Eprints of Rényi Institute

Dense families of countable sets below $c$

Soukup, Lajos (2010) Dense families of countable sets below $c$. (Unpublished)

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We show that it is consistent that 2^\omega is as large as you wish, and for each uncountable cardinal \kappa\le2^\omega, there are a set T\in [R]^\kappa and a family A\subset [T]^\omega with |A|=\kappa such that
(a) |\overline{a}\cap T|=\kappa for each a\in A,
(b) for each X\in [T]^{\omega_1} there is a\in A with a\subset X,
and so
(i) there is an almost disjoint family B\subseteq [\kappa]^\omega with size and chromatic number \kappa,
(ii) there is a locally compact, locally countable T_2 space with cardinality spectrum \{\omega,\kappa\}.

Item Type:Research Paper
Keywords and phrases:almost disjoint, refinement, chromatic number , cardinality spectrum, Cohen reals,
Subjects:54A35 Consistency and independence results
54A25 Cardinality properties (cardinal functions and inequalities, discrete subsets)
03E35 Consistency and independence results
Divisions:Research Divisions > Set theory and general topology
Last Modified:02 Apr 2010 11:18

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