Radó's theorem (Riemann surfaces)
Encyclopedia
In mathematical complex analysis, Rado's theorem, proved by , states that every connected
Riemann surface
is second-countable
(has a countable base for its topology).
The Prüfer surface 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 Rado's theorem in higher dimensions is false: there are 2-dimensional connected complex manifolds that are not second-countable.
Connected space
In topology and related branches of mathematics, a connected space is a topological space that cannot be represented as the union of two or more disjoint nonempty open subsets. Connectedness is one of the principal topological properties that is used to distinguish topological spaces...
Riemann surface
Riemann surface
In mathematics, particularly in complex analysis, a Riemann surface, first studied by and named after Bernhard Riemann, is a one-dimensional complex manifold. Riemann surfaces can be thought of as "deformed versions" of the complex plane: locally near every point they look like patches of the...
is second-countable
Second-countable space
In topology, a second-countable space, also called a completely separable space, is a topological space satisfying the second axiom of countability. A space is said to be second-countable if its topology has a countable base...
(has a countable base for its topology).
The Prüfer surface 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 Rado's theorem in higher dimensions is false: there are 2-dimensional connected complex manifolds that are not second-countable.