Editing Analytic continuation (section)

==The topology of the set of germs==
Let ''g'' and ''h'' be [[Germ (mathematics)|germs]]. If <math>|h_0-g_0|<r</math> where ''r'' is the radius of convergence of ''g'' and if the power series defined by ''g'' and ''h'' specify 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 [[transitive closure|extend]] the relation by [[transitive relation|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 <math>\cong</math>.

We can define a [[topology]] on <math>\mathcal G</math>. Let ''r'' > 0, and let

:<math>U_r(g) = \{h \in \mathcal G : g \ge h, |g_0 - h_0| < r\}.</math>

The sets ''U<sub>r</sub>''(''g''), for all ''r'' > 0 and <math>g\in\mathcal G</math> define a [[basis (topology)|basis of open sets]] for the topology on <math>\mathcal G</math>.

A [[connected space|connected component]] of <math>\mathcal G</math> (i.e., an [[equivalence class]]) is called a ''[[sheaf (mathematics)|sheaf]]''. We also note that the map defined by <math>\phi_g(h) = h_0 : U_r(g) \to \Complex,</math> where ''r'' is the radius of convergence of ''g'', is a [[atlas (topology)#charts|chart]]. The set of such charts forms an [[atlas (topology)|atlas]] for <math>\mathcal G</math>, hence <math>\mathcal G</math> is a [[Riemann surface]]. <math>\mathcal G</math> is sometimes called the ''universal analytic function''.