Extending continuous functions

TL;DR

This paper investigates conditions on compact metrizable spaces that determine when the set of extendable continuous functions from a closed subspace is topologically open or closed in the uniform topology.

Contribution

It characterizes specific conditions on spaces that influence the topological nature of extendable function sets in the uniform topology.

Findings

Identifies conditions for openness of extendable function sets

Identifies conditions for closedness of extendable function sets

Provides a classification for spaces based on extension properties

Abstract

We examine conditions on a (compact metrizable) space $X$ such that for any space $Y$ and closed subspace $Z$, the set of continuous functions from $Z$ to $X$ which extend to $Y$ is either open or closed in the set of continuous functions from $Z$ to $X$ in the uniform topology.