End Closed.

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