Topology MOC

Proper map

Given topological spaces , a map is called proper iff the preïmage of every compact is compact. #m/def/topology

Properties

• If and are locally compact
  • If is second-countable: a continuous map is proper iff every sequence without limit points maps to a sequence without limit points.
  • A continuous map is compact iff the map
  between Alexandroff extensions is continuous.
• If is compact: all continuous maps are proper, since all compact subsets of are closed and all closed subsets of are compact.

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