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

A topological study for the existence of lower-semicontinuous Richter-Peleg multi-utilities