On fiber diameters of continuous maps

TL;DR

This paper proves that for any continuous map from higher to lower dimensions, fibers can have unbounded diameters, with specific boundedness properties for small fibers depending on the target dimension, relevant to data analysis.

Contribution

The paper provides a concise proof of fiber diameter unboundedness for continuous maps and characterizes boundedness of small fibers based on the target dimension.

Findings

Fibers of continuous maps from ^n to ^m can have unbounded diameter when n > m.

When m=1, the union of small fibers is bounded.

When m>1, the union of small fibers may be unbounded.

Abstract

We present a surprisingly short proof that for any continuous map $f : \mathbb{R}^n \rightarrow \mathbb{R}^m$, if $n>m$, then there exists no bound on the diameter of fibers of $f$. Moreover, we show that when $m=1$, the union of small fibers of $f$ is bounded; when $m>1$, the union of small fibers need not be bounded. Applications to data analysis are considered.