# [FOM] P-point/notation

Andreas Blass ablass at umich.edu
Fri May 22 17:12:30 EDT 2009

```	Jan Pax asked why certain ultrafilters on omega are called P-points,
and Thomas Forster provided an approximation to the answer.  The
terminology comes from general topology, where a point x in a space X
is called a P-point iff every intersection of countably many
neighborhoods of x is a neighborhood (not necessarily an open
neighborhood) of x.  There remains the question of why the letter P
is used for this purpose in topology.  Half of the answer is that P
is the first letter of "prime ideal".  The other half is the
following connection between prime ideals and P-points.
For any topological space X, the continuous, real-valued functions
on X form a commutative ring C(X), with the operations of pointwise
addition and multiplication.  For any point x in X, the continuous
functions that vanish at x form a prime (in fact maximal) ideal P_x
in C(X).  In a sufficiently nice space X (I believe complete
regularity suffices, but I haven't checked this), x is a P-point iff
P_x does not properly include any other prime ideal.

Andreas Blass
```