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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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