Let $(X, \ _X\mathscr{O})$ be an analytic space, that is, a ringed space with the property that for all $x \in X$, there is a neighbourhood $U$ of $x$ such that $(U, \ _X \mathscr{O} \vert_U)$ is isomorphic, as a ringed space, to $(Y, \ _Y \mathscr{O})$, where $Y$ is some subdomain of $\mathbb{C}^n$ and $_Y \mathscr{O} = \ _n \mathscr{O}/\mathscr{I}$, where $_n \mathscr{O}$ denotes the sheaf of holomorphic functions on $\mathbb{C}^n$ and $\mathscr{I}$ denotes the ideal sheaf of $_n \mathscr{O}$.
Equip $(X, \ _X \mathscr{O})$ with a separable topology such that the set of regular points $\mathscr{R}(X)$ is a complex manifold with a countable number of connected components. It should be made clear that a point $x \in X$ is said to be regular if there exists a neighbourhood $U$ of $x$ such that $(U, \ _X \mathscr{O} \vert_U)$ is a complex manifold.
On page 155 of Gunning and Rossi, they may make the claim that the closure of the connected components are analytic subvarieties, referred to as the global irreducible branches of $X$.
I agree that the closure of the connected components are analytic subvarieties, but I can't seem to understand the motivation behind calling them irreducible branches. Can we think of Riemann surfaces in this context?
I mean of course, Riemann surfaces are analytic spaces, etc... But I was wondering if someone could be a little more solid with this notion of an irreducible branch. Do we have a covering space type (conceptual) image?