### Introduction to Topology and Geometry, Second Edition

This is part of his discussion of methods to concoct an iterated function system that ap- proximates a previously given picture. Julia sets are named for Gaston Julia. He and Pierre Fatou are both credited with creating the theory of iteration in the complex plane at about the same time There was a sometimes bitter priority dispute between the two of them about this material.

The fractal sets associated with complex number systems are discussed, for example, in , , . The statement of Exercise 1. Exercises 1. It is possible that both alternatives occur: there is some number with more than one expansion, while there is another number with no expan- sion. For example, modify the Eisenstein number system Exercise 1. Thanks to Peter Hinow for this example. Here is a hint for Exercise 1.

Chapter 2 is a more technical chapter. Have patience! It really is useful for the understanding of the rest of the book.

Mathematics students will eventually learn almost everything in this chap- ter in the normal course of their studies. So many readers may be able to skip this chapter completely; but it is here for those who need it. Many of the proofs and exercises are merely the usual real-number proofs adapted to the setting of metric spaces.

So a student who has experience dealing with the proofs of ordinary calculus will see many familiar ideas. Metric topology is, in fact, important for a lot of modern mathemat- ics. The selection of topics for this chapter was determined by what is required later in the book; so this chapter is a bit peculiar as an introduction to metric topology. Examples Let us consider a few examples of metric spaces.

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Theorem 2. Exercise 2. When is the triangle inequality actually an equality in the metric space R? If d is a positive integer, then Rd is the set of all ordered d-tuples of real numbers. In order to show that this is a metric space , I will prove two basic in- equalities. Corollary 2. But a square is nonnegative, so this means that all terms must be 0. So all that remains is the triangle inequality. The diameter of A is the distance between the two most distant points of A, if such points exist.

Even though no two points of A have distance exactly 1, there are pairs x, y of points of A with distance as close as we like to 1; and there are no pairs x, y of points of A with distance greater than 1. Note that this is not a metric, for example because the triangle inequality fails. Let S be a metric space, and let A be a subset. An open ball Br x is an open set. So y is an interior point of Br x. Let S be a metric space. So S is an open set. Suppose U and V are both open. Therefore V is an open set. Proposition 2. A closed ball Br x is a closed set. Therefore y is not an accumulation point of Br x. This shows that Br x is a closed set. Note that in Proposition 2. The properties of an ultrametric space may seem strange if you are familiar only with Euclidean space and its subsets.

Here are a few examples to help you understand the situation. Let S be an ultrametric space. We will discuss functions on metric spaces and sequences in metric spaces. What are the isometries of the Euclidean plane R2 into itself?

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Some ex- amples are pictured in Fig. In fact, the maps of these types are the only isometries of R2. A detailed argument along these lines may be found in [12, Chap. An alternate term is similitude. The number r is the ratio of h. Describe all of the similarities of two-dimensional Euclidean space R2 onto itself. Let S and T be metric spaces. A thorough understanding of it will be useful to you not only in the study of fractal geometry, but also in much of the other mathematics you will study. Isometries and similarities are continuous functions. Continuity can be phrased in terms of open sets.

First, suppose that h is continuous. Let V be an open set in T. I must show that h is continuous at x. So h is continuous.

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Let B be a base for the open sets of T. We may even write simply xn. Also, x is called the limit of the sequence xn. Let xn be a sequence in a metric space S. First, suppose h is continuous. For the other direction, I will prove the contrapositive. Suppose h is not continuous. This means that the sequence xn converges to x, but the image sequence h xn does not converge to h x.

So the convergence property fails. This is called a subsequence of xn. The point x is a cluster point of the sequence xn if and only if x is the limit of some subsequence of xn.

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Suppose x is a cluster point of xn. Let k1 be such an n. Let k2 be such an n. Thus x is the limit of the subsequence xkj of xn. Therefore x is a cluster point of the sequence xn.

## An Illustrated Introduction to Topology and Homotopy

Let A be a subset of a metric space S. A sequence in R is convergent if and only if it is a Cauchy sequence. Every convergent sequence is a Cauchy sequence. I will show that xn is a Cauchy sequence. The number 2 is irrational. This is a Cauchy sequence in the metric space S that does not converge in S. Three-dimensional Euclidean space R3 is complete. Suppose S is an ultrametric space. Completeness is a metric property, but not a topological property.

The closure of a set A is the set A, consisting of A together with all of its accumulation points. It is a closed set. The boundary of A is the set of all boundary points of A.