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Part IB | Metric and Topological Spaces Based on lectures by J. Rasmussen Notes taken by Dexter Chua Easter 2015 These notes are not endorsed by the lecturers, and I have modified them (often significantly) after lectures. They are nowhere near accurate representations of what was actually lectured, and in particular, all errors are almost surely mine. Metrics Definition and examples. Limits and continuity. Open sets and neighbourhoods. Characterizing limits and continuity using neighbourhoods and open sets. [3] Topology Definition of a topology. Metric topologies. Further examples. Neighbourhoods, closed sets, convergence and continuity. Hausdorff spaces. Homeomorphisms. Topologi- cal and non-topological properties. Completeness. Subspace, quotient and product topologies. [3] Connectedness Definition using open sets and integer-valued functions. Examples, including inter- vals. Components. The continuous image of a connected space is connected. Path- connectedness. Path-connected spaces are connected but not conversely. Connected open sets in Euclidean space are path-connected. [3] Compactness Definition using open covers. Examples: finite sets and [0, 1]. Closed subsets of compact spaces are compact. Compact subsets of a Hausdorff space must be closed. The compact subsets of the real line. Continuous images of compact sets are compact. Quotient spaces. Continuous real-valued functions on a compact space are bounded and attain their bounds. The product of two compact spaces is compact. The compact subsets of Euclidean space. Sequential compactness. [3] 1 Contents IB Metric and Topological Spaces Contents 0 Introduction 3 1 Metric spaces 4 1.1 Definitions . .4 1.2 Examples of metric spaces . .6 1.3 Norms . .7 1.4 Open and closed subsets . 10 2 Topological spaces 15 2.1 Definitions . 15 2.2 Sequences . 18 2.3 Closed sets . 19 2.4 Closure and interior . 19 2.4.1 Closure . 19 2.4.2 Interior . 21 2.5 New topologies from old . 22 2.5.1 Subspace topology . 22 2.5.2 Product topology . 23 2.5.3 Quotient topology . 25 3 Connectivity 27 3.1 Connectivity . 27 3.2 Path connectivity . 29 3.2.1 Higher connectivity* . 30 3.3 Components . 30 3.3.1 Path components . 30 3.3.2 Connected components . 31 4 Compactness 34 4.1 Compactness . 35 4.2 Products and quotients . 38 4.2.1 Products . 38 4.2.2 Quotients . 39 4.3 Sequential compactness . 40 4.4 Completeness . 41 2 0 Introduction IB Metric and Topological Spaces 0 Introduction The course Metric and Topological Spaces is divided into two parts. The first is on metric spaces, and the second is on topological spaces (duh). In Analysis, we studied real numbers a lot. We defined many properties such as convergence of sequences and continuity of functions. For example, if (xn) is a sequence in R, xn ! x means (8" > 0)(9N)(8n > N) jxn − xj < ": Similarly, a function f is continuous at x0 if (8" > 0)(9δ > 0)(8x) jx − x0j < δ ) jf(x) − f(x0)j < ": However, the definition of convergence doesn't really rely on xn being real numbers, except when calculating values of jxn − xj. But what does jxn − xj really mean? It is the distance between xn and x. To define convergence, we don't really need notions like subtraction and absolute values. We simply need a (sensible) notion of distance between points. Given a set X, we can define a metric (\distance function") d : X × X ! R, where d(x; y) is the distance between the points x and y. Then we say a sequence (xn) in X converges to x if (8" > 0)(9N)(8n > N) d(xn; x) < ": Similarly, a function f : X ! X is continuous if (8" > 0)(9δ > 0)(8x) d(x; x0) < δ ) d(f(x); f(x0)) < ": Of course, we will need the metric d to satisfy certain conditions such as being non-negative, and we will explore these technical details in the first part of the course. As people studied metric spaces, it soon became evident that metrics are not really that useful a notion. Given a set X, it is possible to find many different metrics that completely agree on which sequences converge and what functions are continuous. It turns out that it is not the metric that determines (most of) the properties of the space. Instead, it is the open sets induced by the metric (intuitively, an open set is a subset of X without \boundary", like an open interval). Metrics that induce the same open sets will have almost identical properties (apart from the actual distance itself). The idea of a topological space is to just keep the notion of open sets and abandon metric spaces, and this turns out to be a really good idea. The second part of the course is the study of these topological spaces and defining a lot of interesting properties just in terms of open sets. 3 1 Metric spaces IB Metric and Topological Spaces 1 Metric spaces 1.1 Definitions As mentioned in the introduction, given a set X, it is often helpful to have a notion of distance between points. This distance function is known as the metric. Definition (Metric space). A metric space is a pair (X; dX ) where X is a set (the space) and dX is a function dX : X × X ! R (the metric) such that for all x; y; z, { dX (x; y) ≥ 0 (non-negativity) { dX (x; y) = 0 iff x = y (identity of indiscernibles) { dX (x; y) = dX (y; x) (symmetry) { dX (x; z) ≤ dX (x; y) + dX (y; z) (triangle inequality) We will have two quick examples of metrics before going into other important definitions. We will come to more examples afterwards. Example. { (Euclidean \usual" metric) Let X = Rn. Let v u n uX 2 d(v; w) = jv − wj = t (vi − wi) : i=1 This is the usual notion of distance we have in the Rn vector space. It is not difficult to show that this is indeed a metric (the fourth axiom follows from the Cauchy-Schwarz inequality). { (Discrete metric) Let X be a set, and ( 1 x 6= y dX (x; y) = 0 x = y To show this is indeed a metric, we have to show it satisfies all the axioms. The first three axioms are trivially satisfied. How about the fourth? We can prove this by exhaustion. Since the distance function can only return 0 or 1, d(x; z) can be 0 or 1, while d(x; y) + d(y; z) can be 0, 1 or 2. For the fourth axiom to fail, we must have RHS < LHS. This can only happen if the right hand side is 0. But for the right hand side to be 0, we must have x = y = z. So the left hand side is also 0. So the fourth axiom is always satisfied. Given a metric space (X; d), we can generate another metric space by picking out a subset of X and reusing the same metric. Definition (Metric subspace). Let (X; dX ) be a metric space, and Y ⊆ X. Then (Y; dY ) is a metric space, where dY (a; b) = dX (a; b), and said to be a subspace of X. 4 1 Metric spaces IB Metric and Topological Spaces Example. Sn = fv 2 Rn+1 : jvj = 1g, the n-dimensional sphere, is a subspace of Rn+1. Finally, as promised, we come to the definition of convergent sequences and continuous functions. Definition (Convergent sequences). Let (xn) be a sequence in a metric space (X; dX ). We say (xn) converges to x 2 X, written xn ! x, if d(xn; x) ! 0 (as a real sequence). Equivalently, (8" > 0)(9N)(8n > N) d(xn; x) < ": Example. k { Let (vn) be a sequence in R with the Euclidean metric. Write vn = 1 k 1 k k i i (vn; ··· ; vn), and v = (v ; ··· ; v ) 2 R . Then vn ! v iff (vn) ! v for all i. 1 { Let X have the discrete metric, and suppose xn ! x. Pick " = 2 . 1 Then there is some N such that d(xn; x) < 2 whenever n > N. But if 1 d(xn; x) < 2 , we must have d(xn; x) = 0. So xn = x. Hence if xn ! x, then eventually all xn are equal to x. Similar to what we did in Analysis, we can show that limits are unique (if exist). Proposition. If (X; d) is a metric space, (xn) is a sequence in X such that 0 0 xn ! x, xn ! x , then x = x . Proof. For any " > 0, we know that there exists N such that d(xn; x) < "=2 if 0 0 0 n > N. Similarly, there exists some N such that d(xn; x ) < "=2 if n > N . Hence if n > max(N; N 0), then 0 ≤ d(x; x0) 0 ≤ d(x; xn) + d(xn; x ) 0 = d(xn; x) + d(xn; x ) ≤ ": So 0 ≤ d(x; x0) ≤ " for all " > 0. So d(x; x0) = 0, and x = x0. Note that to prove the above proposition, we used all of the four axioms. In the first line, we used non-negativity to say 0 ≤ d(x; x0). In the second line, we used triangle inequality. In the third line, we used symmetry to swap d(x; xn) with d(xn; x). Finally, we used the identity of indiscernibles to conclude that x = x0. To define a continuous function, here, we opt to use the sequence definition. We will later show that this is indeed equivalent to the " − δ definition, as well as a few other more useful definitions. Definition (Continuous function). Let (X; dX ) and (Y; dY ) be metric spaces, and f : X ! Y .
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