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 X is regular topological space, with character alttext="greater-than German b"> &gt; b encoding="application/x-tex">&gt; \mathfrak {b} X right-arrow left-parenthesis t o p omega plus 1 right-parenthesis Subscript Superscript 1"> → stretchy="false">( t o p ω + 1 stretchy="false">) encoding="application/x-tex">X \rightarrow (top\, \omega + 1)^{1}_{\omega } , then, for all alttext="alpha greater-than α<!-- α <mml:msub> encoding="application/x-tex">\alpha &gt; _1 alpha \alpha )^{1}_{\omega fixing gap in original one. For that we consider decomposition spaces. also define combinatorial principle alttext="normal ♣ upper F"> mathvariant="normal">♣<!-- <mml:mi>F encoding="application/x-tex">\clubsuit _{F} use it prove consistent not-sign C H"> mathvariant="normal">¬<!-- ¬ <mml:mi>C H encoding="application/x-tex">\neg CH alttext="German encoding="application/x-tex">\mathfrak optimal bound . In [Proc. Amer. Math. Soc. 101 (1987), pp. 767–770], this was obtained using ♢"> mathvariant="normal">♢<!-- ♢ encoding="application/x-tex">\diamondsuit