
Continuous surjections
Dear MHF members,
I have the following problem.
Let $\displaystyle \mathbb{D}:=\{(x,y):\ x^{2}+y^{2}\leq1\}$ denote the unit disk in the plane, with interior $\displaystyle \mathbb{D}^{\circ}$ and boundary $\displaystyle \partial\mathbb{D}$.
 Does there exist a continuous surjection $\displaystyle f:\mathbb{D}\to\mathbb{D}^{\circ}$?
 Does there exist a continuous surjection $\displaystyle f:\mathbb{D}^{\circ}\to\mathbb{D}$?
 Does there exist a continuous surjection $\displaystyle h:\mathbb{D}\to\partial\mathbb{D}$, which leaves every point on $\displaystyle \partial\mathbb{D}$ fixed?
Many thanks.
bkarpuz

1) The continuous image of a compact set is compact.
2) Yes. Think about expanding the disk a little, and "capping" the radius of the image at 1.
3) No, but I don't know of a really short proof.