The Topological Version of Fodor’s Theorem

Abstract

The following purely topological generalization is given of Fodor’s theorem from [] (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 cub in X. Then for every neighbourhood assignment U defined on a stationary set S there is a stationary subset T ⊂ S such that $$ \cap \left\{ {U(x):x \in T} \right\} \ne 0. $$.