Read e-book online A Primer on Riemann Surfaces PDF

By A. F. Beardon

ISBN-10: 0521271045

ISBN-13: 9780521271042

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Is any set of the form and Y A x b where respectively. Clearly Z itself A and B are is an open rectangle: also, a finite intersection of open rectangles is an open rectangle. 1 that the class of open rectangles is a base for some topology on Z: we call this the product topology on Z. There are natural coordinate maps P 1 : (x,y) of Z onto X and Y h- x It follows that if h- y respectively and these are continuous because (p )_ 1 (A) = A x Y Plf , P 2^« P 2 : (x,y) , , f : W -*■ Z (p2)_ 1 (B) = X * B .

Our construction depends on finding certain groups of Mobius transformations. 1. Let D be a domain in and let G be a group of Mobius transformations with the properties (1) g(D) = D for every (2) if and g€G g g*I, in G? then the fixed points of (3) for each compact subset {g € G : g(K) n K K of D, g lie outside D? the set * 0} is finite. Then the quotient space Proof. Let G(z). We give D/G D/G inherits a natural analytic atlas. 1. As we see that D/G q(w^) D q z to its is open, continuous and is connected and q is continuous, and ^[(v^) D/G where is Hausdorff.

2 1. Show that of w w contains points of is in Y 3Y if and only if every neighbourhood and points of X-Y. 2. Show that the class of intervals and HR is a topology on 3. For each rvi\ ]R. (-°°,a) j in J, is also a topology on X. By considering the that contain a given class together with 0 Show that in this topology, C let T^ be a topology on of subsets of smallest topology that contains {x} X, T^ is not closed. X. Show that to be those topologies show that there exists a C. 3 HAUSDORFF SPACES In analytic arguments about topological spaces it is essential to be able to distinguish between distinct points by means of open sets and in general, this is not possible.

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A Primer on Riemann Surfaces by A. F. Beardon

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