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General Topology General Topology Tom Leinster 2014{15 Contents A Topological spaces2 A1 Review of metric spaces.......................2 A2 The definition of topological space.................8 A3 Metrics versus topologies....................... 13 A4 Continuous maps........................... 17 A5 When are two spaces homeomorphic?................ 22 A6 Topological properties........................ 26 A7 Bases................................. 28 A8 Closure and interior......................... 31 A9 Subspaces (new spaces from old, 1)................. 35 A10 Products (new spaces from old, 2)................. 39 A11 Quotients (new spaces from old, 3)................. 43 A12 Review of ChapterA......................... 48 B Compactness 51 B1 The definition of compactness.................... 51 B2 Closed bounded intervals are compact............... 55 B3 Compactness and subspaces..................... 56 B4 Compactness and products..................... 58 B5 The compact subsets of Rn ..................... 59 B6 Compactness and quotients (and images)............. 61 B7 Compact metric spaces........................ 64 C Connectedness 68 C1 The definition of connectedness................... 68 C2 Connected subsets of the real line.................. 72 C3 Path-connectedness.......................... 76 C4 Connected-components and path-components........... 80 1 Chapter A Topological spaces A1 Review of metric spaces For the lecture of Thursday, 18 September 2014 Almost everything in this section should have been covered in Honours Analysis, with the possible exception of some of the examples. For that reason, this lecture is longer than usual. Definition A1.1 Let X be a set. A metric on X is a function d: X × X ! [0; 1) with the following three properties: • d(x; y) = 0 () x = y, for x; y 2 X; • d(x; y) + d(y; z) ≥ d(x; z) for all x; y; z 2 X (triangle inequality); • d(x; y) = d(y; x) for all x; y 2 X (symmetry). A metric space is a set together with a metric on it, or more formally, a pair (X; d) where X is a set and d is a metric on X. n Examples A1.2 i. The Euclidean metric d2 on R is given by n 1=2 X 2 d2(x; y) = (xi − yi) i=1 n n for all x = (x1; : : : ; xn) and y = (y1; : : : ; yn) in R . So (R ; d2) is a metric n space. The same formula defines a metric d2 on X for any X ⊆ R . n ii. This is not the only metric on R . For example, there is a metric d1 on Rn given by n X d1(x; y) = jxi − yij i=1 and another, d1, given by d1(x; y) = max jxi − yij: 1≤i≤n 2 n (In fact, there is a metric dp on R for each p ≥ 1; perhaps you can guess what it is from the definitions of d1 and d2. The limit of dp(x; y) as p ! 1 is d1(x; y), hence the name.) iii. Let a; b 2 R with a ≤ b, and let C[a; b] denote the set of continuous functions [a; b] ! R. There are at least three interesting metrics on C[a; b], which again are denoted by d1, d2 and d1. They are defined by Z b d1(f; g) = jf(t) − g(t)j dt; a Z b 1=2 2 d2(f; g) = (f(t) − g(t)) dt ; a d1(f; g) = sup jf(t) − g(t)j: a≤t≤b If you do the Linear Analysis or Fourier Analysis course, you'll get very used to the idea of spaces whose elements are functions. iv. Let A be any set, which you might think of as an alphabet. Let n 2 N. The Hamming metric d on An is given by d(x; y) = i 2 f1; : : : ; ng : xi 6= yi n for x = (x1; : : : ; xn) and y = (y1; : : : ; yn) in A . In other words, the Ham- ming distance between two strings or `words' (x1; : : : ; xn) and (y1; : : : ; yn) is the number of coordinates in which they differ. The Hamming metric is often used in the theory of information and com- munication. For instance, if I write the word `needle' on the blackboard and you mistakenly copy it down as `noodle', the Hamming distance be- tween the words is 2, which is the number of errors of communication. v. An informal example: consider any region of space X, such as the area within the King's Buildings accessible by foot. (This excludes the space occupied by trees, walls, etc.) We can certainly use the Euclidean metric on X, which is the distance as the crow flies. But in practical terms, we are often more interested in the `shortest path metric', that is, the distance by foot. This is indeed a metric; you should be able to persuade yourself that the three axioms hold. This example is made precise on problem sheet 1. Strictly speaking, we should write metric spaces as pairs (X; d), where X is a set and d is a metric on X. But usually, I will just say `a metric space X', using the letter d for the metric unless indicated otherwise. Definition A1.3 Let X be a metric space, let x 2 X, and let " > 0. The open ball around x of radius ", or more briefly the open "-ball around x, is the subset B(x; ") = fy 2 X : d(x; y) < "g of X. Similarly, the closed "-ball around x is B¯(x; ") = fy 2 X : d(x; y) ≤ "g: 3 A good exercise is to go through Examples A1.2 and work out what the open and closed balls are in each of the examples given. Now comes an extremely important definition. Definition A1.4 Let X be a metric space. i. A subset U of X is open in X (or an open subset of X) if for all u 2 U, there exists " > 0 such that B(u; ") ⊆ U. ii. A subset V of X is closed in X if X n V is open in X. Thus, U is open if every point of U has some elbow room|it can move a little bit in each direction without leaving U. Warning A1.5 Closed does not mean `not open'! Subsets are not like doors. A subset of a metric space can be: • neither open nor closed, such as [0; 1) in R • both open and closed, such as R in R • open but not closed, such as (0; 1) in R • closed but not open, such as [0; 1] in R. Warning A1.6 Another warning: properly, there's no such thing as an `open set', only an open subset. In other words, we should never say `U is open'; we should always say `U is open in X'. This can matter. For instance, [0; 1) is not open in R, but it is open in [0; 2]. (Why?) In practice, it's often clear which space X we're operating inside, and then it's generally safe to speak of sets simply being `open' without mentioning which space they're open in. (Wade's book is often casual in this way.) Nevertheless, it's important to realize that this is a casual use of language, and can lead to errors if you're not careful. 4 Remark A1.7 Open balls are open and closed balls are closed. For a proof, see Remark 10.9 of Wade's book, or try it as an exercise. Closed subsets of a metric space can be characterized in terms of convergent sequences, as follows. 1 Definition A1.8 Let X be a metric space, let (xn)n=1 be a sequence in X, and let x 2 X. Then (xn) converges to x if d(xn; x) ! 0 as n ! 1: Explicitly, then, (xn) converges to x if and only if: for all " > 0, there exists N ≥ 1 such that for all n ≥ N, d(xn; x) < ". This generalizes the definition you're familiar with for R. Lemma A1.9 Let X be a metric space and V ⊆ X. Then V is closed in X if and only if: for all sequences (xn) in V and all x in X, if (xn) converges to x then x 2 V . A proof very similar to the following can also be found in Wade (Theo- rem 10.16). Proof Suppose that V is closed, and let (xn) be a sequence in V converging to some point x 2 X. We must show that x 2 V . Suppose for a contradiction that x 2 X n V . Since X n V is open in X, there is some " > 0 such that B(x; ") ⊆ X n V . Now (xn) converges to x, so there exists N such that d(xn; x) < " for all n ≥ N. In particular, d(xN ; x) < ", that is, xN 2 B(x; "); hence xN 2 X n V . This contradicts the hypothesis that (xn) is a sequence in V . Now suppose that V is not closed. We must show that the given condition does not hold, in other words, that there exists a sequence (xn) in V converging to a point of X not in V . Since V is not closed, X n V is not open. Hence there is some point x 2 X n V with the property that for all " > 0, the ball B(x; ") has nonempty intersection with V . For each n ≥ 1, choose an element xn 2 B(x; 1=n) \ V . Then (xn) is a sequence in V converging to x 2 X n V , as required. Next we state some fundamental properties of open and closed subsets. In order to do this, we'll need to recall some basic set theory. Remark A1.10 Let X be a set. A family (Ai)i2I of subsets of X is a set I together with a subset Ai ⊆ X for each i 2 I.
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