Interpolating functions

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• sets with auxiliary relations studied in domain theory; roximities and their simplification, Urysohn relations; and ioms assumed by Katětov and by Lane to originally show some of these results. polation 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.