Products of ultrafilters and irresolvable spaces

V. I. Malykhin

Sbornik. Mathematics, Tome 19 (1973) no. 1, pp. 105-115

Résumé

A space dense in itself is said to be $k$-resolvable if there exists a system of cardinality $k$ of disjoint dense subsets. The main results of the paper can be formulated as follows: 1. If there exists a countably-centered free ultrafilter, then there are dense in themselves $T_1$-spaces whose product is irresolvable. 2. Any sets $X$ and $Y$ support irresolvable $T_1$-topologies whose product is maximally resolvable. 3. Assuming the continuum hypothesis, an ultrafilter whose cartesian square is dominated by only three ultrafilters is constructed on a countable set. 4. If a set of uncountable cardinality supports an ultrafilter whose square is dominated by exactly three ultrafilters, then its cardinality is measurable. A number of problems are posed. Bibliography: 9 titles.