Analytic Continuation - The Topology of The Set of Germs

The Topology of The Set of Germs

Let g and h be germs. If |h₀ − g₀| < r where r is the radius of convergence of g and if the power series that g and h define identical functions on the intersection of the two domains, then we say that h is generated by (or compatible with) g, and we write g ≥ h. This compatibility condition is neither transitive, symmetric nor antisymmetric. If we extend the relation by transitivity, we obtain a symmetric relation, which is therefore also an equivalence relation on germs (but not an ordering). This extension by transitivity is one definition of analytic continuation. The equivalence relation will be denoted .

We can define a topology on . Let r > 0, and let

The sets U_r(g), for all r > 0 and g ∈ define a basis of open sets for the topology on .

A connected component of (i.e., an equivalence class) is called a sheaf. We also note that the map φ_g(h) = h₀ from U_r(g) to C where r is the radius of convergence of g, is a chart. The set of such charts forms an atlas for, hence is a Riemann surface. is sometimes called the universal analytic function.