Ultrafilter convergence in ordered topological spaces

We characterize ultrafilter convergence and ultrafilter compactness in linearly ordered and generalized ordered topological spaces. In such spaces, and for every ultrafilter $D$, the notions of $D$-compactness and of $D$-pseudocompactness are equivalent. Any product of initially $\lambda$-compact generalized ordered topological spaces is still initially $\lambda$-compact. On the other hand, preservation under products of certain compactness properties are independent from the usual axioms for set theory.Comment: v. 2: some additions and some improvement

(kappa,theta)-weak normality

We characterize the situation of small cardinality for a product of cardinals divided by an ultrafilter. We develop the notion of weak normality. We include an application to Boolean Algebras.Comment: 13 page

Compactness of powers of \omega

We characterize exactly the compactness properties of the product of \kappa\ copies of the space \omega\ with the discrete topology. The characterization involves uniform ultrafilters, infinitary languages, and the existence of nonstandard elements in elementary estensions. We also have results involving products of possibly uncountable regular cardinals.Comment: v2 slightly improve

Generalizing random real forcing for inaccessible cardinals

The two parallel concepts of "small" sets of the real line are meagre sets and null sets. Those are equivalent to Cohen forcing and Random real forcing for aleph_0^aleph_0; in spite of this similarity, the Cohen forcing and Random Real Forcing have very different shapes. One of these differences is in the fact that the Cohen forcing has an easy natural generalization for lambda 2 while lambda greater than aleph 0, corresponding to an extension for the meagre sets, while the Random real forcing didn't seem to have a natural generalization, as Lebesgue measure doesn't have a generalization for space 2 lambda while lambda greater than aleph 0. In work [1], Shelah found a forcing resembling the properties of Random Real Forcing for 2 lambda while lambda is a weakly compact cardinal. Here we describe, with additional assumptions, such a forcing for 2 lambda while lambda is an Inaccessible Cardinal; this forcing is less than lambda-complete and satisfies the lambda^+-c.c hence preserves cardinals and cofinalities, however unlike Cohen forcing, does not add an undominated real