A Primer on Riemann Surfaces by A. F. Beardon

By A. F. Beardon

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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. Accordingly, we say that the two curves y : [a,b] + X, a : [c,d] -* X are equivalent if and only if there is a strictly increasing map [a,b] onto Cc,d] such that y(t) = a(h(t)) for all t in h [a,b]. of Such a map (and its inverse) is necessarily continuous and this is an equivalence 25 relation on the class of all curves. In general, we do not distinguish between equivalent curves.

In general, we do not distinguish between equivalent curves. We say that final point If v in y : [a,b] X. a : Cc,d] -»■ X X Note that joins v to joins its initial point y (a+b-t) w, then by a^(t) = a(t+c-b): the curve defined as on Cb,b+d-c] now joins u to w in then joins a v u to u to its in is equivalent to y(t) on [a,b] X. 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.

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. If B is a topology, then (1) and (2) must hold. Now * suppose that (1) and (2) hold. 1) we see that 0 is in B and * ** * that B is closed under arbitrary unions (B = B ) . If C and D are * in B , then C n D is a union of sets of theform B n B with B.

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