View source for Weak P-point

A point in a [[topological space]] that is not an [[accumulation point]] of any countable subset of the space. Every [[P-point|$P$-point]] is a weak $P$-point. Weak $P$-points were introduced by K. Kunen [[#References|[a2]]] in his proof that $\mathbf{N}^*$, the remainder $\beta\mathbf{N}\setminus\mathbf{N}$ of the [[Stone–Čech compactification]] of the natural numbers, is not homogeneous (cf. [[Cech-Stone compactification of omega]]). In fact, Kunen proved that $\mathbf{N}^*$ contains points that are very much like $P$-points, so-called $\mathfrak{c}$-OK points: A point is $\mathfrak{c}$-OK if for every sequence $(U_n)_{n\in\mathbf{N}}$ of neighbourhoods there is a $\mathfrak{c}$-sequence $(V_\alpha)_{\alpha<\mathfrak{c}}$ of neighbourhoods such that $\cap_{\alpha\in F} V_\alpha \subseteq U_n$ whenever $F$ has $n$ elements. A $\mathfrak{c}$-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 $P$-points and similar types of points have been used to give so-called  "effective"  proofs that many spaces are not homogeneous [[#References|[a3]]], [[#References|[a4]]]. These proofs are generally considered simpler than the proof by Z. Frolík [[#References|[a1]]] of the non-homogeneity of $\mathbf{N}^*$, which takes a countably infinite discrete subset $D$ of $\mathbf{N}^*$ (whose closure is homeomorphic to $\beta\mathbf{N}$) and shows that a point $p$ of $\mathbf{N}^*$ cannot be mapped by any auto-homeomorphism of $\mathbf{N}^*$ to its copy in the closure of $D$.  "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====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  Z. Frolík,   "Non-homogeneity of $\beta P-P$"  ''Comment. Math. Univ. Carolinae'' , '''8'''  (1967)  pp. 705–709  {{ZBL|0163.44601}}</TD></TR>
<TR><TD valign="top">[a2]</TD> <TD valign="top">  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}}</TD></TR>
<TR><TD valign="top">[a3]</TD> <TD valign="top">  J. van Mill,   "Weak $P$-points in Čech–Stone compactifications"  ''Trans. Amer. Math. Soc.'' , '''273'''  (1982)  pp. 657–678  {{ZBL|0498.54022}}</TD></TR>
<TR><TD valign="top">[a4]</TD> <TD valign="top">  J. van Mill,   "Sixteen topological types in $\beta\omega-\omega$"  ''Topol. Appl.'' , '''13''' :  1  (1982)  pp. 43–57  {{ZBL|0489.54022}}</TD></TR>
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