One of the most flexible versions of the Tietze extension principle is Katetov-Tong's theorem [E], [Ja]. This asserts that when X is normal, f: X → ℝ and g: X → ℝ are respectively lower and upper semicontinuous while f(x) ≥ g(x) for all x, then there is a continuous mapping h: X → ℝ with f(x) ≥ h(x) ≥ g(x). There are two relevant refinements of Katetov-Tong's theorem for X paracompact (and Hausdorff). First, if f is allowed to take the value +∞ and g the value —∞ the result still holds and is due in the metric setting to Hahn [Sir]. Second, if actually f(x) > g(x) for all x, there is a continuous mapping h: X → ℝ with f(x) > h(x) > g(x). This is due to Dowker [Str], [Du]. In this paper we allow f and g to take extended values in a partially ordered vector space (Y,S) where S is an ordering convex cone, and give versions of Hahn's theorem and of Dowker's theorem in this setting. To do this we make appropriate definitions of semicontinuity for functions and for multifunctions. We are then able to apply Michael's selection theorem to the lower semicontinuous multifunction H(x) := [f(x) — S] ⋂ [g(x) + S] to obtain Hahn-type results. We provide a similar selection result for strongly lower semicontinuous multifunctions which we apply to K(x) := [f(x) — IntS] ⋂ [g(x) + IntS] to obtain Dowker-type results. In each case we place restrictions on S to insure that the selection theorem applies.
Canadian Mathematical Bulletin Vol. 35, Issue 4, p. 463-474