P-point
As defined in [a1], a point in a completely-regular space 
at which any prime ideal of the ring 
of real-valued continuous functions is maximal (cf. also Continuous function; Maximal ideal). A prime ideal 
is "at x" if 
for all 
; thus 
is a 
-point if and only if 
is the only prime ideal at 
. Equivalent formulations are:
1) if 
is a continuous function and 
, then 
vanishes on a neighbourhood of 
; and
2) every countable intersection of neighbourhoods of 
contains a neighbourhood of 
. The latter is commonly used to define 
-points in arbitrary topological spaces.
Of particular interest are 
-points in the space 
, 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 
has 
-points if the continuum hypothesis is assumed; this showed that 
cannot be proved homogeneous (cf. also Homogeneous space), because not every point in an infinite compact space can be a 
-point. Points of 
are identified with free ultrafilters on the set 
(cf. also Ultrafilter). A point or ultrafilter 
is a 
-point if and only if for every sequence 
of elements of 
there is an element 
of 
such that 
for all 
, where 
means that 
is finite. Equivalently, 
is a 
-point if and only if for every partition 
of 
either there is an 
such that 
or there is a 
such that 
is finite for all 
. S. Shelah [a3] constructed a model of set theory in which 
has no 
-points, thus showing that Rudin's theorem is not definitive.
There is continued interest in 
-point ultrafilters because of their combinatorial properties; e.g., 
is a 
-point if and only if for every function 
there is an element 
of 
such that 
is a converging sequence (possibly to 
or 
).
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  -point independence theorem" Israel J. Math. , 43 (1982) pp. 28–48 |
P-point. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=P-point&oldid=42351