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. 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