# Checking whether some sets are clubs in $\aleph_2$ and $\aleph_1$

I'm getting acquainted with the stationary sets and clubs. I don't yet quite get everything well, so I'd appreciate some help with this question:

which of the following sets are clubs, contain a club, are stationary?

1. $\{ \alpha \in \omega_2 : \alpha \in \text{Lim} \text{ and } \text{cf}(\alpha) = \omega \}$

2. $\{\alpha \in \omega_2: \exists~ \beta \text{ s.t. }\alpha = \omega + \beta \}$

3. $\{ \alpha \in \omega_1 : \text{ot}(\alpha \cap \text{Lim}) = \alpha \}$

Where $\text{cf, ot, Lim}$ denote, respectively, cofinality, order type and the class of limit ordinals.

Here are some of my thoughts:

1. I believe that $\sup\limits_{\omega \leq \alpha < \omega_1} \alpha = \omega_1$ and $\text{cf}(\alpha) = \omega$, which would mean that this set is not closed. I'm not sure, however, how to decide if it contains a club. I know that it's stationary, since for any club $C$ we can take a strictly increasing countable sequence $\{c_n\} \subset C$ and its limit belongs to the set.

Is this correct? As for 2 and 3 I am not really sure where to begin. I would appreciate some hints.

• Hint on #2: it has a much cleaner characterization than the written one...(Secondary hint: what is $\omega+\epsilon_0$?) May 17, 2016 at 16:56
• I'm not sure what $\epsilon_0$ is May 17, 2016 at 16:57
• In that case, $\omega + \omega^\omega$? May 17, 2016 at 16:58
• So by $\epsilon_0$ you mean the first ordinal such that $\omega + \epsilon_0 = \omega$? And for sufficiently large $\alpha$ it holds that $\omega + \alpha = \alpha$? May 17, 2016 at 17:12
• $\epsilon_0$ has a different definition than that; it was just a random 'large countable ordinal' pulled out of a hat. But your statement that $\omega+\alpha=\alpha$ for large ordinals is basically correct; more broadly, can you characterize all the ordinals that are of the form $\omega+\alpha$? Cantor normal form will be your friend here... May 17, 2016 at 17:18

Your argument about (1) is correct, and it is really the reason that it cannot contain a club. If $\kappa>\omega$ is regular, and $D\subseteq\kappa$ is a club, then it must have limit points of every cofinality below $\kappa$ (alternatively, simply note that $\{\alpha<\omega_2\mid\operatorname{cf}(\alpha)=\omega_1\}$ is stationary, and therefore the set in (1) cannot be a club.)
In the second case, try to find out which ordinals---in general---cannot be written as $\omega+\beta$ for some $\beta$. You'll find out that there are only countably many of them.
For the third one, it is not hard to prove that the set is closed; to see it is unbounded look at $\varepsilon_0$ and other $\varepsilon$-numbers, and try to understand whether or not they belong to this set.
• But 1. contains $\{\alpha \in \omega_1 \mid Lim(\alpha)\}$, which is club in $\omega_1$. May 17, 2016 at 20:52
• But it's not a club in $\omega_2$, because $\omega_1<\omega_2$ and therefore it is bounded there. May 17, 2016 at 21:00
• Sure, it isn't club in $\omega_2$, but I thought th equestion might be a bit of a "trick" question in that it doesn't specify "club in what?". May 17, 2016 at 21:20