A Primer on Riemann Surfaces by A. F. Beardon

By A. F. Beardon

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If this is so, is an open subset of S. If D is also compact, then it is a closed subset of the Hausdorff space S. As S is connected we find that D = S and we have proved the next result. 3. A compact surface has no non-trivial extension. 6). 4. A surface is arcwise connected. Proof. For can be joined to x parametric disc joined to x Q x on S, let by a curve on with centre y. S. Cx]denote For any S, of the connected space S can be. Thus and so S = [x] [x] S which construct a Clearly every point of by a curve or no point of Q are open subsets the points on y on Q can be and S-[x] as required.

1) the latter because we may elect to choose the empty collection. Now let T * be a topology on X. We say that B is a base for set is open if and only if it is a union of sets in T if B). B = T (so a The open balls in a metric space are a base for the metric topology. Given a class B, it is * of interest to know when B is a topology (necessarily with base B) and this is easily answered. 1. The class B is a topology on if (1) X € B (2) if and B 1 and B 2 are in B then B^ n b2 is in * B . X if and only X 29 Proof.

In general, we shall define concepts relating to a Riemann surface only when these remain invariant under pre and post applications of univalent holomorphic mappings. As a typical example, let us define the angle between two smooth curves which cross at a point surface Let the two curves crossing at in the parametric region curves, say y and on a Riemann R. a y^ and are smooth if x we can apply be a y and c. If x lies and obtain two image crossing at x^ y^ are continuously differentiable and a (= <|>a (x)) in (D.

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