Proper map

Given topological spaces , a map is called proper iff the preïmage of every compact is compact. 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.