After finishing my undergrad degree earlier this year, I've decided to go back and type up the sum total of all my math notes in one huge TeX document, with the goal of putting together a single cohesive reference for myself of everything I learned. It's a big endeavor, and in the process of trying to make the whole thing fairly rigorous, I've started to notice that mathematicians often seem to play fairly fast and loose with the distinction between object-variables and meta-variables when defining notation. It's never something I had noticed before, but it occurred to me when I went back to try and precisely define the notation I was using for indexed sets. Here's the quote:
Let $X$ be an arbitrary set. Let $I$ be a set and $\alpha:I\to X$ be a surjective function. Then we say that $\alpha$ is an indexing of $X$ by $I$, and that $I$ is acting as an index set of $X$. For the sake of convenience, we may also simply say that $X$ is indexed by $I$ and omit any explicit mention of the function $\alpha$, when context permits. The elements of $I$ are known as indices. For a given index $i\in I$, we often denote the element $\alpha(i)\in X$ by $\boldsymbol{x}_i$, for some contextually appropriate symbol $\boldsymbol{x}.$
With this in mind, we use the notation $\{\boldsymbol{x}_{\boldsymbol{i}}\}_{\boldsymbol{i}\in I}$ to refer to the set $X$ packaged with the additional information of a specified indexing. Furthermore, if $f$ is some function with $X$ as its domain, then we may similarly write $\{f(\boldsymbol{x}_{\boldsymbol{i}})\}_{\boldsymbol{i}\in I}$ to refer to the set $f(X)$ as indexed by $I$ via the indexing function $f\circ \alpha.$
Indexed sets allow us to introduce some new notation for union and intersection. Let $X=\{x_i\}_{i\in I}$ be an indexed set, and $\alpha$ be its indexing. Given a subset $U\in I$, we write $\bigcup_{i\in U}x_i$ and $\bigcap_{i\in U}x_i$ for $\cup \alpha(U)$ and $\cap \alpha(U),$ respectively.
I'm using bold to indicate meta-variables, or at least, where I think metavariables should go. But I'm really not sure. What about when giving notation for union and intersection in the last paragraph? What about the function $f$? None of these use meta-variables, but I think maybe they should?
There is a slightly obsessive part of me that feels like I won't be satisfied unless I figure this out and do it rigorously, and an alternate lazy part of me that just doesn't want to bother with it. In any case, can anyone check my use of meta-variables here? Is there a reason why the distinction between object- and meta-variables isn't more widely made in math when defining notation?
