a question on stationary sets

Question

$S\subset \lambda$ is called a stationary set if for any closed unbounded set $E$ of $\lambda$, then $S\cap E \neq \emptyset.$ Why do people give the name "stationary set" for the sets which has such property? Could someone tell me the the background of stationary sets?

Answer 1

The definition of stationary sets were given by Bloch in 1953, the name comes from Fodor's lemma stating:

Let $\kappa$ be a regular uncountable cardinal. $S$ is stationary if and only if for every $f:S\to\kappa$ such that $f(\alpha)<\alpha$ there is some $\gamma$ such that $f^{-1}(\gamma)$ is stationary.

This can be seem a bit strange, however we can rewrite the definition of a club set, and have something even nicer:

Let $\kappa$ be a regular uncountable cardinal. $A\subseteq\kappa$ is closed and unbounded if and only if there exists a normal function $f:\kappa\to\kappa$ such that $A=Rng(f)$.

(By normal I mean strictly increasing and continuous $f(\alpha)=\bigcup_{\beta<\alpha} f(\beta)$ for a limit $\alpha$.)

Using this definition we can redefine stationary sets:

Let $\kappa$ be a regular uncountable cardinal. $S\subseteq\kappa$ is stationary if and only if for every normal function $f:\kappa\to\kappa$ there is some $\alpha\in S$ such that $f(\alpha)=\alpha$.

Such a point $\alpha$ is stationary with respect to this $f$, and the set $S$ has a stationary point for every normal function, thus the name stationary set.