A criterion for rings of analytic functions

Abstract

corresponds to an open set in terms of the given topology on Q. The conformal structure is determined by the requirement that the topological local uniformizers (locally univalent functions) be analytic. This completes the proof of Theorem 1. Proof of Theorem 2. This is almost a corollary of the above proof. For every topological disk in Q is also a disk in the sense of Definition 2 (once the assumptions (1) and (2) in Theorem 2 are made). What we need to show is that every disk is open in the topology of Q. Choose a topological disk (see Definition 2a; here f maps A bicontinuously onto {zI l1}). Let Dr(cA and KrcA be the inverse images under f of {IzI f(p) for allf c Ri. We derive the topology on Q? via Lemma 3; this method is closely related to a theorem of Heins [7, Theorem A].) 4. Counterexamples. EXAMPLE 1. Let Q be the real line and let R be the ring of all complex-valued real-analytic functions on Q. EXAMPLE 2. Let Q be the real line. Let R be the ring of all complex-valued functions f on Q such that (a) f is real-analytic except at a finite set of points {xi}, This content downloaded from 157.55.39.136 on Thu, 19 May 2016 04:56:54 UTC All use subject to http://about.jstor.org/terms 1967] CRITERION FOR RINGS OF ANALYTIC FUNCTIONS 529 where {xjl depends onf; (b) near each x,,f is of the formf(x) = at(x) + b,(x)(x -Xi)13 + c1(x)(x x,)213 with ai, bi, ci real-analytic. The rings in Examples 1 and 2 satisfy Conditions (A) and (B) of Theorem 1, but of course they possess no disks. Note that in Example 2, there is no manifold structure on El in terms of which all the functions in R become analytic (in contrast to the situation for surfaces; cf. Corollary 1). EXAMPLE 3. Let Q be the euclidean space En. Let R be the ring of all complexvalued functions f on Q such that (a) f is C1 except at a finite set of points {xi}, where {xj depends on f; (b) near each xi, f is of the form