By A. F. Beardon

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As D. A topology on a non-empty set X K . n f (X,T) where T is a topology on X: this is an implicit reference A metric space is a topological space, a set being open if and only if it is a union of open balls: this is the metric topology induced by the metric. In general, A-B denotes the set of points in A but not in B. 22 A subset Y of a topological space X is closed if and only if X -Y is open. Thus a finite union of closed sets is closed: any intersection of closed sets is closed. The interior of Y of Y: this is the largest open subset of closure of Y Y is the union of all open subsets O and is denoted by Y .

4 THE EXTENDED COMPLEX PLANE In this section we realise the unit sphere S = {(x,y,t) in ]R^ : x 2 + y 2 + t 2 =1} as a Riemann surface in three superficially different (but equivalent) ways. The most natural way to regard S as a surface is to project some 43 neighbourhood of each x on S normally onto the tangent plane at x. This does not realise S as a Riemann surface, however, for the correspondÂ ing transition functions do not preserve angles. 1. 1. Now define homeomorphisms * ! (X ,y,t) = and give S the atlas U by As we see that the transition maps are S on *2 (x,yft> = â€” , (|>1 (x,y,t) (j>2(x,y,t) = 1 atlas,

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.