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 $c = {\aleph_{\omega_1}}$.
@article{119429,
author = {Antonio de Padua Franco-Filho},
title = {Topological characterization of the small cardinal $i$},
journal = {Commentationes Mathematicae Universitatis Carolinae},
volume = {44},
year = {2003},
pages = {745-750},
zbl = {1098.54003},
mrnumber = {2062891},
language = {en},
url = {http://dml.mathdoc.fr/item/119429}
}
Franco-Filho, Antonio de Padua. Topological characterization of the small cardinal $i$. Commentationes Mathematicae Universitatis Carolinae, Tome 44 (2003) pp. 745-750. http://gdmltest.u-ga.fr/item/119429/
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