(** * Topology on [R]² *)
Definition locally_2d (P : R -> R -> Prop) x y :=
exists delta : posreal, forall u v, Rabs (u - x) < delta -> Rabs (v - y) < delta -> P u v.

Lemma locally_2d_align :
forall (P Q : R -> R -> Prop) x y,
( forall eps : posreal, (forall u v, Rabs (u - x) < eps -> Rabs (v - y) < eps -> P u v) ->
forall u v, Rabs (u - x) < eps -> Rabs (v - y) < eps -> Q u v ) ->
locally_2d P x y -> locally_2d Q x y.