Combinatorial and model-theoretical principles related to regularity of ultrafilters and compactness of topological spaces. VI

TL;DR

This paper explores the principles underlying the regularity of ultrafilters and the compactness properties of topological spaces, focusing on accumulation points, uniformity, and various forms of compactness in product spaces.

Contribution

It generalizes previous results on accumulation points and compactness, extending the theory to include uniformity and $ ext{ extbackslash kappa}$-$( ext{ extbackslash lambda})$-compactness, complementing earlier parts.

Findings

Established new connections between ultrafilter regularity and topological compactness.

Generalized results on accumulation points in product spaces.

Clarified relationships between different forms of compactness.

Abstract

We discuss the existence of complete accumulation points of sequences in products of topological spaces. Then we collect and generalize many of the results proved in Parts I, II and IV. The present Part VI is complementary to Part V to the effect that here we deal, say, with uniformity, complete accumulation points and $ \kappa $-$(\lambda)$-compactness, rather than with regularity, $[ \lambda, \mu ]$-compactness and $ \kappa $-$ (\lambda, \mu)$-compactness. Of course, if we restrict ourselves to regular cardinals, Parts V (for $ \lambda = \mu$) and Part VI essentially coincide.