Let $U \subseteq C$ be an open set. We say that a closed curve $γ : [t_{0}, t_{1}] \to U$ is homologous to zero if 1;; $n_{γ} (a) = 0$ for all $a \in C - U$ .

Example

Any curve which is null homotopic is homologous to 0. This does not go in the opposite direction.

This is a curve in $\mathbb{C}-\{ -1,1 \}$ which is not null homotopic but is homologous to zero.