Homeomorphism between punctured disk and circle Is there a homeomorphism between the circle $S^1$ and the punctured disk $D^2 \setminus \\{p\\}$, $p\in D^2$ ? Intuitively it seems that when you puncture the disk you can then deform it into a circle but I can't see how this would be done.

No, the two spaces can't be homeomorphic. $S^1$ is compact but the punctured disk is not. Also, you can disconnnect $S_1$ by removing two points, but that's not possible for the punctured disc.

You can continuously deform the punctured disc to get $S_1$ by using the retraction mapping: Just walk every point in the punctured disc away from the center along a radius until you reach the boundary.