Editing Compact space (section)

=== Functions and compact spaces ===

Since a [[continuous function (topology)|continuous]] image of a compact space is compact, the [[extreme value theorem]] holds for such spaces: a continuous real-valued function on a nonempty compact space is bounded above and attains its supremum.<ref>{{harvnb|Arkhangel'skii|Fedorchuk|1990|loc=Corollary 5.2.1}}</ref> 
(Slightly more generally, this is true for an upper semicontinuous function.) As a sort of converse to the above statements, the pre-image of a compact space under a [[proper map]] is compact.