Require Export TopologicalSpaces Nets FilterLimits Homeomorphisms SeparatednessAxioms SubspaceTopology.
Set Asymmetric Patterns.
Definition compact (X:TopologicalSpace) :=
forall C:Family (point_set X),
(forall U:Ensemble (point_set X), In C U -> open U) ->
FamilyUnion C = Full_set ->
exists C':Family (point_set X),
Finite C' /\ Included C' C /\
FamilyUnion C' = Full_set.

Lemma compact_closed: forall (X:TopologicalSpace)
(S:Ensemble (point_set X)), Hausdorff X ->
compact (SubspaceTopology S) -> closed S.