Low distortion embeddings between $C(K)$ spaces

TL;DR

This paper investigates the conditions under which certain function spaces, specifically $C([0,eta])$ spaces, can be embedded into each other with low distortion, revealing a link between the embedding distortion and the topological properties of the space.

Contribution

It establishes a precise criterion relating the embeddability with distortion less than 2 to the non-emptiness of the $ ext{alpha}$-th derived set of the compact space $K$ for spaces $C([0,eta])$ with $eta< ext{omega}_1$.

Findings

Embeddings with distortion less than 2 are impossible unless $K^{(eta)}$ is non-empty.

The result characterizes the geometric structure of $C(K)$ spaces via topological properties.

Provides a new connection between metric embeddings and ordinal topological complexity.

Abstract

We show that, for each ordinal $\alpha<\omega_1$, the space $C([0,\omega^\alpha])$ does not embed into $C(K)$ with distortion strictly less than $2$ unless $K^{(\alpha)}\neq \emptyset$.