Abstract:We show that the small cardinal number $i = \min \{\vert \Cal A \vert : \Cal A$ is a maximal independent family\} has the following topological characterization: $i = \min \{\kappa \leq c: \{0,1\}^{\kappa }$ has a dense irresolvable countable subspace\}, where $\{0,1\}^{\kappa }$ denotes the Cantor cube of weight $\kappa $. As a consequence of this result, we have that the Cantor cube of weight $c$ has a dense countable submaximal subspace, if we assume (ZFC plus $i=c$), or if we work in the Bell-Kunen model, where $i = {\aleph _{1}}$ and $c = {\aleph _{\omega _1}}$.
Keywords: independent family, irresolvable, submaximal
AMS Subject Classification: Primary 54A05, 54A35, 54C25; Secondary 54A25, 54B05, 54B10