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_ex_dec :
forall (x : T) (P : T -> Prop),
(forall x, P x \/ ~P x) ->
locally x P ->
{d : posreal | forall y, ball x d y -> P y}.