Let $M$ be a complex manifold (or a complex analityc space) and $Z$ be an analytic subset of $M$. By Noetherianity of the rings of germs of analytic functions at a point we know that $Z$ has finitely many irreducible germs at any $p \in Z$.
What about the "global" irreducible components?
How does one prove that a decomposition into irreducible analytic sets exists? (I have looked into Gunning and Rossi but couldn't find the relevant reference off-hand.)
I also have a related question: is it possible that an analytic set is connected but has infinitely many irreducible components?