Complete separation in the random and Cohen models

It is shown that in the model obtained by adding κ many random reals, where κ is a supercompact cardinal, every C⁎-embedded subset of a first countable space (even with character smaller than κ) is C-embedded. It is also proved that if two ground model sets are completely separated after adding a random real then they were completely separated originally but CH implies that the Cohen poset does not have this property.