Tubular neighborhood theorem

This fact is an application of the following pivotal fact/result/idea: inverse function theorem
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This fact is an application of the following pivotal fact/result/idea: existence of smooth partitions of unity
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This fact is an application of the following pivotal fact/result/idea: Lebesgue number lemma
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Statement

Let $M$ be a submanifold (differential sense) of ${\mathbb{R}}^n$, of dimension $m$. Then, there exists $ϵ>0$ such that for any point at distance at most $ϵ$ from $M$, there is a unique expression of the point as a sum $p+v$ where $p\in M$ and $v$ is a normal at $p$, with $\left|v\right|<ϵ$.

If we define $U$ as the open subset of ${\mathbb{R}}^n$ comprising those points of ${\mathbb{R}}^n$ at distance less than $ϵ$ from $M$, then $U$ can be viewed as a concrete realization, in the ambient space ${\mathbb{R}}^n$, of the normal bundle to $M$ in ${\mathbb{R}}^n$. In the situations where the normal bundle to $M$ is trivial, we see that this gives a natural diffeomorphism $U\cong M\times {\mathbb{R}}^{n-m}$.