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.