analytic space
A Hausdorff topological space $X$ is said to be an analytic space if:
 1.

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 ${\u2102}^{n}.$

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^{} $\mathcal{G}$, 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 $({V}_{\iota},{\phi}_{\iota},{Y}_{\iota})\in \mathcal{G}$ then $(V,\phi ,Y)\in \mathcal{G}.$ 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 if there exists (at least one) a coordinate system $({V}_{\iota},{\phi}_{\iota},{Y}_{\iota})\in \mathcal{G}$ 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^{}.
References
 1 Hassler Whitney. . AddisonWesley, Philippines, 1972.
Title  analytic space 

Canonical name  AnalyticSpace 
Date of creation  20130322 17:41:43 
Last modified on  20130322 17:41:43 
Owner  jirka (4157) 
Last modified by  jirka (4157) 
Numerical id  4 
Author  jirka (4157) 
Entry type  Definition 
Classification  msc 32C15 
Synonym  complex analytic space 
Related topic  LocallyCompactGroupoids 