Analysis and Topology ? Zhiyuan Bai Compiled on July 11, 2021 This document serves as a set of revision materials for the Cambridge Math- ematical Tripos Part IB course Analysis and Topology in Michaelmas 2019. However, despite its primary focus, readers should note that it is NOT a verba- tim recall of the lectures, since the author might have made further amendments in the content. Therefore, there should always be provisions for errors and typos while this material is being used. Contents 1 Uniform Convergence 2 Uniform Continuity 6 3 Metric Space 7 4 Completeness 14 5 Topological Spaces 17 6 Connectedness 24 7 Compactness 27 8 Differentiation 31 1 1 Uniform Convergence Definition 1.1. A complex sequence xn is said to converge to a complex number x if Ve > 0, EN E N such that Vn > N, x - xn| < €. Definition 1.2. Let S be a set and let fn : S -> C be a sequence of functions. Let f : S -> C be a function. We say fn -> f pointwise if for any x, fn(x) -> f(x). In other words, Va E S, VE > 0, IN E N, Vn > N, If(x) - fn(x)| < c. Example 1.1. Let S be the closed interval [0, 1] and fn(x) = x", then fn > f pointwise where 10,iptheryise "Based on the lectures under the same name taught by Prof. A. Zsák in Michaelmas 2019. 1 ..... ...
Note that in this example, despite the fact that all of fn are continuous, even smooth, the resulting limit f needs not be continuous. Here is another example: Example 1.2. Let S = R20 and let fn (x) = r2e"na, then fn -> 0 pointwise, since 2 1 + nx + 2 + 3 + 3 ... 22 = 0 ? \fn(x) = enx x ? nx n 2 = 2 ?0 as n > 00. There is another form of convergence, called uniform convergence, which is defined as follows: Definition 1.3. Let S be a set and let fn : S -> C be a sequence of functions. Let f : S -> C be a function. We say fn -> f uniformly if Ve > 0, 3N E N, Vn > N, Vx E S, |f (x) - fn(x)| < c. Note that the only difference between pointwise and uniform convergence is that the large integer N does not depend on x if the convergence is uniform. Note also that uniform convergence implies pointwise convergence, but not the other way around. Although it does not seem to be such a great difference in definition, in practice, it makes all the difference in the world. Proposition 1.1. The sequence in the first example, i.e. fn : [0, 1] -> R with fn(x) = x", does not converge uniformly. Proof. We can take for example € = 1/2. Then for any n E N, we can take x = " P 2/3, so that