I've been trying to teach my partner some set theory, and I got thrown for a loop while trying to give her a precise definition of some basic terminology. So we've heard of a set being described as "closed under" an operation, as well as an operation being "closed over" a set.
First, correct me if I'm wrong, but my impression is that these things mean the same thing.
Second, does this terminology have a precise definition? For instance we'd normally say that set $A$ is closed under $f:X\rightarrow Y$ if $A\subseteq X$ and for any $x\in A$, $f(x)\in A$. But it's also common to extend the definition such that $X$ isn't a superset of $A$ but of $A^n$ for some $n$. For instance, we'd say $A$ is closed under addition if the addition operator maps elements of $A^2$ to $A$. Does this extend to anything bigger than $n$-tuples? Infinite sequences, for instance?
Third, what's the motivation for this terminology and is it in any way related to the normal topological definition of closedness? There's a way that closed sets are related to sets that are closed under a particular operation in certain topological spaces (if you allow the usage that includes infinite sequences as above), but I haven't come up with a general relation between the two concepts. What's the motivation for saying an operation is closed over a set though, or is that just a corruption of the former? Does anyone know the historical reasoning behind any of this terminology?