Complex Analytic Subset vs. Complex Analytic Set In my complex geometry class we have introduced two concepts of analytic sets. Let $M$ be a complex manifold.

*

*A subset $A\subset M$ is called complex analytic subset, if for each $p\in M$ there exists an open neighborhood $U\subset M$ of $p$ and finitely many holomorphic functions $f_{1},...,f_{k}:U\rightarrow \mathbb{C}$ such that

$$
A\cap U = \{ q\in U \mid f_{1}(q)=...=f_{k}(q)=0 \}. 
$$


*A subset $A\subset M$ is called complex analytic set, if for each $p\in A$ (!!!) there exists an open neighborhood $U\subset M$ of $p$ and finitely many holomorphic functions $f_{1},...,f_{k}:U\rightarrow \mathbb{C}$ such that

$$
A\cap U = \{ q\in U \mid f_{1}(q)=...=f_{k}(q)=0 \}. 
$$
I do not understand the difference. Moreover, I do not understand the following implication: A domain $D\subset \mathbb{C}^{n}$ (open connected subset) is a complex analytic set but not a complex analytic subset. If it were a complex analytic subset then we necessary have that $D=\mathbb{C}^{n}$. This I do not understand. Would be thankfull for explication.
Greetings,
Nina
 A: You've already pointed out the difference with a "(!!!)": the first definition quantifies over $M$, while the second quantifies over $A$. Let's explore a little about what that means:

*

*In the first definition, $A$ must be closed in $M$: writing $U_p$ for the neighborhood of $p$ guaranteed by the definition, we have $A^c = \bigcup_{p\in M} U_p\setminus (U_p\cap A)$, an open subset.

*In the second definition, $A$ is locally closed - the intersection of an open set and a closed set. This is because (using the same notation as before) we have that $A\subset \bigcup_{p\in A} U_p$, and $\bigcup_{p\in A} U_p$ is open in $M$ while $A$ is closed in $\bigcup_{p\in A} U_p$ by the same argument as before. Therefore $A=U\cap V$ for some $U\subset M$ open and $V\subset M$ closed.

This should resolve your issues in the final paragraph: as $\Bbb C^n$ is connected, the only sets which are open (as assumed in the definition of a domain) and closed (as in our bullet point above) are the empty set and the whole space.
(Let me comment that these definitions are not insisted upon quite so strictly as one progresses in the literature - I can't remember seeing this sort of distinction drawn anywhere in my work. Authors will usually refer to a complex analytic space and then talk about how it's embedded in some other complex analytic space, which gets around this stuff that's causing problems for you.)
