Tame submanifold

This article defines a property of a submanifold inside a manifold

Definition

Let $M$ be a manifold of dimension $m$ and $N$ a submanifold of dimension $n$. Then $N$ is termed tame in $M$ if for every point $x\in N$, there exists a neighbourhood $U$ of $x$ in $M$ such that the pair $(U,U\cap N)$ is homeomorphic to the pair $(\mathbb {R} ^{m},\mathbb {R} ^{n})$ where $\mathbb {R} ^{n}$ is viewed as a linear subspace of $\mathbb {R} ^{m}$.

Another way of saying this is that the local codimension at each point, equals the codimension of the submanifold as a whole.

Facts

An example of a submanifold which is not tame is the Alexander horned sphere in $\mathbb {R} ^{3}$.