(** locally' *)

Definition locally' {T : UniformSpace} (x : T) :=
  within (fun y => y <> x) (locally x).Global Instance locally'_filter :
  forall {T : UniformSpace} (x : T), Filter (locally' x).Definition at_point (a : T) (P : T -> Prop) : Prop := P a.Global Instance at_point_filter (a : T) :
  ProperFilter (at_point a).Definition open (D : T -> Prop) :=
  forall x, D x -> locally x D.

Lemma locally_open : forall (D E : T -> Prop), open D -> (forall x : T, D x -> E x) -> forall x : T, D x -> locally x E.