space _ from Dow, Gubbi, and Szyma
44ski's work, resolving an open problem.

Findings

Constructed lonely points in -star space.

Used the space _ to demonstrate existence.

Answered an open question by P. Simon.

Abstract

In our previous paper, Lonely points, we introduced the notion of a lonely point, due to P. Simon. A point $p\in X$ is lonely if it is a limit point of a countable dense-in-itself set, not a limit point a countable discrete set and all countable sets whose limit point it is, form a filter. We use the space ${\mathcal G}_\omega$ from a paper of A. Dow, A.V. Gubbi and A. Szyma\'nski (Rigid Stone spaces within ZFC, Proc. Amer. Math. Soc. 102 (1988), no. 3, 745--748) to construct lonely points in $\omega^*$. This answers the question of P. Simon posed in our paper Lonely points (Lonely points in $\omega^*$, Topology Appl. 155 (2008), no. 16, 1766--1771).