# Download A Primer on Riemann Surfaces by A. F. Beardon PDF

By A. F. Beardon

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

**Example text**

By taking B f are open and g open, we see that ifhis open then so is h then so is g. Finally, = gf. Exercise 2,7 1, Let generated by x that the map g G (x) b- exp(ix) G x + 2irand let of X/G into W be the group of translations of the real line f : X -► X/G be the natural map. Show = iz : |z| = l) defined by is properly defined and 1-1. Show that the composite map h : x h- exp(ix) is open, continuous and surjective and deduce that is homeomorphic to X/G W. 8 A BASE FOR A TOPOLOGY * Let B be a class of subsets of of those subsets of X of sets chosen from B.

It is a theorem (much later) and not an assumption that these conditions hold for a Riemann surface: this example merely serves to emphasize this result and is of no other importance. 6. First, define f : JR^ ‘►JR'* (x,y,z) if y < O? (x+^-,y,0) if y > O . f (x,y,z) Next, partition JR into the horizontal planes = i(x,y,z) e JR and define : z = t) by 38 S = f(IR3) = {(x,y,z) elR3 : y < 0} U PQ . Observe that f restriction of induced by so A f ffc. is acts on S to and let We give T-open as the identity map.

If the charts S S is a surface and has a natural concept of (for example, if