  Scott continuity is a concept from domain theory that had an unexpected
previous life in the theory of von Neumann algebras. Scott-continuous states
are known as normal states, and normal states are exactly the states coming
from density matrices. Given this, and the usefulness of Scott continuity in
domain theory, it is natural to ask whether this carries over to generalized
probabilistic theories. We show that the answer is no - there are
infinite-dimensional convex sets for which the set of Scott-continuous states
on the corresponding set of 2-valued POVMs does not recover the original convex
set, but is strictly larger. This shows the necessity of the use of topologies
for state-effect duality in the general case, rather than purely order
theoretic notions.
