A Primer on Riemann Surfaces by A. F. Beardon

By A. F. Beardon

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Y It is common to use to denote the corresponding point set as well although, for emphasis, we sometimes use Cyl = (y(t) The curve y final point : a < t < b}. is closed if the initial point y(b). The curve y is simple y(a) if y coincides with the is 1-1 on [a,b]: it is a simple closed curve if it is closed and if it is 1-1 on the interval Ea,b) . We can think of the curve as t y passes through the time interval is the order of the points taken by as a point [a,b]. y(t) y(t) moving in X In most applications, it and not the 'speed' of y(t) that is important: thus we identify curves which travel the same path but at different speeds.

Given and a^(t) X. 5 1. Show that the initial point, the final point, the range [y] and the properties 'simple, 'closed' and 'simple closed' are all defined independently of the choice of y from within its equivalence class. 6 CONNECTED SETS A topological space open sets A and B non-empty open sets X is disconnected if it can be split by in the sense that A and X is the union of the disjoint B: equivalently, if is open, closed, non-empty and not X. X The space is not disconnected. More generally, a subset of has a subset X X A which is connected if it is connected if it is a connected space with the subspace topology.

T2 * 1 ) z I-»- 1/z on ^p (z) = z £00 on 4^ = p so using p to transfer we arrive at the atlas £ , a 0(z) = 0p(z) = 1 /z on £ - {0} z z 00 on £ .

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