Type: Article

Independent families and resolvability

S. García-Ferreira; M. Hrušák; Á. Tamariz-Mascarúa;

Abstract:

Let tau and gamma be infinite cardinal numbers with tau <= gamma. A subset Y of a space X is called C-tau-compact if f vertical bar Y vertical bar is compact for every continuous function f : X -> R-tau. We prove that every C-tau-compact dense subspace of a product of gamma non-trivial compact spaces each of them of weight <= tau is 2(tau)-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 sigma-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 lambda), with lambda = 2(omega 1), which is not maximally resolvable. Moreover, we improve a result by W. Hu (2006) [17] by showing that if maximal theta-independent families do not exist, then every dense subset of square(theta) {0, 1}(gamma) is omega-resolvable for a regular cardinal number theta with omega(1) <= theta <= gamma. Finally, if there are no maximal independent families on kappa of cardinality gamma, then every Baire dense subset of {0,1}(gamma) of cardinality <= kappa and every Baire dense subset of [0, 1](gamma) of cardinality <= kappa are omega-resolvable.