I am reading McDuff and Salamon's "J-holomorphic curves and symplectic topology". I am stuck at a (probably trivial) point. On page 32 of the book, the authors take an equivalence relation $\Gamma_0$ on $\Sigma\setminus X$ (where $\Sigma$ is a compact Riemann surface and $X$ a finite subset of it), thought of as a subset of $\Sigma\setminus X\times\Sigma\setminus X$ and consider its closure $\Gamma$ in $\Sigma\times\Sigma$. It is known that $\Gamma_0$ is closed in $\Sigma\setminus X\times\Sigma\setminus X$ and that $\Gamma\setminus\Gamma_0\subseteq X\times X$. They then claim that $\Gamma$ is the graph of an equivalence relation on $\Sigma$. I'm able to see that $\Gamma$ is reflexive and symmetric. I'm not able to see why transitivity of $\Gamma$ is true. I'm not sure of how much of the preceding part of the proof (on page 31) is necessary for the proof of this assertion.
In summary, can someone help me see that $\Gamma$ is indeed transitive?