## Lonely points in $$\omega ^*$$.(English)Zbl 1152.54021

A point $$p$$ in a space $$X$$ is lonely iff (i) it is not the limit point of a countable discrete set, (ii) it is a limit point of a countable set with no isolated points, and (iii) if it is a limit point of two countable sets $$A, B$$ then it is a limit point of their intersection. The two main theorems are: 1. There is a lonely point in $$\omega^*$$ iff there is a countable, open hereditarily irresolvable, extremally disconnected, zero-dimensional space with a remote weak $$P$$-point. 2. There is a large subspace of $$\omega^*$$ with a lonely point. More specifically, there is a countable set $$S \subset \omega^*$$ and a point $$p \in\operatorname{cl}S\operatorname{so}\{p\} \cup \omega^* \setminus (\operatorname{cl} S \setminus S)$$ contains a lonely point. Whether there is a lonely point in $$\omega^*$$ remains open.

### MSC:

 54D35 Extensions of spaces (compactifications, supercompactifications, completions, etc.) 06E15 Stone spaces (Boolean spaces) and related structures 54G05 Extremally disconnected spaces, $$F$$-spaces, etc.
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