# Why stationary sets were named such?

Do you know why stationary sets were named such?

Going over the following MO question about the intuition behind stationary sets, the only compelling argument I can think of is Fodor's lemma.

Is this the reason?

• Ioannis, this was addressed in MSE. They were also referred early on as Mahlo sets. – Andrés E. Caicedo Jul 26 '13 at 0:16
• @Andres: Thank you for the link. It indeed answers the question. I never seem to look MSE for some reason. – Ioannis Souldatos Jul 26 '13 at 13:39

• Indeed. Neumer's theorem states that if $S$ is a stationary subset of a cardinal $\rho$ of uncountable cofinality, and $f:S\to\rho$ is regressive, then $f$ is bounded on a cofinal set, that is, there is an $\alpha<\rho$, and a $D\subset S$, $D$ cofinal, such that $f(\beta)<\alpha$ for all $\beta\in D$. The reference is W. Neumer. Verallgemeinerung eines Satzes von Alexandroff und Urysohn, Math. Z., 54, (1951), 254—261. MR0043860 (13,331a). – Andrés E. Caicedo Jul 26 '13 at 6:36
• Actually, Bloch's precise statement is a bit more cumbersome than I mentioned. He uses a notion similar to club, that he calls a band: Instead of requiring that $C\subset\rho$ is cofinal, he requires that it has order type $\rho$. Such $C$ he calls full parts, and bands are sets that are closed full parts. The theorem mentioned in the title of Neumer's paper is a particular case of a result of Dushnik from the early 1930s. (The version I stated above is a modern reformulation. Juhász proved in the 70s that it is essentially a topological result.) – Andrés E. Caicedo Jul 26 '13 at 6:40