  In the present paper we study necessary and sufficient conditions for the
existence of a semicontinuous and finite Richter-Peleg multi-utility for a
preorder. It is well know that, given a preorder on a topological space, if
there is a lower (upper) semicontinuous Richter-Peleg multi-utility, then the
topology of the space must be finer than the Upper (resp. Lower) topology.
However, this condition does not guarantee the existence of a semicontinuous
representation.
  We search for finer topologies which are necessary for semicontinuity, as
well as that they could guarantee the existence of a semicontinuous
representation. As a result, we prove that Scott topology (that refines the
Upper one) must be contained in the topology of the space in case there exists
a finite lower semicontinuous Richter-Peleg multi-utility. However, as it is
shown, the existence of this representation cannot be guaranteed.
