In mathematical complex analysis, Radó's theorem, proved by , states that every
connected Riemann surface is
second-countable
In topology, a second-countable space, also called a completely separable space, is a topological space whose topology has a countable base. More explicitly, a topological space T is second-countable if there exists some countable collection \mat ...
(has a countable base for its topology).
The
Prüfer surface Pruefer or Prüfer is a surname of German derivation, and may refer to:
* Heinz Prüfer, German Jewish mathematician (1896-1934)
* Kevin Prufer, American poet (1969)
* Gustav Franz Pruefer
Gustav, Gustaf or Gustave may refer to:
* Gustav (name), a ...
is an example of a surface with no countable base for the topology, so cannot have the structure of a Riemann surface.
The obvious analogue of Radó's theorem in higher dimensions is false: there are 2-dimensional connected complex manifolds that are not second-countable.
References
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{{DEFAULTSORT:Rado's theorem (Riemann surfaces)
Riemann surfaces
Theorems in complex analysis