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.: $codom(d)=[0,\infty]$)?



Terminology: Co-completion of Met?
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.


