Difference between revisions of "Weak P-point"
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| − | A point in a [[ | + | 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 | + | 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. |
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Latest revision as of 20:01, 21 November 2017
A point in a topological space that is not an accumulation point of any countable subset of the space. Every $P$-point is a weak $P$-point. Weak $P$-points were introduced by K. Kunen [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 [a3], [a4]. These proofs are generally considered simpler than the proof by Z. Frolík [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
| [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 |
How to Cite This Entry:
Weak P-point. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Weak_P-point&oldid=42356
Weak P-point. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Weak_P-point&oldid=42356
This article was adapted from an original article by K.P. Hart (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article