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 $X\to (top\omega +1{)}_{\omega }^1$, then, for all $\alpha <{\omega }_1$, $X\to (top\alpha {)}_{\omega }^1$, fixing a gap in the original one. For that we consider a new decomposition of topological spaces. We also define a new combinatorial principle ${♣}_F$, 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 $♢$.