Geodesic
Topological characterization of the small cardinal $i$
Commentationes Mathematicae Universitatis Carolinae, Tome 44 (2003) no. 4, pp. 745-750 Cet article a éte moissonné depuis la source Czech Digital Mathematics Library

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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 $c = {\aleph_{\omega_1}}$.
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 $c = {\aleph_{\omega_1}}$.
Classification : 54A05, 54A25, 54A35, 54B05, 54B10, 54C25
Keywords: independent family; irresolvable; submaximal 
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     author = {Franco-Filho, Antonio de Padua},
     title = {Topological characterization of the small cardinal $i$},
     journal = {Commentationes Mathematicae Universitatis Carolinae},
     pages = {745--750},
     year = {2003},
     volume = {44},
     number = {4},
     mrnumber = {2062891},
     zbl = {1098.54003},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/CMUC_2003_44_4_a18/}
}
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Franco-Filho, Antonio de Padua. Topological characterization of the small cardinal $i$. Commentationes Mathematicae Universitatis Carolinae, Tome 44 (2003) no. 4, pp. 745-750. http://geodesic.mathdoc.fr/item/CMUC_2003_44_4_a18/