Editing Proper map (section)

==Properties==

* Every continuous map from a compact space to a [[Hausdorff space]] is both proper and [[Closed map|closed]].
* Every [[Surjective map|surjective]] proper map is a compact covering map.
** A map <math>f : X \to Y</math> is called a '''{{em|compact covering}}''' if for every compact subset <math>K \subseteq Y</math> there exists some compact subset <math>C \subseteq X</math> such that <math>f(C) = K.</math> 
* A topological space is compact if and only if the map from that space to a single point is proper.
* If <math>f : X \to Y</math> is a proper continuous map and <math>Y</math> is a [[compactly generated Hausdorff space]] (this includes Hausdorff spaces that are either [[First-countable space|first-countable]] or [[Locally compact space|locally compact]]), then <math>f</math> is closed.<ref name=palais>{{cite journal|last=Palais|first=Richard S.|author-link=Richard Palais| title=When proper maps are closed|journal=[[Proceedings of the American Mathematical Society]]|year=1970|volume=24|issue=4|pages=835–836|doi=10.1090/s0002-9939-1970-0254818-x|doi-access=free|mr=0254818 }}</ref>