In Miranda's , he wrote such a proposition:
proposition 1.6 of chapter 1: Let $X,Y$ be 2 Riemann surfaces. $U\subset X$, $V\subset Y$ are their open subsets. Suppose $\phi: U\rightarrow V$ is an isomorphism of Riemann surfaces, then there's a unique complex structure on $Z:=X\amalg Y/\phi$ such that the 2 natural embeddings $j_X: X\rightarrow Z$ and $j_Y: Y\rightarrow Z$ are holomorphic maps. In particular, if $Z$ itself is Hausdorff, then we obtain a Riemann surface.
My question is: for example, we can glue 2 torus (of genus 1) together by choosing $U,V$ as 2 disks small enough. Of course in this case $Z$ is a Hausdorff space. But what's the genus of it? What makes me puzzled is that topologically this identification space doesn't look like a closed surface! (it looks like the "tube" in the middle of the genus 2 torus has been closed!) Can anybody explain this space in detail to me? thanks in advance!
Best Answer
"Of course in this case $Z$ is a Hausdorff space" - of course not. The point on boundary of $U$ in the first torus, and "corresponding" point on boundary of $V$ in the second torus (corresponding meaning they are limits of the identified sequences of points in $U$ and in $V$) will have no disjoint open neighborhoods in $Z$. (You can see the same thing when trying to glue two segments into a Y shape by identifying along a open subsegments; there are two points at the center of the Y, and they don't have disjoint neighborhoods.)