The Topological Version of Fodor’s Theorem

The following purely topological generalization is given of Fodor’s theorem from [F1] (also known as the “pressing down lemma”): Let X be a locally compact, non-compact T2 space such that any two closed unbounded (c u b) subsets of X intersect [of course, a set is bounded if it has compact closure]; call S ⊂ X stationary if it meets every c u b in X. Then for every neighbourhood assignment U defined on a stationary set S there is a stationary subset T ⊂ S such that ∩{U(x) : x ∈ T} 6 = ∅. Just like the “modern” proof of Fodor’s theorem, our proof hinges on a notion of diagonal intersection of c u b’s, definable under some additional conditions. We also use these results to present an (alas, only partial) generalization to this framework of Solovay’s celebrated stationary set decomposition theorem. AMS classification: Primary: 04A10; 54D30; Secondary: 54C60