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 filter_cluster_point_impl_compact:
forall X:TopologicalSpace,
(forall F:Filter (point_set X), exists x0:point_set X,
filter_cluster_point F x0) -> compact X.