Simple closed curve ($\gamma :S^1\hookrightarrow X$)

A topological embedding $\gamma :S^1\to X$ of a circle into a topological space. On an oriented surface, an essential such curve is noncontractible; on a punctured surface, nonperipheral means that it does not surround only one puncture.

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 $\pm 1$, proving that the original curve represents a primitive homology class.