$ \mathcal{D}$-forced spaces: a new approach to resolvability

István Juhász, Lajos Soukup and Zoltán Szentmiklóssy

We introduce a ZFC method that enables us to build spaces (in fact special dense subspaces of certain Cantor cubes) in which we have "full control" over all dense subsets.

Using this method we are able to construct, in ZFC, for each uncountable regular cardinal $ \lambda$ a 0-dimensional $ T_2$, hence Tychonov, space which is $ \mu$-resolvable for all $ \mu < \lambda$ but not $ \lambda$-resolvable. This yields the final (negative) solution of a celebrated problem of Ceder and Pearson raised in [3]: Are $ \omega$-resolvable spaces maximally resolvable?

This method enables us to solve several other open problems as well, like [1, Question 4.4], [2, Problem 8.6], [5, Questions 3.4, 3.6, 4.5], and [4, Question 3.11].


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