Let U be a open subset of $\mathbb{C}$. We say a cycle $\gamma$ in $U$ is null-homologous in $U$ if the winding number of the cycle at each point $p\in \mathbb{C}-U$ is zero.
Now, the definition seems a bit confusing, since we are talking about winding number of some cycle inside $U$ at points outside of the set $U$, doesnt that means winding number of that cycle at any point outside $U$ will be zero? since the cycle will not wind around any point outside the set it lives in?
The text I am reading gives some examples. But I cant picture any cycle inside $U$ whos winding number is not zero for points outside of $U$.