By A. F. Beardon

ISBN-10: 0521271045

ISBN-13: 9780521271042

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

**Example text**

If such an f f ^ : Y exists, we say that X f and X X onto exists Y if is open and Y are homeomorphic and this is an equivalence relation on the class of all topological spaces. In general, we identify homeomorphic spaces (just as we identify isomorphic groups). The function f ^(B) thus If is a neighbourhood of f X f : X -*■ Y x is continuous at a point whenever B x in is a neighbourhood of X f(x): is continuous if and only if it is continuous at every point in and Y in the familiar if X. are metric spaces, then continuity can be defined (equivalently) (e,6) manner.

The remaining cases satisfy = b^ = 0, In this case, we take r. If f(U) (3/y/S) and s * t U f(V) and to be horizontal open discs, each of radius r <1/2: r of u and and r v respectively with y >0 and +1) is chosen small enough to violate this inequality, then f(V) (a,y,t), then t - s = |y| -1oc-81 ^ r(|a1 - b j so if s < t). have a common point, then there are points within a distance We insist that V (say f(U) are disjoint. 1 1. Show that if a sequence lies in a compact subset of a surface, then it has a convergent subsequence.

We then obtain explicit examples of surfaces. 1 SURFACES A surface S is a topological space, each point of which appears (topologically) to lie in an open subset of (C. Although one naturally thinks of surfaces as lying within JR^, we shall not insist that this is so: indeed, the essential idea is to define a surface in its own right without it being necessary to discuss the possible existence of any larger space containing it. 1. A surface S is a topological space together with a family (1) A={