What Is Left of CH After You Add Cohen Reals?
by
István Juhász
Coauthors: L. Soukup, Z. Szentmiklossy

In this talk we formulate a principle concerning elementary submodels of size \omega₁, that we call CH* . It is valid in any generic extension obtained by adding an arbitrary number of Cohen reals to a ground model satisfying CH, in particular, CH implies CH*. It turns out that CH* has a large number of interesting topological consequences, many but not all of which were known to follow from CH. Here is a (partial) list of these consequences:

- - Every initially \omega₁ compact countably tight T₃ space is compact.

- - Let X be a countably tight compact T₂ space, then

(i) if S is a G_\delta-dense subset of X then every point of X is the limit of a convergent \omega₁ sequence of points of S;

(ii) if Y is any subspace of X with s(Y) < \omega₂ then h(Y) < \omega₂ as well;

(iii) X contains no complete binary tree of closed sets of height \omega₂.

- - If X is a compact T₂ space with small diagonal then X is metrisable.

All this supports the long standing philosophical observation that CH is not simply a quantitative statement about the the size of the power set of \omega but also a qualitative one deeply affecting its structure.

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-48.