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)).
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Infinitely far, yet very close

One thought on “Infinitely far, yet very close

  1. Given an extended metric space, “being at finite distance” is an equivalence relation. Let’s call its equivalence classes “finite distance components”. Obviously, the restriction of the (extended) metric to every finite distance component is a metric (in the standard sense), and the open subsets of every finite distance component are exactly the open subsets of the induced metric. Therefore, as topological spaces, all extended metric spaces are disjoint unions of metric spaces.

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