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independent family; irresolvable; submaximal
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 $c = {\aleph_{\omega_1}}$.
[ASTTW] Alas O.T., Sanchis M., Tkačenko M.G., Tkachuk V.V., Wilson R.G.: Irresolvable and submaximal spaces: homogeneity vs ${\sigma}$-discreteness and new ZFC examples. Topology Appl. 107 (2000), 259-278. MR 1779814
[BK] Bell M., Kunen K.: On the Pi-character of ultrafilters. C.R. Math. Rep. Acad. Sci. Canada 3 (1981), 351-356. MR 0642449 | Zbl 0475.54001
[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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