There are several competing definitions of a "proper function". Some authors call a function ${\displaystyle f:X\to Y}$  between two topological spaces proper if the preimage of every compact set in ${\displaystyle Y}$  is compact in ${\displaystyle X.}$  Other authors call a map ${\displaystyle f}$  proper if it is continuous and closed with compact fibers; that is if it is a continuous closed map and the preimage of every point in ${\displaystyle Y}$  is compact. The two definitions are equivalent if ${\displaystyle Y}$  is locally compact and Hausdorff.

If ${\displaystyle X}$  is Hausdorff and ${\displaystyle Y}$  is locally compact Hausdorff then proper is equivalent to universally closed. A map is universally closed if for any topological space ${\displaystyle Z}$  the map ${\displaystyle f\times \operatorname {id} _{Z}:X\times Z\to Y\times Z}$  is closed. In the case that ${\displaystyle Y}$  is Hausdorff, this is equivalent to requiring that for any map ${\displaystyle Z\to Y}$  the pullback ${\displaystyle X\times _{Y}Z\to Z}$  be closed, as follows from the fact that ${\displaystyle X\times _{Y}Z}$  is a closed subspace of ${\displaystyle X\times Z.}$ 

An equivalent, possibly more intuitive definition when ${\displaystyle X}$  and ${\displaystyle Y}$  are metric spaces is as follows: we say an infinite sequence of points ${\displaystyle \{p_{i}\}}$  in a topological space ${\displaystyle X}$  escapes to infinity if, for every compact set ${\displaystyle S\subseteq X}$  only finitely many points ${\displaystyle p_{i}}$  are in ${\displaystyle S.}$  Then a continuous map ${\displaystyle f:X\to Y}$  is proper if and only if for every sequence of points ${\displaystyle \left\{p_{i}\right\}}$  that escapes to infinity in ${\displaystyle X,}$  the sequence ${\displaystyle \left\{f\left(p_{i}\right)\right\}}$  escapes to infinity in ${\displaystyle Y.}$ 

• Every continuous map from a compact space to a Hausdorff space is both proper and closed.
• Every surjective proper map is a compact covering map.
  • A map ${\displaystyle f:X\to Y}$  is called a compact covering if for every compact subset ${\displaystyle K\subseteq Y}$  there exists some compact subset ${\displaystyle C\subseteq X}$  such that ${\displaystyle f(C)=K.}$ 
• A topological space is compact if and only if the map from that space to a single point is proper.
• If ${\displaystyle f:X\to Y}$  is a proper continuous map and ${\displaystyle Y}$  is a compactly generated Hausdorff space (this includes Hausdorff spaces that are either first-countable or locally compact), then ${\displaystyle f}$  is closed.^[2]

It is possible to generalize the notion of proper maps of topological spaces to locales and topoi, see (Johnstone 2002).

• Almost open map – Map that satisfies a condition similar to that of being an open map.
• Open and closed maps – A function that sends open (resp. closed) subsets to open (resp. closed) subsets
• Perfect map – Continuous closed surjective map, each of whose fibers are also compact sets
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