Definition is_filter_lim (F : (T -> Prop) -> Prop) (x : T) :=
forall P, locally x P -> F P.

(* Specifications: *)
Lemma is_filter_lim_filter_le :
forall {F G} (x : T),
filter_le G F ->
is_filter_lim F x -> is_filter_lim G x.
Lemma is_filter_lim_close {F} {FF : ProperFilter F} (x y : T) :
is_filter_lim F x -> is_filter_lim F y -> close x y.
Lemma is_filter_lim_locally_close (x y : T) :
is_filter_lim (locally x) y -> close x y.