## Tsukuba Journal of Mathematics

### 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.

#### Article information

Source
Tsukuba J. Math., Volume 30, Number 1 (2006), 225-236.

Dates
First available in Project Euclid: 30 May 2017

https://projecteuclid.org/euclid.tkbjm/1496165038

Digital Object Identifier
doi:10.21099/tkbjm/1496165038

Mathematical Reviews number (MathSciNet)
MR2248293

Zentralblatt MATH identifier
1128.54010

#### Citation

Terada, Toshiji. Continuity of interpolations. Tsukuba J. Math. 30 (2006), no. 1, 225--236. doi:10.21099/tkbjm/1496165038. https://projecteuclid.org/euclid.tkbjm/1496165038