I am reading the book Introduction to complex analytic geometry by Stanislaw Lojasiewicz. In Chapter II, we have that a subset $Z$ of a complex manifold $M$ is called an analytic subset of $M$ if every point of $M$ has an open neighbourhood $U$ such that the set $Z \cap U$ is a globally analytic subset of $U$ i.e. there exist finitely many holomorphic functions $f_1, \dotsc, f_k$ in $U$ such that
$Z \cap U =\{x \in U \ : \ f_1(x)=\dotsc =f_k(x)=0\}.$
Is it true that the connected components of $Z$ are analytic subsets of $M$ and is open in $Z$?
I think saw some result in this direction in the book but it seems confusing. Any good reference for the same would be a great help ! Thanks in Advance.