= Nonseparating curve
A <simple closed curve> in a connected oriented <topological surface> is \b[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 $\pm1$. Consequently the original curve represents a nonzero <primitive homology class>.
On a \b[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.
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