P-point

As defined in [a1], a point in a completely-regular space $X$ at which any prime ideal of the ring $C(X)$ of real-valued continuous functions 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 [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 ultrafilters 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 [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

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