(** 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.

(* Specifications: *)
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.
Lemma open_ext : forall D E : T -> Prop, (forall x, D x <-> E x) -> open D -> open E.
Lemma open_and : forall D E : T -> Prop, open D -> open E -> open (fun x => D x /\ E x).
Lemma open_or : forall D E : T -> Prop, open D -> open E -> open (fun x => D x \/ E x).
Lemma open_true : open (fun x : T => True).
Lemma open_false : open (fun x : T => False).
Lemma open_comp : forall {T U : UniformSpace} (f : T -> U) (D : U -> Prop), (forall x, D (f x) -> filterlim f (locally x) (locally (f x))) -> open D -> open (fun x : T => D (f x)).