Let $\kappa$ be an infinite cardinal.
We know that the notion of being closed and unbounded (club) in $\kappa$ is strictly stronger than being stationary in $\kappa$.


My question: Do you know any notion that is strictly between being a club and being a stationary set?



Notion strictly between stationary and club
Here are a few intermediate notions:


* "Being in the club filter" is strictly stronger than stationary and strictly weaker than club.
* More generally, "being in $F$", for some fixed normal filter $F$ on $\kappa$. By normality, $F$ contains the club filter and so all such sets are stationary, but needn't be club.
* "Being $\omega$-closed" (that is, containing all limits of countable sequences) and unbounded. Every such set is stationary, but when $\kappa\gt\omega\_1$, there are $\omega$-clubs that are not club, such as the set of ordinals with cofinality $\omega$.
* Similarly, "being $\delta$-club" or "being ${\lt}\delta$-club" for some specific $\delta\lt\kappa$ is intermediate.
* More generally, "agreeing with a club on $E$" for some fixed stationary set $E$. That is, $B$ agrees with a club on $E$ if there is some club $C$ such that $C\cap E=B\cap E$.


In the large cardinal context, there are several more:


* When $\kappa$ is a measurable cardinal, then "having normal measure one," meaning that the set is in *some* normal measure on $\kappa$, is an example. All such sets are stationary, but not all are club or even in the club filter, and if there is more than one normal measure, these sets do not even form a filter.
* Similarly, when $\kappa$ is even stronger, say, supercompact, then "having normal measure one with respect to a normal measure induced by a $\kappa^{++}$-supercompactness ultrafilter (or whatever, as you like). That is, $\kappa\in j(A)$ for an embedding $j:V\to M$ of the desired strong type.
* When $\kappa$ is a weaker large cardinal, say, just weakly compact, then one can say "having weak compactness measure one." A set $A\subset\kappa$ has this property when for every $\kappa$-model $M$ there is a weak compactness embedding $j:M\to N$ with $\kappa\in j(A)$.


