Dense families of countable sets below $c$

Abstract

We show that it is consistent that the continuum is as large as you wish, and for each uncountable cardinal $\kappa$ below the continuum, there are a subset $T$ of the reals and a family $A$ of countable subsets of $T$ such that (1) both $T$ and $A$ have cardinality $\kappa$, (2) $|\bar{a}\cap T|=\kappa$ for each $a\in A$, (3) for each uncountable subset of $T$ contains some elements of $A$, and so (i) there is an almost disjoint family of subsets of the reals with size and chromatic number $\kappa$, (ii) there is a locally compact, locally countable $T_2$ space with cardinality spectrum $\{\omega,\kappa\}$.