Smooth approximation theorem for differential manifolds

This fact is an application of the following pivotal fact/result/idea: Stone-Weierstrass theorem
View other applications of Stone-Weierstrass theorem OR Read a survey article on applying Stone-Weierstrass theorem

This article gives a result on smooth approximation; a result stating that a continuous map can be replaced by a smooth map satisfying similar properties

Statement

Let $M$ and $N$ be differential manifolds and define $C^0(M,N)$ to be the space of all continuous maps from $M$ to $N$, endowed with the compact-open topology. Let $C^{\mathrm{\infty }}(M,N)$ be the space of all smooth maps from $M$ to $N$, viewed as a subset of $C^0(M,N)$. Then $C^{\mathrm{\infty }}(M,N)$ is dense in $C^0(M,N)$.

Applications

• Smooth homotopy theorem: This theorem has two parts: any continuous map of differential manifolds is homotopy-equivalent to a smooth map, and that if two continuous maps are homotopic, then they are smoothly homotopic.