More Results on Regular Ultrafilters in ZFC

We prove, in ZFC alone, some new results on regularity and decomposability of ultrafilters. We also list some problems, and furnish applications to topological spaces and to extended logics.Comment: 57 page

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