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Lonely points revisited

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 ${\mathcal G}_ω$ from a paper of A. Dow, A.V. Gubbi and A. Szymański (Rigid Stone spaces within ZFC, Proc. Amer. Math. Soc. 102 (1988), no. 3, 745--748) to construct lonely points in $ω^*$. This answers the question of P. Simon posed in our paper Lonely points (Lonely points in $ω^*$, Topology Appl. 155 (2008), no. 16, 1766--1771).