Intersection of topological manifolds. A condition for the intersection of two smooth manifolds to be a smooth manifold is that they intersect transversally. Is this only an obstruction because of the smooth structure? **Question:** Is the intersection of two topological manifolds always a topological manifold? If not, are there conditions which can be added to the manifold(Not the smooth+transversal intersection) so that it is true?

Actually, the intersection of two topological manifolds in a Euclidean space can be almost anything. Here's an example to show how bad things can get. Let $C$ be any closed subset of $\mathbb R^n$ whatsoever, and let $f\colon \mathbb R^n \to \mathbb R$ be the function $$ f(x) = \operatorname{dist}(x,C). $$ Thus $f$ is continuous, and $f(x)=0$ if and only if $x\in C$.

Let $M\subset\mathbb R^{n+1}$ be the graph of $f$, and let $N\subset\mathbb R^{n+1}$ be the graph of the zero function (i.e., $N = \mathbb R^n\times \\{0\\}$). Then $M$ and $N$ are both topological submanifolds of $\mathbb R^{n+1}$, and $M\cap N = C \times \\{0\\}$.