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:
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.
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