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I'm just second-marking some exam scripts, and I wanted to leap on a question and made the following pedantic remark concerning the model answers: "if the metric space is empty then this proof doesn't work because something which is supposed to be finite is $-\infty$. Hence this proof is incomplete -- it's missing the line "If the space is empty then the result is trivial".

But then another question made me wonder whether in fact the lecturer of the course had actually put as part of the definition of metric space, that it be non-empty. A quick trip to Wikipedia revealed that there also the definition required the space to be non-empty.

Why?

I certainly don't want to require that a topological space be non-empty, for example. There is presumably some sensible reason why the general convention for topological spaces has been to allow the empty set (this I understand!) but the general convention for metric spaces appears to be not to allow it...

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    I wonder about that $-\infty.$ One could say $\sup\varnothing=-\infty,$ but within the set of possible distances, $[0,+\infty),$ one should say $\sup\varnothing=0$ and $\inf\varnothing$ doesn't exist, but within $[0,+\infty]$ one would have $\inf\varnothing=+\infty.$ That last is useful if the metric space is a manifold that is not connected: the distance between two points is the infimum of all path length, so if there is no path, the distance would be $+\infty.$ Maybe that disqualifies it from being a metric space by some definitions, but maybe the definitions should be adjusted. $\qquad$2018-06-10

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Rudin doesn't require that a metric space be non-empty. I agree that there is no good reason for a convention which says otherwise. For example, for those of you who are convinced by such reasons, we want subspaces of metrizable spaces to be metrizable.

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    @ever$y$one: we are actually a bit late to this debate: see en.wikipedia.org/wiki/Diameter.2011-06-13