End Compactness.
(** * Specific instances of compactness for 1D and 2D spaces *)

(* Specifications: *)
Lemma compactness_value_1d :
forall a b (delta : R -> posreal),
{ d : posreal | forall x, a <= x <= b -> ~~ exists t, a <= t <= b /\ Rabs (x - t) < delta t /\ d <= delta t }.
Lemma compactness_value_2d :
forall a b a' b' (delta : R -> R -> posreal),
{ d : posreal | forall x y, a <= x <= b -> a' <= y <= b' ->
~~ exists u, exists v, a <= u <= b /\ a' <= v <= b' /\ Rabs (x - u) < delta u v /\ Rabs (y - v) < delta u v /\ d <= delta u v }.