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◃YD, then f−1[C]◃Xg−1[D]. We show that there is then a function h:X→Y such that whenever C◃YD, then f−1[C]◃Xh−1[D], h−1[C]◃Xh−1[D] and h−1[C]◃Xg−1[D]. Since any function h that satisfies h−1[C]◃Xh−1[D] whenever C◃YD, 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.