Prove a figure eight is not homeomorphic to a circle

For the sake of lowering the number of unanswered questions:

If there was a homeomorphism between both figures, say $\;f: S^1\vee S^1\to S^1\;$ , then also $\;\overline f: \left(S^1\vee S^1\right)\setminus\;\{x_0\}\to S^1\setminus\{f(x_0)\}$ , would be a homeomorphism (why?).

But if $\,x_0=$ the intersection (in fact, the tangent) point of the two circles, then $\;\left(S^1\vee S^1\right)\setminus\{x_0\}\;$ is not connected, yet $\;S^1\setminus\{f(x_0)\}\;$ still is connected, no matter what $\;f(x_0)\in S^1\;$ is.