Is the collection of reduced points in a complex space is open?

Question

A point $x$ in a complex (analytic) space $(X,\mathcal{O}_{X})$ is said to be reduced if $\mathcal{O}_{X.x}$ is a reduced ring i.e. it doesn't have any non-zero nilpotent elements. Defined the set $S=\{x\in X\mid x\,\text{is a reduced point of}\,(X,\mathcal{O}_{X})\}$. Is $S$ necessarily an open subset of $X$?

Answer 1

Yes it is!

For exact, the set of non-reduced points of a complex analytic space $(X,\mathcal{O}_X)$ is an analytic set, and these sets are canonically closed complex analytic subspace of $X$; so the set of reduced points of $X$ is open.

See H. Grauert, R. Remmert (1984) Coherent Analytic Sheaves, Springer-Verlag; chapter 4, §1.1 and §3.1.