# Why is the unit circle not homeomorphic to the closed unit disk?

I know that the unit circle = $$\{(x,y): x^2+y^2 =1\}$$ is not homeomorphic to the closed unit disk = $$\{(x,y): x^2+y^2 \leq 1\}$$, but I'm not sure how to prove it. I've tried with arguments with cut-points and with (path)connectedness, but still not getting a good argument. Any help?

The unit disk $$B=\{x^2+y^2\leq1\}$$ has fundamental group $$\pi_1(B)=\{0\}$$, while the circle $$S^1=\{x^2+y^2=1\}$$ has fundamental group $$\pi_1(S^1)=\Bbb Z$$. And a necessary condition for two topological spaces to be homeomorphic is that they have the same fundamental group (or better isomorphic groups).