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In the definition of the support of a real function $f$ on $X$, why is it important to consider the closure of the set $S=\{x\in X:f(x)\neq0\}$ and not just $S$ itself?

Why is the closure of $S$ called the "support" of $f$ or how did this name come about?

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I guess it's defined to be the closure, as saying "functions of precompact support" is much more tedious than saying "functions of compact support". –  M. Luethi Jan 17 at 15:24
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The name seems rather intuitive to me. It's the smallest subset of the domain of $f$ which 'holds $f$ up'. That is, if you consider $f$ to be a subset of the space $X\times\mathbb{R}$ in the obvious way, then $supp(f)\times\{0\}$ is the smallest closed subspace which $f$ is 'built on'. Taking the closure is probably simply for notational convenience as closed sets are more well behaved when doing the usual things that one might do with supports, such as infinite intersections. –  Daniel Rust Jan 17 at 15:25
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