By A. F. Beardon
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Extra info for A Primer on Riemann Surfaces
In Thus an open subset of 3Rn C(x), [y] is open we have is connected if and only if it is arcwise connected. 61 3 2 1. Show that if connected. Deduce that each 2. Let D E is a connected subset of C(x) points X then E is is closed. be a domain in 3Rn . can be joined by a simple curve in 3. Let X, Show that any two points in D D. be the plane set consisting of the origin and the (x, sin 1/x), x > 0. Show that [o] = Co} * C(O). Deduce that X is connected but not arcwise connected. 7 QUOTIENT SPACES Let If we give Y T be a topology on a topology, then a largest topology T^ on Y f X and let f : X Y be any function.
We define a Riemann surface as an abstract surface rather than an intuitive model for some many-valued function. We then obtain explicit examples of surfaces. 1 SURFACES A surface S is a topological space, each point of which appears (topologically) to lie in an open subset of (C. Although one naturally thinks of surfaces as lying within JR^, we shall not insist that this is so: indeed, the essential idea is to define a surface in its own right without it being necessary to discuss the possible existence of any larger space containing it.
5) The continuous image of a compact set is compact. (6) A continuous bijection of a compact space onto a Hausdorff space is a homeomorphism. As an illustration of the ideas involved, we pause to prove (6) and give a particularly attractive application of it. First, we give the proof. Suppose that and that Y X is compact, that f : X -»■ Y is Hausdorff. It is only necessary to prove that sets to open sets. If A is an open subset of and hence, by (3), compact. By (5), f(B) closed. Thus f find that is a continuous bijection Y - f(B) Y - f(B) is is open.