Nonseparating curve
= Nonseparating curve
A simple closed curve in a connected oriented <surface> is nonseparating when cutting along it leaves a connected <surface>. A separating curve bounds a subsurface and represents zero in first <homology>. For a nonseparating curve, an arc between the two new boundary components closes to a curve meeting it once. Its <algebraic intersection number of curves on an oriented surface> is $\pm1$, proving that the original curve represents a <primitive homology class>.