Section CPO_THEORY.
#[local] Existing Instance direct_product_of_two_Prosets.
#[local] Existing Instance subProset.
#[local] Existing Instance pi_isProset.
#[local] Existing Instance Scott_topology.
Definition U {D : Type} {PROSET : isProset D} (x : D) : ensemble D :=
fun z : D => ~ z =< x.

Lemma Scott_topology_T0' {D : Type} {PROSET : isProset D} (a : D) (b : D)
(NE : ~ a == b)
: exists O : ensemble D, isOpen O /\ ((a \in O /\ ~ b \in O) \/ (b \in O /\ ~ a \in O)).