Weak P-point

A point in a topological space that is not an accumulation point of any countable subset of the space. Every -point is a weak -point. Weak -points were introduced by K. Kunen [a2] in his proof that , the remainder of the Stone–Čech compactification of the natural numbers, is not homogeneous (cf. Cech-Stone compactification of omega). In fact, Kunen proved that contains points that are very much like -points, so-called -OK points: A point is -OK if for every sequence of neighbourhoods there is a -sequence of neighbourhoods such that whenever has elements. A -OK point cannot be an accumulation point of any set that satisfies the countable chain condition (cf. Chain condition), hence it is not an accumulation point of any countable set either.

Weak -points and similar types of points have been used to give so-called "effective" proofs that many spaces are not homogeneous [a3], [a4]. These proofs are generally considered simpler than the proof by Z. Frolík [a1] of the non-homogeneity of , which takes a countably infinite discrete subset of (whose closure is homeomorphic to ) and shows that a point of cannot be mapped by any auto-homeomorphism of to its copy in the closure of . "Simpler" does not necessarily mean that the proof is easier, but that the properties used to distinguish the point are of a simpler nature.

References

[a1]	Z. Frolík, "Non-homogeneity of $\beta P-P$" Comment. Math. Univ. Carolinae , 8 (1967) pp. 705–709 Zbl 0163.44601
[a2]	K. Kunen, "Weak $P$-points in $\mathbf{N}^*$" Á. Császár (ed.) , Topology (Proc. Fourth Colloq., Budapest, 1978) , II , North-Holland (1980) pp. 741–749 Zbl 0435.54021
[a3]	J. van Mill, "Weak $P$-points in Čech–Stone compactifications" Trans. Amer. Math. Soc. , 273 (1982) pp. 657–678 Zbl 0498.54022
[a4]	J. van Mill, "Sixteen topological types in $\beta\omega-\omega$" Topol. Appl. , 13 : 1 (1982) pp. 43–57 Zbl 0489.54022