Section Closed.
Context {T : UniformSpace}.
Definition closed (D : T -> Prop) :=
forall x, not (locally x (fun x : T => not (D x))) -> D x.
(* Lemma closed_or : forall D E : T -> Prop, closed D -> closed E -> closed (fun x => D x \/ E x). *)
End Closed.

Lemma closed_filterlim_loc :
forall {T} {U : UniformSpace} {F} {FF : ProperFilter' F} (f : T -> U) (D : U -> Prop),
forall y, filterlim f F (locally y) ->
F (fun x => D (f x)) ->
closed D -> D y.