Find a homeomorphism between two open discs in $\mathbb{R}^{2}$. The title says it all, but let me explain my plight a bit as I am trying to learn about how to find homeomorphisms and am a rookie. 
Suppose we have two open discs in $\mathbb{R}^{2}$. Call them $D_1$ and $D_2$. My first reaction was "Well, imagine it's like Photoshop. I have two discs and I use the resize tool. So I take $D_1$ and just resize it until it's the same size as the other $D_2$." In more math terms, if I think in polar coordinates, $D_1$ can be turned into $D_2$ if I just scale its radius. 
Here's my problem: I don't know how to state that in a way that allows me to prove it's a homeomorphism. All I have is these images in my mind. 
Remark: This is something I have encountered in another problem with finding a homeomorphism between a circle and a square. Just because I can imagine it doesn't mean I know how to state that thought in words so I can prove it.
 A: Your idea of polar coordinates is a good one. However, the discs may not be centred at the origin, so describing them in terms of polar coordinates can be quite complicated. We can rectify this though; we can translate both discs so that they are centred at the origin. More precisely, $B(x_0, r)$ is homeomorphic to $B(0, r)$ with homeomorphism given by $x \mapsto x - x_0$. So our two given discs $D_1$ and $D_2$ are homeomorphic to discs $\widehat{D_1}$ and $\widehat{D_2}$ which are centred at the origin. If we can show that $\widehat{D_1}$ and $\widehat{D_2}$ are homeomorphic, then it follows that $D_1$ and $D_2$ are homeomorphic. 
As $\widehat{D_1}$ and $\widehat{D_2}$ are centred at the origin, $\widehat{D_1} = B(0, r_1)$ and $\widehat{D_2} = B(0, r_2)$ for some radii $r_1$ and $r_2$. What you want to do now is "resize" one of the discs, say $\widehat{D_2}$, as you mention in your answer. Pick a point in $\widehat{D_2}$ other than the origin. What happens to the point in the resizing process? If you were to write this point in polar coordinates $(r, \theta)$, what would happen to $r$? What about $\theta$? Answering these questions will help you to write down the map explicitly.
A: Both (nonzero) dilation and translation are homeomorphisms in the plane. You are composing two homeomorphisms; the result is another.
