View source for P-point

As defined in [[#References|[a1]]], a point in a [[completely-regular space]] $X$ at which any [[prime ideal]] of the ring $C(X)$ of real-valued [[continuous function]]s is maximal. A prime ideal $\mathfrak{P}$ is  "at $x$"  if $f(x)=0$ for all $f \in \mathfrak{P}$; thus $x$ is a $P$-point if and only if $\mathfrak{M}_x = \{f \in C(X) : f(x) = 0 \}$ is the only prime ideal at $x$. Equivalent formulations are:

1) if $f$ is a continuous function and $f(x)=0$, then $f$ vanishes on a neighbourhood of $x$; and

2) every countable intersection of neighbourhoods of $x$ contains a neighbourhood of $x$. The latter is commonly used to define $P$-points in arbitrary topological spaces.

Of particular interest are $P$-points in the space $\mathbf{N}^* = \beta\mathbf{N}\setminus\mathbf{N}$, the remainder in the [[Stone–Čech compactification]] of the space of natural numbers (cf. [[Cech-Stone compactification of omega]]). This is so because W. Rudin [[#References|[a2]]] proved that the space $\mathbf{N}^*$ has $P$-points if the [[continuum hypothesis]] is assumed; this showed that $\mathbf{N}^*$ cannot be proved homogeneous (cf. also [[Homogeneous space]]), because not every point in an infinite compact space can be a $P$-point. Points of $\mathbf{N}^*$ are identified with free [[ultrafilter]]s on the set $\mathbf{N}$. A point or ultrafilter $u$ is a $P$-point if and only if for every sequence $(U_n)$ of elements of $u$ there is an element $U$ of $u$ such that $U \stackrel{*}{\subseteq} U_n$ for all $n$, where $A \stackrel{*}{\subseteq} B$ means that $A\setminus B$ is finite. Equivalently, $u$ is a $P$-point if and only if for every partition $\{A_n\}$ of $\mathbf{N}$ either there is an $n$ such that $A_n \in u$ or there is a $U\in u$ such that $U \cap A_n$ is finite for all $n$. S. Shelah [[#References|[a3]]] constructed a model of set theory in which $\mathbf{N}^*$ has no $P$-points, thus showing that Rudin's theorem is not definitive.

There is continued interest in $P$-point ultrafilters because of their combinatorial properties; e.g., $u$ is a $P$-point if and only if for every function $f : \mathbf{N} \rightarrow \mathbf{R}$ there is an element $U$ of $u$ such that $f[U]$ is a converging sequence (possibly to $\infty$ or $-\infty$).

====References====
<table>
<TR><TD valign="top">[a1]</TD> <TD valign="top">  L. Gillman,   M. Henriksen,   "Concerning rings of continuous functions"  ''Trans. Amer. Math. Soc.'' , '''77'''  (1954)  pp. 340–362</TD></TR>
<TR><TD valign="top">[a2]</TD> <TD valign="top">  W. Rudin,   "Homogeneity problems in the theory of Čech compactifications"  ''Duke Math. J.'' , '''23'''  (1956)  pp. 409–419; 633</TD></TR>
<TR><TD valign="top">[a3]</TD> <TD valign="top">  E. Wimmers,   "The Shelah $P$-point independence theorem"  ''Israel J. Math.'' , '''43'''  (1982)  pp. 28–48</TD></TR>
</table>

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