# The Complexity of the Classification Problems of Finite-Dimensional Continua

## TL;DR

This paper investigates the complexity of classifying finite-dimensional continua, demonstrating that for dimensions two and higher, the problem is more complex than classifying countable graphs, and compares various equivalence relations.

## Contribution

It establishes the strict complexity hierarchy between continuum classification and countable graph isomorphism for dimensions two and above, and analyzes related equivalence relations.

## Key findings

- Classification of n-dimensional continua is more complex than countable graph isomorphism for n ≥ 2.
- The paper compares the relative complexity of various equivalence relations.
- Results show a hierarchy in the complexity of classification problems for continua.

## Abstract

We consider the homeomorphic classification of finite-dimensional continua as well as several related equivalence relations. We show that, when $n \geq 2$, the classification problem of $n$-dimensional continua is strictly more complex than the isomorphism problem of countable graphs. We also obtain results that compare the relative complexity of various equivalence relations.