View source for Weak P-point

A point in a [[Topological space|topological space]] that is not an [[Accumulation point|accumulation point]] of any countable subset of the space. Every [[P-point|<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130030/w1300302.png" />-point]] is a weak <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130030/w1300303.png" />-point. Weak <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130030/w1300304.png" />-points were introduced by K. Kunen [[#References|[a2]]] in his proof that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130030/w1300305.png" />, the remainder <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130030/w1300306.png" /> of the [[Stone–Čech compactification|Stone–Čech compactification]] of the natural numbers, is not homogeneous (cf. [[Cech-Stone compactification of omega]]). In fact, Kunen proved that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130030/w1300307.png" /> contains points that are very much like <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130030/w1300308.png" />-points, so-called <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130030/w1300309.png" />-OK points: A point is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130030/w13003011.png" />-OK if for every sequence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130030/w13003012.png" /> of neighbourhoods there is a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130030/w13003013.png" />-sequence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130030/w13003014.png" /> of neighbourhoods such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130030/w13003015.png" /> whenever <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130030/w13003016.png" /> has <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130030/w13003017.png" /> elements. A <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130030/w13003018.png" />-OK point cannot be an accumulation point of any set that satisfies the countable chain condition (cf. [[Chain condition|Chain condition]]), hence it is not an accumulation point of any countable set either.

Weak <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130030/w13003019.png" />-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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130030/w13003020.png" />, which takes a countably infinite discrete subset <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130030/w13003021.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130030/w13003022.png" /> (whose closure is homeomorphic to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130030/w13003023.png" />) and shows that a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130030/w13003024.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130030/w13003025.png" /> cannot be mapped by any auto-homeomorphism of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130030/w13003026.png" /> to its copy in the closure of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130030/w13003027.png" />.  "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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130030/w13003028.png" />"  ''Comment. Math. Univ. Carolinae'' , '''8'''  (1967)  pp. 705–709</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  K. Kunen,   "Weak <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130030/w13003029.png" />-points in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130030/w13003030.png" />"  Á. Császár (ed.) , ''Topology (Proc. Fourth Colloq., Budapest, 1978)'' , '''II''' , North-Holland  (1980)  pp. 741–749</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  J. van Mill,   "Weak <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130030/w13003031.png" />-points in Čech–Stone compactifications"  ''Trans. Amer. Math. Soc.'' , '''273'''  (1982)  pp. 657–678</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  J. van Mill,   "Sixteen topological types in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130030/w13003032.png" />"  ''Topol. Appl.'' , '''13''' :  1  (1982)  pp. 43–57</TD></TR></table>