Insertion theorems for maps to ordered topological vector spaces

For an ordered topological vector space Y   and a,b∈Ya,b∈Y, we write a≪ba≪b if b−ab−a is an interior point of the positive cone. Modifying the earlier results of Borwein–Théra, in this paper, ≪ is extended over Y••=Y∪{∞}∪{−∞} and a natural topology on Y•• is introduced. For a topological space X, and a non-trivial separable ordered topological vector space Y with an interior point of the positive cone, we show the following: X   is normal and countably paracompact if and only if for every lower semi-continuous map f:X→Y•• and every upper semi-continuous map g:X→Y•• with g≪fg≪f, there exists a continuous map h:X→Yh:X→Y such that g≪h≪fg≪h≪f.