A Hausdorff topological space $X$ is said to be an analytic space if:

1. There exists a countable number of open sets $V_j$ covering $X.$
2. For each $V_j$ there exists a homeomorphism $\varphi_j \colon Y_j \to V_j ,$ where $Y_j$ is a local complex analytic subvariety in some ${\mathbb{C}}^n .$
3. If $V_j$ and $V_k$ overlap, then $\varphi_j^{-1} \circ \varphi_k$ is a biholomorphism.

Usually one attaches to $X$ a set of coordinate systems $\mathcal{G}$ which is a set (now uncountable) of triples $(V_\iota,\varphi_\iota,Y_\iota)$ as above, such that whenever $V$ is an open set, $Y$ a local complex analytic subvariety, and a homeomorphism $\varphi \colon Y \to V$ such that $\varphi_\iota^{-1} \circ \varphi$ is a biholomorphism for some $(V_\iota,\varphi_\iota,Y_\iota) \in \mathcal{G}$ then $(V,\varphi,Y) \in \mathcal{G} .$ Basically $\mathcal{G}$ is the set of all possible coordinate systems for $X$

We can also define the singular set of an analytic space. A point $p$ is simple, regular or nonsingular if there exists (at least one) a coordinate system $(V_\iota,\varphi_\iota,Y_\iota) \in \mathcal{G}$ with $p \in V_\iota$ and $Y_\iota$ a complex manifold. All other points are the singular points.

Any local complex analytic subvariety is an analytic space, so this is a natural generalization of the concept of a subvariety.

Bibliography

1

Hassler Whitney. Complex Analytic Varieties. Addison-Wesley, Philippines, 1972.