In the book Algebraic Curves and Riemann Surfaces, Miranda explains how to resolve the node of a plane curve by plugging the hole at the node using hole charts. The idea is at a node $p$, the surface $X$ is the locus of $gh$, we delete $p$ from $X$ so that $X \setminus \{p\}$ is the union of $X_g \setminus \{p\}$ and $X_h \setminus \{p\}$. We now have two hole charts at $X_g$ and $X_h$ and we resolve the node at $p$ by plugging in these two hole charts.
My question is how is the surface that we get after plugging in the hole charts Hausdorff? This construction looks to me like the line with 2 origins.