analytic space

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 ${\phi }_j:Y_j\to V_j,$where $Y_j$ is a local complex analytic subvariety in some ${ℂ}^n.$

3. If $V_j$ and $V_k$ overlap, then ${\phi }_j^{-1}\circ {\phi }_k$ is a biholomorphism.

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

We can also define the singular set of an analytic space. A point $p$ isif there exists (at least one) a coordinate system $(V_{\iota },{\phi }_{\iota },Y_{\iota })\in 𝒢$ 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.