Topological characterization of the small cardinal $i$

Franco-Filho, Antonio de Padua

Commentationes Mathematicae Universitatis Carolinae, Tome 44 (2003) no. 4, pp. 745-750.

Résumé

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 $c = {\aleph_{\omega_1}}$.

Détail

Classification : 54A05, 54A25, 54A35, 54B05, 54B10, 54C25
Keywords: independent family; irresolvable; submaximal