Is there a name or symbol for this set relation? Given two sets $A,B$ we say $A\leq B$ if for each $a\in A$ there is some $b\in B$ with $a\subseteq b$.
So for instance $\mathbb{2^N<2^Z<2^Q<2^R}$, $\{\emptyset\}<\{\{0\},\{1\}\}<\{\{0,1\}\}$, or more general for any set of sets $X$ we have $\{\emptyset\}\leq X\leq \bigcup X$ and even! $\bigcup X=2^{\bigcup X}$.
 A: $A$ is finer than $B$ and $B$ is coarser than $A$. $A$ is a refinement of $B$.
This is borrowed from the language of partitions and covers, but there seems to be no reason it can't apply to arbitrary sets. (Besides, I'm pretty sure all sets can be considered to be covers, anyway.)
Source (for partitions): https://en.wikipedia.org/wiki/Partition_of_a_set#Refinement_of_partitions
A: There is (sort of) but it's adapted from a more general setting. Given a preorder $(X,\preceq)$, a subset $B\subseteq X$ is cofinal in $X$ $\!\iff\!$ for all $x\in X$ there is $b\in B$ with $x\preceq b$. Given $A\subseteq X$, if for all $a\in A$ there is $b\in B$ with $a\preceq b$, then we can say $B$ is cofinal with respect to $A$; or, borrowing terms from other subjects in math, $B$ dominates $A$, or $B$ covers $A$ (with respect to $\preceq$).
Of course, it's clunky & verbose to say "$B$ is cofinal with respect to $A$ with respect to set inclusion $\subseteq$". The last option, "covers", is probably the best choice for the relation between $A$ and $B$ you define. But it's not a standard term for this relation, just a plausible one. If you need a name for your relation, the options I've mentioned provide some choices, but in any case you'll need to state a definition before using your chosen term.
