Continuity of interpolations

Abstract

An interpolation function for a set of finite input-output data is a function which fits the data. Let us say that a topological space $X$ has a continuous interpolation if interpolation functions can be selected continuously, more precisely, if there is a continuous map from a certain subspace of the hyperspace $F(X x \mathbb{R})$ of finite subsets of $X x \mathbb{R}$ to the Banach space $C(X)$ of bounded real-valued continuous functions on $X$. The concept of weakly continuous interpolation is also introduced. The real line has a continuous interpolation. Every metrizable space has a weakly continuous interpolation. On the other hand, $\omega_{1}$ and $\beta \omega$ do not have weakly continuous interpolations.