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 topological surface is nonseparating when cutting along it leaves a connected topological surface. In that case an arc joining the two new boundary components closes to a simple closed curve meeting the original once. Their algebraic intersection number of curves on an oriented surface is $\pm 1$. Consequently the original curve represents a nonzero primitive homology class.

On a closed oriented topological surface, a separating simple closed curve bounds a compact subsurface, whose fundamental class exhibits it as a boundary in homology. Thus on a closed oriented topological surface, separation is equivalent to vanishing of its integral first homology class.

The closed-surface condition cannot be omitted. In the open annulus $S^1\times \mathbb{R}$, the core $S^1\times \{0\}$ separates the two ends, yet generates $H_1(S^1\times \mathbb{R};\mathbb{Z})\cong \mathbb{Z}$. Either side is noncompact and supplies no finite singular chain bounding that core.