Partitions of topological spaces and a new club-like principle

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We give a new proof of the following theorem due to W. Weiss and P. Komjath: if X X is a regular topological space, with character > b > \mathfrak {b} and X → ( t o p ω + 1 ) ω 1 X \rightarrow (top\, \omega + 1)^{1}_{\omega } , then, for all α > ω 1 \alpha > \omega _1 , X → ( t o p α ) ω 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 ♣ F \clubsuit _{F} , and use it to prove that it is consistent with ¬ C H \neg CH that b \mathfrak {b} is the optimal bound for the character of X X . In [Proc. Amer. Math. Soc. 101 (1987), pp. 767–770], this was obtained using ♢ \diamondsuit .