Insertion theorems for maps to ordered topological vector spaces (Q890082)

Fix an ordered topological vector space \(Y\). For functions \(f,g:X\to Y\) write \(f\ll g\) provided that for each \(x\in X\) there are neighbourhoods \(U\), \(V\) of \(f(x)\), \(g(x)\) such that \(U<V\). Various insertion theorems, such as the following are proved. Assume \(Y\) is non-trivial and separable such that the positive cone has an interior point. Then a Hausdorff topological space \(X\) is normal and countably compact if and only if for each l.s.c. map \(f:X\to Y\) and each u.s.c. map \(g:X\to Y\) with \(g\ll f\) there is a continuous \(h:X\to Y\) such that \(g\ll h\ll f\).