Complex analytic variety

Remarks

For points $a \notin A$ the second condition holds with $U = X \setminus A$ and $f_1 = 1$, so it only matters at points of $A$. Example: in $\C^2$, $A = \set{z_1 z_2 = 0}$ is an analytic variety, with $U = \C^2$ and $f_1 = z_1 z_2$; it is the union of the two coordinate axes. Non-example: the closed unit disc $\set{\abs{z} \le 1}$ in $\C$ is closed but not analytic. Let $a$ be a point with $\abs{a} = 1$ and let $U$ be a connected neighbourhood of $a$. A holomorphic function on $U$ that vanishes on the part of the disc in $U$ vanishes on an open set, hence on all of $U$. So the common zero set would be $U$, not the part of the disc in $U$. A set that satisfies condition 2 only at its own points is called locally analytic. It need not be closed, for example the punctured line $\set{z_1 = 0} \setminus \set{0}$ in $\C^2$. Some authors reserve "variety" for irreducible analytic sets; here no irreducibility is required. The empty set and $X$ itself are analytic varieties, and finite unions and arbitrary intersections of analytic varieties are analytic varieties.