Interpolating Functions

Abstract

Let X, Y be sets with quasiproximities (sic)x and (sic)y (where A (sic) B is interpreted as "B isa neighborhood of A"). Let f.g : X -> Y be a pair of functions such that whenever C (sic)y D. then f(-1) vertical bar C vertical bar (sic)x g(-1)vertical bar D vertical bar. We show that there is then a function h : X -> Y such that whenever C (sic)y D. then f(-1)vertical bar C vertical bar (sic)x h(-1)vertical bar D vertical bar, h(-1)vertical bar C vertical bar (sic)x h(-1)vertical bar D vertical bar and h(-1)vertical bar C vertical bar (sic)x g(-1)vertical bar D vertical bar. Since any function It that satisfies h(-1) vertical bar C vertical bar (sic)x h(-1)vertical bar D vertical bar whenever C (sic)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 Katetov 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. (C) 2010 Elsevier B.V. All rights reserved.