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Is the intersection of a decreasing sequence of countably many stationary subsets of $\omega_1$ always stationary?

It seems elementary, but I fail to find the answer in textbooks.

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The answer is no. Here is an example: Take $T_n$ for $n<\omega$ disjoint stationary sets. Let $S_m$, for $m<\omega$, be the union of $T_n$ with $n\ge m$.

That there are $\omega$ (in fact, $\omega_1$) disjoint stationary subsets of $\omega_1$ is an immediate consequence of the existence of Ulam matrices. (See for example this blog post of mine. You can also find this in Kunen's book or in Jech's book.)

I think the thing is that the construction suggests itself: Suppose $S_0\supseteq S_1\supseteq\dots$ are stationary sets. If each $S_i\setminus S_{i+1}$ is non-stationary, then $\bigcap_n S_n$ is stationary; in fact, $S_0\setminus \bigcap_n S_n$ is a non-stationary set. So we would need that infinitely often we have $S_i\setminus S_{i+1}$ stationary. By passing to a subsequence if necessary, we may as well assume that it happens for all $i$. But now the construction above is immediate.

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Thank your very much! It is really elementary :-) – sonicyouth Dec 18 '10 at 4:55

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