Interpolating functions

Abstract

Let X , Y be sets with quasiproximities ◃ X and ◃ Y (where A ◃ B is interpreted as “ B is a neighborhood of A”). Let f , g : X → Y be a pair of functions such that whenever C ◃ Y D , then f − 1 [ C ] ◃ X g − 1 [ D ] . We show that there is then a function h : X → Y such that whenever C ◃ Y D , then f − 1 [ C ] ◃ X h − 1 [ D ] , h − 1 [ C ] ◃ X h − 1 [ D ] and h − 1 [ C ] ◃ X g − 1 [ D ] . Since any function h that satisfies h − 1 [ C ] ◃ X h − 1 [ D ] whenever C ◃ Y D , is continuous, many classical “sandwich” or “insertion” theorems are corollaries of this result. The paper is written to emphasize the strong similarities between several concepts • the posets with auxiliary relations studied in domain theory; • quasiproximities and their simplification, Urysohn relations; and • the axioms assumed by Katětov and by Lane to originally show some of these results. Interpolation results are obtained for continuous posets and Scott domains. We also show that (bi-)topological notions such as normality are captured by these order theoretical ideas.