February 12th
Today I learned the definition of support in topology. In terms of set theory, the support of a function $f:X\to\RR$ (or $f:X\to\CC$) is the set\[\op{supp}(f):=\{x\in X:f(x)\ne0\}.\]In general, we expect this to be "large'' because the set of $0$s of a function are frequently "small.'' However, this set is somewhat poorly-behaved in the general case, so we define the support topologically as\[\op{supp}(f):=\op{Cl}(\{x\in X:f(x)\ne0\}),\]the closure of the zero set. We remark that if $f$ is continuous, then $\{x\in X:f(x)\ne0\}=f^{-1}(\RR\setminus\{0\})$ is an open set.
Note that I think we expect $\op{supp}(f)$ to usually be isolated points anyways, so this is doesn't usually make much of a difference. But of course, if we do something like\[f(x)=\max\{x,0\},\]then $\{x\in\RR:f(x)\ne0\}=(0,\infty).$ We would like this set to be closed, so we have defined the support to be $\overline{(0,\infty)}=[0,\infty)$ to fix this.
Of note are the implications of being in or out of the support because these are somewhat confusing. The idea to remember is that $\op{supp}(f)$ is potentially a bit bigger than the nonzero set of a function (as it is defined). For example, we do have\[f(x)\ne0\implies x\in\op{supp}(f),\]but the converse does not hold: some $x$ with $f(x)=0$ could sneak in because of the closure. Explicitly, see $f(x)=\max\{x,0\}$ from before. Taking contrapositives, we also have\[x\notin\op{supp}(f)\implies f(x)=0,\]but again the converse does not hold for the same reason. Here, we're saying that $X\setminus\op{supp}(f)$ is a restricted version of $f(x)=0.$
I will also mention the definition of compact support, which is actually why I had to look up the definition of support. A function has "compact support'' merely if its support (topologically) is also compact. Our $f(x)$ from before does not work because $[0,\infty)$ is too big to be compact, but\[f(x)=\max\left\{1-x^2,0\right\}\]has support $\overline{(-1,1)}=[-1,1]$ and therefore has compact support. In general, I'm thinking about compact support as meaning some strongish form of very small—the nonzero set must be bounded for compact support.