Approximation by zero-free continuous maps
General Topology
2026-01-12 v2
Abstract
We prove that if E a subset of an n-dimensional manifold, then every continuous R^n-valued map on E that is zero-free on the interior of E can be approximated in the fine topology, and hence, in particular, in the uniform topology, by a continuous R^n-valued map that is zero-free on all of E.
Cite
@article{arxiv.2508.05931,
title = {Approximation by zero-free continuous maps},
author = {Alexander J. Izzo},
journal= {arXiv preprint arXiv:2508.05931},
year = {2026}
}
Comments
The main theorem has been strengthened in two ways: (i) The theorem now applies to subsets of an n-manifold rather than just subsets of Euclidean n-space. (ii) The conclusion has been strengthened from approximation in the uniform topology to approximation in the fine topology