Some spaces related to topological inequalities proven by the Erdős-Rado theorem

Abstract:

The Erdös-Rado theorem is very useful in proving cardinal inequalities in topology. It has been suggested that certain of these inequalities might be strengthened. We note that trees constructed by Jensen and Gregory using various extra axioms of set theory yield several counterexamples to these suggestions; for example, a space $X,|X| = {\omega _2},c(X) = {\omega _1},\chi (X) = \omega$, answering a question of Hajnal and Juhász. We consider the apparently similar relation between $|X|,e(X)$, and $d\Psi (X)$ of Ginsburg and Woods. Using combinatorial consequences of $V = L$, we construct ${G_\delta }$ tree families, and establish that, assuming $V = L$, an infinite cardinal $\kappa$ is weakly compact ${\text {iff}} d\Psi (X) < \kappa ,{e_a}(X) \subset \kappa {\text {imply}}|X| < \kappa$. We consider products of countable chain condition spaces, and show that, using Cohen forcing that (${2^\omega }$ can be anything allowed by König’s theorem and there are spaces $X,Y,c(X) = c(Y) = \omega ,c(X \times Y) = {2^\omega }$). A variation is a space W with the property $c({W^n}) = {\omega _{n - 1}}$.