Partitions of topological spaces and a new club-like principle

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 [Proc. Amer. Math. Soc. 101 (1987), pp. 767–770], this was obtained using \diamondsuit .