A topological space \(X\) is \(\theta\)-resolvable if it possesses \(\theta\)-many mutually disjoint dense subspaces and is resolvable if it is \(2\)-resolvable; \(X\) is maximally resolvable if it is \(\Delta(X)\)-resolvable, where \(\Delta(X)\) denotes the minimum cardinality of the non-empty open subsets of \(X\). A space is almost-\(\omega\)-resolvable if it possesses a countable cover \(\{X_n:n\in\omega\}\) such that for each \(m\in\omega\), int\((\bigcup\{X_k:k\leq m\})=\emptyset\). It is known that resolvability and almost-\(\omega\)-resolvability are equivalent in the class of Baire spaces, that under \(V=L\), every Baire space is \(\omega\)-resolvable and that countably compact Tychonoff spaces are \(\omega\)-resolvable.
The natural questions motivating this paper are then: Is every Tychonoff pseudocompact space resolvable (respectively, almost-\(\omega\)-resolvable) in ZFC? Among other results, it is shown that every pseudocompact dense subspace of a product of non-trivial metrizable spaces is \(2^\omega\)-resolvable; using a maximal \(\sigma\)-independent family on \(\lambda=2^{\omega_1}\) – the existence of which is equiconsistent with that of a measurable cardinal – an example is given of a dense pseudocompact subspace of \(\{0,1\}^{2^\lambda}\) which is \(\omega_1\)-resolvable but not maximally resolvable. In the final section the authors prove that if there is no maximal independent family of cardinality \(\lambda\) on a cardinal \(\kappa\), then every Baire dense subset of \([0,1]^\lambda\) and every Baire dense subset of \(\{0,1\}^\lambda\) of cardinality at most \(\kappa\) is \(\omega\)-resolvable.