A. Tamariz-Mascarua         Independent Families and Resolvability

Joint work with S. Garcia-Ferreira and M. Hrusak

Let $\tau$ and $\gamma$ be infinite cardinal numbers with $\tau \leq \gamma$. A subset $Y$ of a space $X$ is called $C_\tau$-compact if $f[Y]$ is compact for every continuous function
$f : X \to \R^\tau$. We prove that every
$C_\tau$-compact dense subspace of a product of $\gamma$ non-trivial compact spaces each of them of weight $\leq \tau$ is $2^\tau$-resolvable. In particular, every
pseudocompact dense subspace of a product of non-trivial metrizable compact spaces
is $\frak{c}$-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 \cite{wahu} by showing that if maximal $\theta$-independent families do not exist, then
every dense subset of $\Box_{\theta}\{0,1\}^\gamma$ is $\omega$-resolvable
for a regular cardinal number $\theta$ with $\omega_1 \leq \theta \leq \gamma$. Finally, if there are no maximal independent
families on $\kappa$ of cardinality $\gamma$, then every Baire dense subset of $\{0,1\}^\gamma$of cardinality $\leq \kappa$ and every Baire dense subset of $[0,1]^\gamma$ of cardinality $\leq \kappa$ are $\omega$-resolvable.