Definition of "a topological manifold with corners".

Solution 1:

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."

Solution 2:

Manifolds with corners are defined similarly to manifolds with boundary. The only difference that in addition to local model space equal to $R^n$, one also allows products $H_{k,n}=(R_+)^k \times R^{n-k}$. Here $R_+=[0,\infty)$. Thus, a topological $n$-manifold with corners is a Hausdorff 2nd countable topological space $X$ equipped with a maximal atlas, whose charts are described below: For every $x\in X$ there exists a neighborhood $U$ of $x$ in $X$ homeomorphic to some $H_{k,n}$ via a prescribed homeomorphism $f: U\to H_{n,k}$ (a "chart"); here $k$ depends on $x$. It is a requirement that the transition maps between charts $$ h=g\circ f^{-1}: V\subset H_{k,n}\to H_{k,n} $$ preserve the corners, in the sense that $$ h(0\times R^{n-k} \cap V)\subset 0\times R^{n-k}. $$ (You do not need this requirement in the case of manifolds with boundary.) This defines a stratification of $X$ into $n-k$-dimensional strata $X_k$: $$ X_k=\{x\in X: f(x)=0\times R^{n-k}\subset H_{n,k}\} $$ for some chart $f$. if $X=X_0$ then $X$ is simply a topological manifold; if $X=X_0\cup X_1$ then $X$ is a manifold with boundary.

See also here.

Edit: Lastly, you do not need the invariance of domain theorem to define corners. Furthermore, a homeomorphism between two manifolds with corners need not preserve corners. Just think about self-homeomorphism $(R_+)^2\to (R_+)^2$: They need not set the origin to itself. However, the category of manifolds with corners has its own notion of isomorphisms; such isomorphisms will preserve corners by definition.