The complement of a simple closed curve in the Euclidean plane 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?

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?