$S\subset \lambda$ 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?
a question on stationary sets
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0I don’t know who first used the term, but I’d not be surprised if it stems from the fact that any pressing-down (regressive) function on a stationary subset $S$ of an uncountable regular cardinal is constant (‘stationary’, so to speak) on a stationary subset of $S$. This is the [Pressing-Down Lemma](http://en.wikipedia.org/wiki/Fodor%27s_lemma), Lemma II.6.15 in Kunen. – 2012-01-14
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0I removed [notation] and [homework] since I don't think that these are very relevant to the question. John, feel free to correct me otherwise. – 2012-01-14
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0@Brain: Mmm..., by your remarks, I find Pressing-Down Lemma is a little interesting:) – 2012-01-14
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0@Asaf Karagila : You are right. Thanks for your attention. – 2012-01-14
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0For additional information, see [here](http://mathoverflow.net/q/137792/6085). – 2013-07-26
1 Answers
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.
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0What's meaning of $Rng(f)$?:P – 2012-01-14
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0@John: $\{f(x)\mid x\in\kappa\}$ (the image of $f$) – 2012-01-14
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0We always let $ran(f)$ denote the image of $f$. So a stationary set is with respect to some $f$. I see:) – 2012-01-14