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We show that the small cardinal number $i=min\left\{\right|𝒜|:𝒜$ is a maximal independent family} has the following topological characterization: $i=min\{\kappa \le c:{\{0,1\}}^{\kappa }$ 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 $c={\aleph }_{{\omega }_1}$.