$X$ and $Y$ be smooth affine plane curves defined by $y^2 = h(x)$ and $w^2 = k(z)$ respectively, where $h(x)$ is of degree $2g + 1 + \epsilon$ and $k(x) = z^{2g + 2}h(1/z)$. Let $U=\{(x, y) \in X\mid x \ne 0\}$ and $V = \{(z, w) \in Y\mid z \ne 0\}$. Define an isomorphism $\phi : U \rightarrow V$ by $$\phi(x,y) = (z, w) = (1/x, y/x^{g+1})$$ Let $Z$ be Riemann surface obtained by glueing $X$ and $Y$ together along $U$ and $V$ via $\phi$.
In the proof of lemma 1.7 author states that one can readily check that $Z$ is Haudorff. But I could not show $\{(x,y)\}$ ($(x, y) \in X \setminus U$) and $\{(z,w)\}$ ($(z, w) \in Y\setminus V$) satisfy Hausdorff condition in $Z$.