TITLE: Definition of "a topological manifold with corners".
QUESTION [11 upvotes]: How can we define a topological manifold with corners and its corners? Then, do we use "invariance of domain" to define corners, as we really need this theorem in order to define "boundaries of a manifold"?

REPLY [27 votes]: Actually, a topological manifold with corners is identical to a topological manifold with boundary.  A typical definition of an $n$-dimensional topological manifold with boundary is a second-countable (or paracompact, take your pick) Hausdorff space in which each point has a neighborhood homeomorphic to an open subset of $[0,\infty) \times \mathbb R^{n-1}$. The natural way to define a topological manifold with corners would be to replace the "model space" by $[0,\infty)^k \times \mathbb R^{n-k}$ for some $1\le k \le n$.  However, it turns out that any two such model spaces are homeomorphic. Thus corner points and boundary points are not topologically distinguishable from each other. 
The only time it makes sense to distinguish between "manifolds with corners" and "manifolds with boundary" is if a smooth structure is involved. Because $[0,\infty) \times \mathbb R^{n-1}$ is not diffeomorphic to $[0,\infty)^k \times \mathbb R^{n-k}$ when $k>1$, the smooth compatibility condition for charts depends on which model space is used, and a "smooth manifold with corners" need not be a "smooth manifold with boundary."