The Topological Version of Fodor's Theorem

I. Juhász and A. Szymanski

The following purely topological generalization is given of Fodor's theorem from [F] (also known as the ``pressing down lemma''):

Let $ X$ be a locally compact, non-compact $ T_2$ space such that any two closed unbounded (cub) subsets of $ X$ intersect [of course, a set is bounded if it has compact closure]; call $ S \subset 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 \subset S$ such that

$\displaystyle \cap \{U(x) : x\in T\} \neq \emptyset.$

Just like the ``modern'' proof of Fodor's theorem, our proof hinges on a notion of diagonal intersection of cub'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.


Key words and phrases:pressing down lemma; locally compact space; ideal of bounded sets; stationary set, stationary set decomposition

2000 Mathematics Subject Classification: Primary: 04A10; 54D30; Secondary: 54C60

References:
[F] G. Fodor, Eine Bemerkung zur Theorie der regressiven Funktionen, Acta Sci. Math. (Szeged) 17 (1956), 139-142.

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