Results on Resolvability
by
Istvan Juhasz
Renyi Institute, Budapest/UNCG

Our aim is to present resolvability related results in three different areas. All this is joint work with L. Soukup and Z. Szentmiklóssy.

I. Our method of constructing D-forced spaces, all of them dense and NODEC subspaces of certain Cantor cubes, enabled us to get a large number of examples of spaces with various resolvability properties. Thus we got e. g. the following

Theorem. For any uncountable regular cardinal l and for any k ≥ l there is a 0-dimensional CCC, NODEC, Hausdorff space X with |X| = D(X) = k that is m-resolvable for all m < l but is not l-resolvable.

II. We improve a recent result of O. Pavlov by proving the following

Theorem. If D(X) ≥ k = cf(k) and X has no discrete subspace of size k then X is k-resolvable. In particular, any space of countable spread and uncountable dispersion character is maximally resolvable.

III. We obtained a number of interesting resolvability results concerning monotonically normal spaces. Thus we have

Theorem. Every crowded monotonically normal space is w-resolvable, but a measurable k yields a monotonically normal space of dispersion character k that is not w₁-resolvable. Moreover, any monotonically normal space of cardinality less than ℵ_w is maximally resolvable.

Date received: February 22, 2006