The range of maps on classical insertion theorems

A classical insertion theorem due to Kat??tov?CTong (or Dowker?CKat??tov, Michael) reads that ? can be a test space for the range of maps on the insertion theorem which characterizes the domain to be normal (or normal and countably paracompact, perfectly normal). It is known that the range ? in the Kat??tov?CTong insertion theorem is not necessarily replaced by a non-trivial separable Banach lattice. We show that the range ? in the Dowker?CKat??tov and Michael insertion theorems can be replaced by any non-trivial separable Banach lattice.