analytic space

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

  1. 1.

    There exists a countableMathworldPlanetmath number of open sets Vj covering X.

  2. 2.

    For each Vj there exists a homeomorphismMathworldPlanetmath φj:YjVj, where Yj is a local complex analytic subvariety in some n.

  3. 3.

    If Vj and Vk overlap, then φj-1φk is a biholomorphism.

Usually one attaches to X a set of coordinate systemsMathworldPlanetmath 𝒢, which is a set (now uncountable) of triples (Vι,φι,Yι) as above, such that whenever V is an open set, Y a local complex analytic subvariety, and a homeomorphism φ:YV, such that φι-1φ is a biholomorphism for some (Vι,φι,Yι)𝒢 then (V,φ,Y)𝒢. Basically 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 (Vι,φι,Yι)𝒢 with pVι and Yι a complex manifold. All other points are the singular pointsMathworldPlanetmathPlanetmath.

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


  • 1 Hassler Whitney. . Addison-Wesley, Philippines, 1972.
Title analytic space
Canonical name AnalyticSpace
Date of creation 2013-03-22 17:41:43
Last modified on 2013-03-22 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