# Infinitely far, yet very close

How much does the theory of metric spaces change if we let the distance function take the value $+\infty$? Some natural examples of “extended metric spaces”:

• The extended real numbers.
• Any function space $Y^X$ where $Y$ is a metric space (or an extended metric space) with the usual supremum distance: $d(f,g) = \sup_{x\in X} d(f(x),g(x))$.