The Topological Version of Fodor’s Theorem I.Juh´asz 1 and A. Szymanski 2 Abstract 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 T 2 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 } = ∅. 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) gener- alization to this framework of Solovay’s celebrated stationary set de- composition theorem. AMS classification: Primary: 04A10; 54D30; Secondary: 54C60 Key words and phrases: pressing down lemma; locally compact space; ideal of bounded sets; stationary set, stationary set decomposition 1. Introduction One of the most frequently used results in set-theory is Fodor’s theorem (also known as the pressing down lemma) from [F1]: Theorem 1. Let α be an ordinal of uncountable cofinality. If S ⊂ α is stationary in α [i.e. S ∩ C = ∅ for every closed unbounded (in short: c u b) 1 Research supported by OTKA grant no. 37758. 2 Research supported by Charles University and the Czech Academy of Sciences. 1