A Primer on Riemann Surfaces by A. F. Beardon

By A. F. Beardon

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1 with an important example. 2. Let D = (t and let G be the group of transla­ tions z n- z + nX + my where X and y (m,n € Z) are linearly independent over 3R (that is, they lie on different lines through the origin). 1 is applicable so C/G is a Riemann surface. We shall show that topologically this is a torus. One approach is to construct the parallelogram with vertices 0,A,y and A+y and observe that corresponding points on opposite sides are in the same G-orbit. As the parallelogram contains at least one point from each orbit it seems likely that we can construct D/G by identifying the paired sides of the parallelogram.

Is a topology, that that this is the largest topology which makes the quotient topology induced by f f is continuous and continuous. We call T^ f. An important example of a quotient topology arises from a group G of homeomorphisms of a space in X X onto itself. The orbit of a point x is G (x) = {g (x) : g € G} and the space of orbits is denoted by f : X that X/G X/G taking each x X/G. to its orbit There is a natural map G(x) : we shall assume always inherits the quotient topology induced by matically continuous.

We say that is (1) open if (2) continuous if f(A) is open whenever f 1 (B) A is open; is open whenever B is open. The composition of open functions is open: the composition of continuous functions is continuous. If f is a bijection of as a function: we say then that both f and f ^ f X onto Y then is a homeomorphism of are continuous (equivalently, if continuous) . If such an f f ^ : Y exists, we say that X f and X X onto exists Y if is open and Y are homeomorphic and this is an equivalence relation on the class of all topological spaces.

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