Topological characterization of the small cardinal $i$

Franco-Filho, Antonio de Padua. "Topological characterization of the small cardinal $i$." Commentationes Mathematicae Universitatis Carolinae 44.4 (2003): 745-750. <http://eudml.org/doc/249187>.

@article{Franco2003,
abstract = {We show that the small cardinal number $i = \min \lbrace \vert \mathcal \{A\} \vert : \mathcal \{A\}$ is a maximal independent family\} has the following topological characterization: $i = \min \lbrace \kappa \le c: \lbrace 0,1\rbrace ^\{\kappa \}$ has a dense irresolvable countable subspace\}, where $\lbrace 0,1\rbrace ^\{\kappa \}$ 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 $c = \{\aleph _\{\omega _1\}\}$.},
author = {Franco-Filho, Antonio de Padua},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {independent family; irresolvable; submaximal; independent family; irresolvable subspace; submaximal subspace; Cantor cube},
language = {eng},
number = {4},
pages = {745-750},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {Topological characterization of the small cardinal $i$},
url = {http://eudml.org/doc/249187},
volume = {44},
year = {2003},
}

TY - JOUR
AU - Franco-Filho, Antonio de Padua
TI - Topological characterization of the small cardinal $i$
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 2003
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 44
IS - 4
SP - 745
EP - 750
AB - We show that the small cardinal number $i = \min \lbrace \vert \mathcal {A} \vert : \mathcal {A}$ is a maximal independent family} has the following topological characterization: $i = \min \lbrace \kappa \le c: \lbrace 0,1\rbrace ^{\kappa }$ has a dense irresolvable countable subspace}, where $\lbrace 0,1\rbrace ^{\kappa }$ 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 $c = {\aleph _{\omega _1}}$.
LA - eng
KW - independent family; irresolvable; submaximal; independent family; irresolvable subspace; submaximal subspace; Cantor cube
UR - http://eudml.org/doc/249187
ER -