A characterization of the $2$-sphere in terms of Jordan curve separation

Abstract:

We show that, up to homeomorphism, the $2$-sphere is the only compact arcwise connected metric space $X$ satisfying (i) If $J$ is a simple arc in $X$, then $X\backslash J$ is arcwise connected; (ii) If $J$ is a simple closed curve in $X$, then $X\backslash J$ is disconnected (in the sense that $X\backslash J$ contains a proper nonempty subset that is both open and closed).