# Continuous surjections

• May 4th 2011, 01:38 PM
bkarpuz
Continuous surjections
Dear MHF members,

I have the following problem.
Let $\mathbb{D}:=\{(x,y):\ x^{2}+y^{2}\leq1\}$ denote the unit disk in the plane, with interior $\mathbb{D}^{\circ}$ and boundary $\partial\mathbb{D}$.

1. Does there exist a continuous surjection $f:\mathbb{D}\to\mathbb{D}^{\circ}$?
2. Does there exist a continuous surjection $f:\mathbb{D}^{\circ}\to\mathbb{D}$?
3. Does there exist a continuous surjection $h:\mathbb{D}\to\partial\mathbb{D}$, which leaves every point on $\partial\mathbb{D}$ fixed?

Many thanks.
bkarpuz
• May 4th 2011, 02:55 PM
Tinyboss
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.