Define a extended metric space $X$ to be a metric space $X$ except the distance function $d$ maps from $X \times X\to [0,\infty]$ (Note that $\infty$ is allowed). Looking at this post Metric assuming the value infinity it doesn't seem that any difficulty can arise when switching to an extended metric, as far as theorem go. However, compact sets are not necessarily bounded in extended metric spaces. (consider two points with distance infinity) Is there a general theorem about when theorems in metric spaces can be converted into theorems in extended metric spaces?
It would also be nice to provide other examples of things failing when extended metric is used.
The reason why extended metric spaces are so similar to usual metric spaces is that, in addition to the obvious formal analogy, they share even deeper similarities: given an extended metric $d$ in $X$, define $$ d_1(x,y) = \text{min}\{1,\ d(x,y)\}, \quad \text{for all }x,y\in X. $$ Then $d_1$ is a bona fide metric and the topology defined by $d_1$ is the same as that defined by $d$ (en passant, this is the usual trick for showing that every metric space is equivalent to a bounded one). However, anything related to boundedness will be left out of the analogy as such concepts are not intrinsic to topology.