Abstract
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: If there exists a countably-centered free ultrafilter, then there are dense in themselves T1-spaces whose product is irresolvable. Any sets X and Y support irresolvable T1-topologies whose product is maximally resolvable. Assuming the continuum hypothesis, an ultrafilter whose cartesian square is dominated by only three ultrafilters is constructed on a countable set. 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 items.
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