Possible Duplicate:
Showing $[0,1] \times [0,1]$ is a manifold with boundary
Definition: A manifold with boundary $M$ is a second countable Hausdorff space so that for a $p \in M$ there is an open set $U \subseteq M$ so that there is a homeomorphism $\varphi$ to either (a) an open set $V$ in $H^n \setminus \partial H^n$ or to (b) an open set $V$ in $H^n$ and $\varphi (p) \in \partial H^n$ where $H^n$ is the closed upper half plane.
Can you check my proof please? Thank you for help:
$M = [0,1] \times [0,1]$ is second countable, Hausdorff and locally homeomorphic to $H^n$:
$M$ is second countable because it has subspace topology of $H^2$.
It is Hausdorff because it has subspace topology of $H^2$.
Locally homeomorphic to $H^n$: For point $p \in M$ take set $U$ open in $H^2$ with $p \in U$. Inclusion map $i: U \to U \subseteq H^2$ is local homeomorphism with the property maps to open set and if $p \in \partial M$ then $i(p) \in \partial H^2$.
