By A. F. Beardon

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**Additional resources for A Primer on Riemann Surfaces**

**Sample text**

Let E be a subset of a surface S and suppose that E does not contain any convergent sequence of distinct points. Show that S - E is a surface. 2 RIEMANN SURFACES A surface is a Riemann surface if the change from one coordinate system to another is holomorphic. As this is our major concern, we give a formal definition. 1. A surface is a Riemann surface if the transi tion functions A tga are holomorphic whenever they are defined: the atlas is then called an analytic atlas. A subdomain D of

D. and to the 'same' points in J The functions on w on S are Wj(

Z f (1)(z) dz f(z)-w f b, f Show that is 1-1 and holomorphic on E. Prove that f(z) = az + b. , then the Weierstrass- 20 2 TOPOLOGY Summary. This chapter contains a brief introduction to general topology. The choice of material is determined only by whether or not it will prove useful later in the text. A few other topics will be discussed as they arise. 1 METRIC SPACES A metric on a set (x,y) of points in X X is a real valued function d of pairs with the properties (1) d(x,y) ^ O with equality if and only if (2) d(x,y) = d(y,x) (3) d(x,z) ^ d(x,y) + d(y,z).