# Article

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Keywords:
independent family; irresolvable; submaximal
Summary:
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 $c = {\aleph_{\omega_1}}$.
References:
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[Ma] Malykhin V.I.: Irresolvable countable spaces of weight less than $\frak c$. Comment. Math. Univ. Carolinae 40.1 (1999), 181-185. MR 1715211 | Zbl 1060.54500

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