# analytic space

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

1. 1.

There exists a countable number of open sets $V_{j}$ covering $X.$

2. 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. 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 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.

## References

• 1 Hassler Whitney. . Addison-Wesley, Philippines, 1972.
Title analytic space AnalyticSpace 2013-03-22 17:41:43 2013-03-22 17:41:43 jirka (4157) jirka (4157) 4 jirka (4157) Definition msc 32C15 complex analytic space LocallyCompactGroupoids