Weak $P$-points in Čech-Stone compactifications

Abstract

Let X be a nonpseudocompact space which is either nowhere ccc or nowhere of weight < 2'. Then fiX X contains a point x which is a weak P-point of /3X, i.e. if F C f3X fx} is countable, then x a F. In addition, under MA, if Xis any nonpseudocompact space, then fiX X contains a point x such that whenever F C fiX {x3 is countable and nowhere dense, then x a F. 0. Introduction. All spaces are completely regular and X* denotes 3X X. Frolik's [F] proof that the Cech-Stone remainder of a nonpseudocompact space is not homogeneous is elegant and ingenious, but does not give points which are topologically distinct by an obvious reason. When Kunen [K1] proved that there are Rudin-Keisler incomparable points in fc, Frolik's ideas were used by Comfort [C] and van Douwen [vD2] to show that, respectively, no infinite compact space in which countable discrete subspaces are C*-embedded is homogeneous and that /3X is not homogeneous for any nonpseudocompact space X. These results showed that certain spaces are not homogeneous but not why they are not homogeneous. This suggests an obvious question which has been considered by several authors during the last years. The first promising partial answers to this question were obtained by van Douwen [vD3], who showed that each nonpseudocompact space of countable 'r-weight has a remote point (independently, this was also shown by Chae and Smith [CS]), and that remote points can be used to show that certain Cech-Stone remainders are not homogeneous. Unfortunately, it was soon clear, by examples in van Douwen and van Mill [vDvM1], that this line of attack did not solve the entire problem, since many spaces do not have remote points. Earlier, van Douwen [vD1] had introduced far points and c-far points and showed that these points exist in certain Cech-Stone remainders and that they also could be used to show that a restrictive class of Cech-Stone remainders is not homogeneous. Each remote point is a far point and each far point is an co-far point when we restrict our attention to spaces without isolated points. When it was shown that not every nonpseudocompact space has a remote point, van Douwen's [vD1] question whether every nonpseudocompact space without isolated points has an w-far point again became interesting. The examples in [vDvMl] did not clarify this question since they all have far points. From [vDvM2] it Received by the editors November 12, 1980 and, in revised form, June 24, 1981. 1980 Mathematics Subject Classification. Primary 54D35.