In main-stream mathematical literature, the term metric space is reserved for $(X,d)$ where $X$ is a set and $d:X\times X\rightarrow [0,\infty)$ satisfies the usual properties of a metric. However, at times it is convenient to allow $d$ to take infinite values (for example if we would like to give meaning to a "co-product" of metric spaces). In that case, what is the most standard terminology for such an "extended-real-valued" metric (i.e.: $codom(d)=[0,\infty]$)?
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1$\begingroup$ I'm not sure if there is a standard terminology, but I have seen "infinite-valued metric" used (in Gerald Beer's Topologies on Closed and Closed Convex Sets). $\endgroup$– Logan FoxSep 12, 2020 at 16:37
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$\begingroup$ Interesting, I have seen the same author (in "The Structure of Extended Real-valued Metric Spaces") calling these objects extended real-valued metric spaces... now I'm even more confused. $\endgroup$– ABIMSep 12, 2020 at 16:43
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3$\begingroup$ I think "extended metric" is pretty standard. That's the term I used in my book Lipschitz Algebras. $\endgroup$– Nik WeaverSep 12, 2020 at 17:04
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$\begingroup$ True, yes I have seen that there. Thanks Nik. $\endgroup$– ABIMSep 12, 2020 at 19:00
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I do not think that there is a standard name for such spaces (and hence for such generalised metrics). It is quite common to see the terms '$\infty$-metric space' and 'extended metric space' (or some slight modifications). However, the latter name is also used in a more general sense, where the metric $d$ is allowed to take values in any ordered set. Consequently, the former term can be seen as a special case of the latter, where the ordered set is the one of extended positive real numbers.
answered Sep 12, 2020 at 16:55