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).

Lemma locally_not :
forall (x : T) (P : T -> Prop),
not (forall eps : posreal, not (forall y, ball x eps y -> not (P y))) ->
locally x (fun y => not (P y)).