Prove that for every n>2, neither the line nor the circle is homeomorphic to $\displaystyle S^n$, the n-sphere.
Hint: Use cutpoints.
The hint pretty much says it all. Removing a single point from a line divides it into 1 separated parts. Removing two points from a circle divides it into two separated parts. Neither of those is true for an n-sphere. Remove any finite number of points from an n-sphere and you still have a connected set.