The complement of a simple closed curve in the Riemann sphere has two connected components (Jordan). Schoenflies theorem implies that each of these components is homeomorphic to a disk and hence each has trivial fundamental group. Is there a direct (and simple) way to see that each component has trivial fundamental group without invoking Schoenflies?
Why is a component of complement of Jordan curve 1-connected w/o Schoenflies?
Chris Judge
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