A more detailed statement will be given for subsequent discussions. Let γ γ : [ a , b ] → → R 2 be a piecewise smooth Jordan plane curve. The Jordan curve theorem implies that γ γ divides R 2 into two components, a compact one and another that is non-compact. Let D denote the compact part; then D is bounded by γ γ . It now suffices to transfer this notion of boundary along a continuous map to our surface in R 3 . But we already have such a map: the parametrization of Σ Σ .