Dense families of countable sets below $c$

TL;DR

This paper demonstrates the consistency of having an arbitrarily large continuum with specific families of countable sets below it, leading to new constructions in set theory and topology.

Contribution

It introduces a method to construct families of countable sets with prescribed properties below the continuum, extending understanding of set-theoretic and topological structures.

Findings

Existence of almost disjoint families of size and chromatic number κ

Construction of locally compact, locally countable T2 spaces with spectrum {ω, κ}

Consistency results for large continuum with specific set families

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 $\{\omega,\kappa\}$.