Can anyone help me see why this statement is true

Let $p$ be a point of a Riemann surface $X$, and let $S$ be a subset of $X$ whose closure does not contain $p$, then theres is a closed path $\gamma$ on $X$ with the following properties, $\gamma $ is $1-1$ and its image lies inside the domain $U$ of a chart $\phi:U \rightarrow V$ on $X$, the closed path $\phi \circ \gamma$ has winding number $1$ about $\phi(p)$, no point of $S$ wich lies in the domain of $U$ is mapped to the interior of $\phi\circ \gamma$.

I have tried to since why this is true but couldnt come up with a proof, any help is aprecciated, thanks in advance.