Are the Unit Ball and Any other Ball Topologically Equivalent How would I correctly show that the unit ball $B(0,1)\subset \mathbb{R}^n$ and the ball $B(a,r) \subset \mathbb{R}^n$ are Topologically Equivalent?
I know I need to find a one-to-one function $f: X\rightarrow Y$ for which $f$ and $f^{-1}$ are both continuous to prove it is Topologically Equivalent; however, I am having a hard time finding such a function. 
A function that came to mind is the equation of a circle or sphere. But I wouldn't know if that's correct.
I want to thank you for taking the time to read this question. I greatly appreciate any assistance you provide


A function that came to mind is the equation of a circle or sphere. 

You're on the wrong track here. What you want is not a function that defines a ball, but rather one that transforms one ball (radius $1$, center at the origin) into a different one (radius $r$, center $a$).
Think about how you would do this: you would need to "inflate" (or deflate if $r < 1$) the unit ball (i.e. scale all the points by $r$) and then translate all the points from their original home near the origin to their new location near $a$. How could you write this process as a map?
