## 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.
Full Text:

### References:

  van Douwen, E.K., Remote points, Dissertationes math., 188, (1981) · Zbl 0525.54018  van Douwen, E.K., Applications of maximal topologies, Topology appl., 51, 125-139, (1993) · Zbl 0845.54028  Fine, N.J.; Gillman, L., Remote points in βR, Proc. amer. math. soc., 13, 29-36, (1962) · Zbl 0118.17802  Frolík, Z., Sums of ultrafilters, Bull. amer. math. soc., 73, 87-91, (1967) · Zbl 0166.18602  Hewitt, E., A problem of set-theoretic topology, Duke math. J., 10, 309-333, (1943) · Zbl 0060.39407  Kunen, K., Some points in βN, Math. proc. Cambridge philos. soc., 80, 385-398, (1976) · Zbl 0345.02047  Kunen, K., Weak P-points in $$N^\ast$$, Colloq. math. soc. János bolyai, 23, 741-749, (1980)  van Mill, J., Sixteen types in $$\beta \omega - \omega$$, Topology appl., 13, 43-57, (1982) · Zbl 0489.54022  van Mill, J., Introduction to βω, (), 59-81 · Zbl 0555.54004  Rudin, W., Homogeneity problems in the theory of čech compactifications, Duke math. J., 23, 409-419, (1956) · Zbl 0073.39602  Simon, P., Applications of independent linked families, Colloq. math. soc. János bolyai, 41, 561-580, (1985) · Zbl 0615.54004
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.