Section CompleteSpace1.
Context {T : CompleteSpace}.
Definition lim : ((T -> Prop) -> Prop) -> T := CompleteSpace.lim _ (CompleteSpace.class T).
Definition iota (P : T -> Prop) := lim (fun A => (forall x, P x -> A x)).

Lemma close_iota :
forall P Q : T -> Prop,
(forall x, P x <-> Q x) ->
close (iota P) (iota Q).