# Towards solid abelian groups: A formal proof of N"obeling's theorem

arXiv: https://arxiv.org/abs/2309.07252 · HTML: https://mlsanity.com/paper/2309.07252/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 13 Sep 2023.

## Built on by (most cited first, 1 of 1)

- [LeanProgress: Guiding Search for Neural Theorem Proving via Proof Progress Prediction](https://mlsanity.com/paper/2502.17925/) (2025)

## Abstract (arXiv)

Condensed mathematics, developed by Clausen and Scholze over the last few years, is a new way of studying the interplay between algebra and geometry. It replaces the concept of a topological space by a more sophisticated but better-behaved idea, namely that of a condensed set. Central to the theory are solid abelian groups and liquid vector spaces, analogues of complete topological groups. N"obeling's theorem, a surprising result from the 1960s about the structure of the abelian group of continuous maps from a profinite space to the integers, is a crucial ingredient in the theory of solid abelian groups; without it one cannot give any nonzero examples of solid abelian groups. We discuss a recently completed formalisation of this result in the Lean theorem prover, and give a more detailed proof than those previously available in the literature. The proof is somewhat unusual in that it requires induction over ordinals -- a technique which has not previously been used to a great extent in formalised mathematics.