# Locally flat submanifold

Recently I found the next definition:

Let $M^n$ be an $n-$dimensional topological manifold. Then $N^k\subseteq M^n$ is a locally flat submanifold if for every $x\in N$ there exists an open set $U$ in $M$ such that the pair $(U,U\cap N)$ is homeomorphic to the pair $(R^{n},R^{k})$.

The first thing I noted is that if $N$ is a locally flat submanifold of $M$, then $N$ in fact a submanifold of $M$. The second thing I noted is that in the category of smooth manifolds the notion of locally smooth submanifold and submanifold are equivalent.

I would like to know if there is an example of a topological submanifold of a manifold that is not a locally flat submanifold or if the notions of locally flat submanifold and submanifold are equivalent in the category of topological manifolds?

• Alexander's Horned Sphere is the standard example, see any textbook on knot theory. Rolfsen's book has a detailed example. – Ryan Budney Oct 28 '11 at 21:30
• Then Alexander's Horned Sphere is a submanifold of $\mathbb R^3$. – Ryan Budney Oct 28 '11 at 21:58
• Another example: Take a knot $K$ in $S^3$ and consider the cone on the knot in $B^4$. This is a submanifold of $B^4$ homeomorphic to a disk, but it is not flat at the cone point. – Jim Conant Oct 29 '11 at 1:18
• Some authors use "submanifold" to mean what some other authors mean by "locally flat submanifold". Unfortunate, maybe, but undoubtedly true. – Tom Goodwillie Oct 29 '11 at 1:26
• Please do not ask questions here and on math.stackexchange.com at the same time. math.stackexchange.com/questions/76977/locally-flat-submanifold – Mariano Suárez-Álvarez Oct 29 '11 at 23:03