Homeomorphism of the Disk I'm working through Massey's "Basic Course in AT."  One of the problems is prove that a homeomorphism of the closed disk maps the boundary to the boundary and the interior to the interior.  How would one prove this?  I can't seem to get this one problem.
This answer extends on Chris Eagles comment:

Let $D^n \subset \mathbb R^n$ denote the $n$-dimensional closed unit disk, that is $D^n = \{ x \in \mathbb R^n \;|\; |x|\leq 1 \}$, with boundary $\partial D^n = S^{n-1} = \{ x \in \mathbb R^n \;|\; |x| = 1 \}$ the $(n-1)$-dimensional sphere.

Let $f: D^n \to D^n$ be a homeomorphism that maps $x \in \partial D^n$ to $f(x) \in D^n \setminus \partial D^n$. Obviously $f$ induces a homeomorphism $\tilde{f}: D^n \setminus \{ x\} \to D^n \setminus \{ f(x) \}$.

Since $x \in \partial D^n$, we have that $D^n \setminus \{ x\}$ is convex and therefore homotopy equivalent to a point. On the other hand we can construct a homotopy equivalence $D^n \setminus \{ f(x) \} \simeq \partial D^n = S^{n-1}$ since $D^n$ is compact and radially convex wrt. a neighborhood of $f(x)$. Thus we get $\{pt\} \simeq D^n \setminus \{ x\} \cong D^n \setminus \{ f(x)\} \simeq S^{n-1}$, which is a contradiction by your technique of choice. For example $\pi_{n-1}(\{pt\}) \not \cong \pi_{n-1}(S^{n-1})$.
A homeomorphism $\phi:D \to D$ maps open sets to open sets. Pick a point $P$ in the interior of the disk and a disk $D(P,\delta)$ which lies inside the given disk $D$. If $P$ would be mapped on the boundary of $D$ then $\phi(P)$ would not be in the interior of $\phi(D(P,\delta))$; contradiction with the fact that $\phi(P) \in \phi(D(P,\delta))$ and $\phi(D(P,\delta))$ is open. 

This means that interior points are mapped to the interior of the disk. 

Pick now $Q$ on the boundary of $D$. If $\phi(Q)$ is on the interior of $D$, then by a similar argument with the one above, since $\phi^{-1}$ is also a homeomorphism it follows that $\phi^{-1}(\phi(Q))=Q$ is in the interior of $D$. Contradiction. 

Therefore $\phi$ maps the boundary of the disk onto the boundary of the disk.

[edit] As mentioned in the comments, I used the standard euclidean topology, not the induced one, and that might be a problem. I think that the best solution still remains the one mentioned in the first comment: 

That if the homeomorphism $\phi : D\to D$ maps $P$ to $Q$ then $D\setminus\{P\}$ and $D\setminus \{Q\}$ are homeomorphic. If $P$ is on the boundary and $Q$ is in the interior (or the other way around) we get two domains which are not homeomorphic: one is simply connected, one isn't (it has a "hole")