### 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 be a locally compact, non-compact space such that
any two closed unbounded (cub) subsets of intersect [of
course, a set is bounded if it has compact closure]; call
stationary if it meets every cub in . Then for
every neighbourhood assignment defined on a stationary
set there is a stationary subset
such that

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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