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Let \mathcal{C} be the class of continuous maps of \mathbb{C}_\infty into itself and let \mathcal{R} be the subclass of rational functions. [â¦] Now \mathcal{R} is a closed subset of \mathcal{C}_\infty because if the rational functions R_n converge uniformly to R on the complex sphere, then R is analytic on the sphere and so it too is rational.
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