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Emeritus
OK, let's focus on "distributions" first, not "tempered distributions". We define the test function space (I'll call it $\mathcal D$) as the set of $C^\infty$ functions from $\mathbb R^n$ into $\mathbb C$ that have compact support. I would like to define a "distribution" as a member of the dual space $\mathcal D^*$, defined as the set of continuous linear functions $T:\mathcal D\rightarrow\mathbb C$. But to do that, we need to define what "continuous" means. There are at least two ways to do that.
$T:\mathcal D\rightarrow\mathbb C$ is continuous at $g\in\mathcal D$ if for each $\epsilon>0$ there's a $\delta>0$ such that $\|f-g\|<\delta\implies|T(f)-T(g)|<\epsilon$. T is continous on a set $U\subset\mathcal D$ if it's continuous at each point in U. Alternatively, and equivalently, T is continuous on $U\subset\mathcal D$ if $T^{-1}(E)$ is open for every open $E\subset\mathbb C$.