A complex analytic space is a topological space (say, Hausdorff and second countable) such that each point has an open neighborhood homeomorphic to some zero set $V(f_1,\ldots,f_k)$ of finitely many holomorphic functions in $\mathbb{C}^n$, in a way such that the transition maps (restricted to their appropriate domains) are biholomorphic functions.
I have an embarrassing basic question. It seems weird to me that we are isomorphing the open neighborhoods to the closed sets $V(\ldots)$. I realize that we can think of $V(\ldots)$ as being open in its own induced topology, but I can't really picture what such homeomorphism would look like. Is there possibly a mistake in the definition I read? Should the open neighborhoods be homeomorphic to open subsets of the $V(\ldots)$?