TITLE: Notation for all sets with a given cardinality
QUESTION [6 upvotes]: Is there a notation for the collection of all sets with cardinality $k$, for some integer $k$? This collection is not a set, but it is still a useful collection. For example, suppose this collection is denoted by $\mathcal{S}[k]$. Then, given a set-family $H$, we can write:
$$H\cap \mathcal{S}[k]$$
to denote the elements of $H$ with cardinality $k$.

REPLY [6 votes]: Generally, in set theory we use $[X]^\kappa$ to denote the set of subsets of $X$ of size exactly $\kappa$. Here $\kappa$ can be an integer, or a transfinite cardinal.
If I had to choose a notation which would be recognizable to a set theorist, I'd use $[V]^\kappa$, where $V$ is the set theoretic universe. You can use other letters if $V$ has a different "usual interpretation" in your context.