Partitions of topological spaces and a new club-like principle

Abstract

We give a new proof of the following theorem due to W. Weiss and P. Komjath: if $X$ is a regular topological space, with character $ < \mathfrak{b}$ and $X \rightarrow (top \omega + 1)^{1}_{\omega}$, then, for all $\alpha < \omega_1$, $X \rightarrow (top \alpha)^{1}_{\omega}$, fixing a gap in the original one. For that we consider a new decomposition of topological spaces. We also define a new combinatorial principle $\clubsuit_{F}$, and use it to prove that it is consistent with $\neg CH$ that $\mathfrak{b}$ is the optimal bound for the character of $X$. In \cite{WeissKomjath}, this was obtained using $\diamondsuit$.