Define an analytic space to be a topological space $X$ equipped with a sheaf of rings $\mathcal{O}_X$ such that for every point $x \in X$ there is a neighbourhood $U \subseteq X$ such that $(U, \mathcal{O}_X \vert_U)$ is isomorphic (as a ringed space) to $(Y, \mathcal{O}_Y)$, where $Y \subseteq \mathbb{C}^n$ is some domain and $\mathcal{O}_Y = {}_n \mathcal{O} / \mathscr{I}$, where $_n\mathcal{O}$ denotes the sheaf of germs of analytic functions $f: \mathbb{C}^n \to \mathbb{C}$ and $\mathscr{I}$ denotes the ideal sheaf of ${}_n \mathcal{O}$.
Define a regular point of an analytic space to be a point $x \in X$ which admits a neighbourhood $U$ such that $(U, \mathcal{O}_X \vert_U)$ is a complex manifold (isomorphic as a ringed space to $(Y, \mathcal{O}_Y)$, where $Y \subseteq \mathbb{C}^n$ is some domain and $\mathcal{O}_Y$ denotes the sheaf of analytic functions $f: Y \to \mathbb{C}$.
Can someone provide me an example of an analytic space that has no regular points and is very different to a complex manifold. Intuitively, I think of analytic spaces and complex manifolds in a rather similar boat, but analytic spaces permit sharp points. I am aware that this intuition may be terribly erroneous, but it's what I've picked up over the last month or two of studying analytic spaces.
I was also wondering if someone could provide their example (the example they keep in their head) for the difference between a Stein manifold and Stein space? This is essentially the same question as the one just stated.
Any help/remarks/intuitions is really appreciated.
Cheers.
