Weak P-points, 0-points and Some Other Points of Compactifications
by
A. A. Gryzlov

We regard some types of points of Cech-Stone compactification \beta\tau of a discrete space \tau, namely weak p-points, matrix points, 0-points, and investigate the relations between these types of points. The following two theorems demonstrate the style of the results.

Theorem There is a point x in \omega^* such that:

1. x is a weak p-point of \omega^*;
2. x in [F] \F for some F subset or equal \omega^*, c(F) = \omega, F consists of non 0-points;
3. x not in [F] \F for all F subset or equal \omega^*, c(F) <= \omega, F consists of 0-points.

Let M'_\tau = \cup { U'_\gamma: \gamma < \tau} be a union of a disjoint subsets of \tau, | U'_\gamma | = \tau, M_\tau = \cup { U_\gamma = [U'_\gamma] \\tau: \gamma < \tau},

And let [(M_\tau)\tilde] = [M_\tau] \M_\tau. U(\tau) - a set of regular ultrafilters on \tau.

Theorem In \tau^* there are:

1. a matrix point \xi in [(M_\tau)\tilde] \cap U(\tau) of \tau^* such that \xi not in [ \cup { F_\gamma : \gamma in \tau}] if F_\gamma subset or equal U_\gamma, c(F_\gamma) <= \omega;
2. a matrix point \xi in [(M_\tau)\tilde] \cap U(\tau) of \tau^* such that \xi in [ \cup { F_\gamma : \gamma in \tau}] for some \cup { F_\gamma : \gamma in \tau}, where F_\gamma subset or equal U_\gamma and c(F_\gamma) = \omega, but \xi not in [ \cup { D_\gamma : \gamma in \tau}] if D_\gamma subset or equal U_\gamma and | D_\gamma | = \omega;
3. a matrix point \xi in [(M_\tau)\tilde] \cap U(\tau) of \tau^* such that \xi in [ \cup {F_\gamma : \gamma in \tau}] for some \cup { F_\gamma : \gamma in \tau} such that F_\gamma subset or equal U_\gamma and | F_\gamma | = \omega, but \xi not in [ \cup { D_\gamma: \gamma in \tau}] if D_\gamma subset or equal U_\gamma and |D_\gamma | < \omega;
4. a matrix point \xi in [(M_\tau)\tilde] \cap U(\tau)) such that \xi in [ \cup { D_\gamma : \gamma in \tau}] where | D_\gamma \cap U_\gamma | = 1.

We regard also the R-F-incomparable points of the types, mentioned above.

Date received: June 24, 1996

Copyright © 1996 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Mathematical Conference Abstracts. Document # caah-26.