Definition subfamily (f:family) (D:R -> Prop) : family :=
  mkfamily (intersection_domain (ind f) D) (fun x y:R => f x y /\ D x)
  (restriction_family f D).Definition compact (X:R -> Prop) : Prop :=
  forall f:family,
    covering_open_set X f ->
    exists D : R -> Prop, covering_finite X (subfamily f D).

Lemma family_P1 : forall (f:family) (D:R -> Prop), family_open_set f -> family_open_set (subfamily f D).