Independent families and resolvability

Let τ and γ be infinite cardinal numbers with τ ⩽ γ . A subset Y of a space X is called C τ -compact if f [ Y ] is compact for every continuous function f : X → R τ . We prove that every C τ -compact dense subspace of a product of γ non-trivial compact spaces each of them of weight ⩽τ is 2 τ -resolvable. In particular, every pseudocompact dense subspace of a product of non-trivial metrizable compact spaces is c -resolvable. As a consequence of this fact we obtain that there is no σ-independent maximal independent family. Also, we present a consistent example, relative to the existence of a measurable cardinal, of a dense pseudocompact subspace of { 0 , 1 } 2 λ , with λ = 2 ω 1 , which is not maximally resolvable. Moreover, we improve a result by W. Hu (2006) [17] by showing that if maximal θ-independent families do not exist, then every dense subset of □ θ { 0 , 1 } γ is ω-resolvable for a regular cardinal number θ with ω 1 ⩽ θ ⩽ γ . Finally, if there are no maximal independent families on κ of cardinality γ, then every Baire dense subset of { 0 , 1 } γ of cardinality ⩽κ and every Baire dense subset of [ 0 , 1 ] γ of cardinality ⩽κ are ω-resolvable.