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I added to the statement under Properties; it’s a lot clearer I think to state it first for the one-point compactification.
The article seemed strangely limited in scope, so I generalised it somewhat. Also an Idea section, which motivates the definition.
Actually, in some ways it was too general, since originally $X$ was not required to be locally compact. So if $X$ is the space of rational numbers (as a subspace of the real line), then a function like $x \mapsto \mathrm{e}^{-x^2}$ (taking values in the real line with basepoint $0$) should vanish at infinity, but there are too few compact subspaces of $X$ to see this. The article currently doesn't have a definition that includes this, but at least it doesn't exclude this either.
ETA: Actually, the comment about the Stone–Čech compactification seems to cover this, but I'm not sure how generally this can be made to work.
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