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?
Best Answer
In an ordinary mathematical context I would never say that an operation $\mathcal O$ is closed over a set $A$; I consider that an error for ‘$A$ is closed under $\mathcal O$’. The latter terminology can be properly applied very generally. For starters, if $A \subseteq S$, $n \in \omega$, and $f:S^n \to S$, ‘$A$ is closed under $f$’ means precisely that for every $\langle a_1,\dots,a_n \rangle \in A^n$, $f(a_1,\dots,a_n) \in A$.
This is the most common algebraic usage, I think, but it’s just the beginning. For instance, the non-negative integers $n$ can be replaced by any ordinal $\alpha$, and the operation need not be defined on all members of $S^\alpha$. Suppose that $X$ is a topological space and $A \subseteq X$. ‘$A$ is closed under (the operation of taking) limits of convergent sequences’ then means that if $\langle a_n:n \in \omega \rangle$ is a sequence of points in $A$ that converges as a sequence in the space $X$, its limit point is actually in $A$. Here $\alpha = \omega$, and the operation is defined only for those elements of $X^\omega$ that converge in $X$. This example at least starts to show the relationship between the general notion and that of topological closedness.
Moreover, the terminology is still used when the input to the operation isn’t ordered: it’s perfectly correct to say that a family $\mathcal A$ of sets is closed under (taking) finite intersections, for instance, meaning that if $\mathcal F$ is any finite subfamily of $\mathcal A$, $\bigcap \mathcal F \in \mathcal A$. If $S$ is the underlying set, the operation is $\mathcal O:[(\mathcal P(S)]^{< \omega} \to \mathcal P(S):\mathcal F \mapsto \bigcap \mathcal F$, and $\mathcal A \subseteq \mathcal P(S)$ is closed under it because $\mathcal O$ maps $[\mathcal A]^{< \omega}$ into $\mathcal A$. (Here $[X]^{< \omega}$ denotes the set of finite subsets of $X$.)
It would be a bit difficult to formulate an exhaustive formal definition of the usage, and I’m not at all sure that it would be particularly helpful; it seems more useful to present a variety of examples showing the flexibility of the usage.