# Approximation of maps into spheres by regulous maps

## TL;DR

This paper proves that any continuous map from a compact real algebraic set into an n-sphere can be approximated by regulous maps, extending previous results in the field.

## Contribution

It generalizes and strengthens existing theorems by showing approximation of continuous maps into spheres by regulous maps for compact real algebraic sets.

## Key findings

- Continuous maps into spheres can be approximated by regulous maps.
- The result applies to compact real algebraic sets of any dimension.
- It broadens the scope of approximation theorems in real algebraic geometry.

## Abstract

Let $X$ be a compact real algebraic set of dimension $n$. We prove that every Euclidean continuous map from $X$ into the unit $n$-sphere can be approximated by regulous map. This strengthens and generalizes previously known results.