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 homeomophic to the pair $(R^{n},R^{k})$.
The first thing I noteed is that if $N$ is a locally flat submanifold of $M$, then $N$ in fact a sumbanifold of $M$. The second thing I noteed 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?
