Section Locally.
Context {T : UniformSpace}.
Definition locally (x : T) (P : T -> Prop) :=
exists eps : posreal, forall y, ball x eps y -> P y.
Global Instance locally_filter :
forall x : T, ProperFilter (locally x).
Definition is_filter_lim (F : (T -> Prop) -> Prop) (x : T) :=
forall P, locally x P -> F P.

Lemma is_filter_lim_locally_close (x y : T) :
is_filter_lim (locally x) y -> close x y.