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(2014/15) Geometry and Topology SMSTC (2014/15) Geometry and Topology LECTURE NOTES 1: Metric and topological spaces, fundamental group and covering spaces, higher homotopy groups Brendan Owens, Glasgowa www.smstc.ac.uk The first 4 sections of these notes (1.1 to 1.4) cover some basic point set topology, with an emphasis on examples. The last and longest section (1.5) is an introduction to the fundamental group, covering spaces, and higher homotopy groups. 1.1 Basic topology Sections 1.1 to 1.4 are basic sketchy notes on metric and topological spaces. The first three lectures of the course will cover some of the material in these sections. Many proofs are left as exercises { for the most part these are simply a matter of applying definitions. More detail can be found in standard undergraduate point-set topology books, such as • J. R. Munkres, Topology: a first course, Prentice-Hall (1975). We will also refer to A. Hatcher, Algebraic Topology, Cambridge University Press (2002), freely down- loadable from the author's website. Calculus gives us the concepts of open sets in Euclidean space and continuous functions, as well as properties such as connectedness, compactness and others. In this lecture we will see how these concepts can be generalised first to metric spaces and then to topological spaces. Notation: In these lectures ⊂ has the same meaning as ⊆, and strict inclusion is indicated by the symbol (. 1.1.1 Metric spaces The standard n-dimensional Euclidean space consists of the set n R = f(x1; : : : ; xn) j xi 2 Rg together with the distance function d : Rn × Rn ! R p 2 2 d((x1; : : : ; xn); (y1; : : : ; yn)) = (x1 − y1) + ··· + (xn − yn) : This satisfies the following properties: (1) d(x; y) ≥ 0; and d(x; y) = 0 () x = y (2) d(x; y) = d(y; x) (3) d(x; z) ≤ d(x; y) + d(y; z). More generally, any set M together with a function d : M × M ! R satisfying the above three properties is called a metric space. The following \-δ = epsilon-delta" definition may be familiar from calculus or analysis. a [email protected] 1{1 SMST C: Geometry and Topology 1{2 Definition 1.1 Let f :(M; dM ) ! (N; dN ) be a function from one metric space to another. We say f is continuous at x 2 M if for any > 0 there exists δ > 0 such that dM (x; y) < δ =) dN (f(x); f(y)) < . We say f is continuous if it is continuous at all points x 2 M. Example 1.2 The function f : R ! R defined by ( x if x ≤ 0; f(x) = x + 1 if x > 0; is continuous on R n f0g, but not at 0. (Verify this using -δ ideas.) The concept of an open set will enable us to give a friendlier looking definition of continuity. Roughly speaking a subset U of a metric space is open if for any point x in U, all sufficiently nearby points are also in U. Definition 1.3 The open ball centred at x 2 M with radius r > 0 is the subset B(x; r) = fy 2 M j d(x; y) < rg ⊂ M: Example 1.4 (Manhattan metric) Let M = R2 with d((x1; y1); (x2; y2)) = jx1 − x2j + jy1 − y2j: Sketch the unit ball B(0; 1) about the origin. Example 1.5 (Chessboard metric) Let M = R2 with d((x1; y1); (x2; y2)) = max(jx1 − x2j; jy1 − y2j): Sketch the unit ball B(0; 1) about the origin. Example 1.6 (Railway metric) Let M = R2 with p 2 2 p 2 2 d((x1; y1); (x2; y2)) = x1 + y1 + x2 + y2 provided (0; 0), (x1; y1) and (x2; y2) are not collinear; if they are collinear, set d((x1; y1); (x2; y2)) to be the usual Euclidean distance. Sketch the unit ball B(0; 1) about the origin. Definition 1.7 Let (M; d) be a metric space. A subset U ⊂ M is open if for any x 2 U there exists r > 0 with B(x; r) ⊂ U. A subset U is closed if its complement M n U is open. Exercise 1.8 Verify that a subset U of a metric space is open if and only if U is a union of open balls. Example 1.9 The Euclidean, Manhattan and Chessboard metrics all give the same open sets in R2. The Railway metric has open sets which are not open using any of the previous three metrics. (Verify this.) If (M; d) is a metric space and A ⊂ M is any subset, then A inherits the distance function d. Note that a set V in A is open if and only if V = U \ A for some open set U in M. Let f : M ! N be a function between metric spaces. For a subset U ⊂ N, the preimage of U under f is f −1(U) = fx 2 M j f(x) 2 Ug: Proposition 1.10 Let f : M ! N be a function between metric spaces. Then f is continuous if and only if the preimages of open sets are open. Equivalently, f is continuous if and only if the preimages of closed sets are closed, since f −1(N n U) = M n f −1(U). Proof Exercise. SMST C: Geometry and Topology 1{3 1.1.2 Topological spaces Proposition 1.11 If (M; d) is a metric space, then (1) M and ; are open sets; (2) arbitrary unions of open sets are open; (3) finite intersections of open sets are open. Proof Exercise. Note that infinite intersections of open sets may not be open. For example, in R with its usual metric, 1 \ 1 1 − ; n n n=1 is not open. Definition 1.12 A topological space is a set X together with a collection of subsets called open sets satisfying the three properties in Proposition 1.11: (1) X and ; are open sets; (2) arbitrary unions of open sets are open; (3) finite intersections of open sets are open. The complements of open sets are called closed sets. Examples 1.13 Any set X can be given the discrete topology, in which every subset is open, or the indiscrete topology, in which the only open sets are X and ;. Example 1.14 Given a subset A ⊂ X of a topological space X, the subspace topology on A is formed by taking V ⊂ A to be open if and only if V = U \ A for some open set U in X. Proposition 1.10 now suggests the definition of continuity for functions between topological spaces. Definition 1.15 Let X and Y be two topological spaces. A function f : X ! Y is called continuous (or a continuous mapping) if the preimage of every open set if open, or equivalently if the preimage of every closed set is closed. If A ⊂ X is given the subspace topology, it is straightforward to check that a continuous function f : X ! Y gives rise to a continuous restriction fjA : A ! Y . It is often convenient to specify a topology by giving a base; this is a collection of open sets B such that every open set is a union of elements of B. For example open balls are a base for the topology in any metric space, as are open balls with rational radii. Definition 1.16 Two topological spaces X and Y are homeomorphic (written X =∼ Y ) if there exists f : X ! Y which is a continuous bijection with a continuous inverse. Such an f is called a homeomorphism. This defines a notion of equivalence between topological spaces which satisfies the three properties of an equivalence relation. Also note that for a fixed space X, the set Homeo(X) of homeomorphisms from X to itself is a group under composition. Examples 1.17 Give all subsets of Rn the subspace topologies. (1) The open interval (0; 1) is homeomorphic to (0; 2) via x 7! 2x, and similarly any open interval (a; b) is homeomorphic to (0; 1). (2) The open interval (−π=2; π=2) is homeomorphic to R via x 7! tan x, and hence any open interval (a; b) =∼ R. (3) The sets f0; 1; 2; 3g and f0; 1g (with any choice of topology) are not homeomorphic since there is no bijection between these sets. (4) The sets 1 2 2 2 X = S = f(x; y) 2 R j x + y = 1g and Y = (0; 1) are not homeomorphic: details will be given later. SMST C: Geometry and Topology 1{4 Example 1.18 Let X be any set. The cofinite topology is the topology in which the empty set and complements of finite sets are open. If X = R this is also called the Zariski topology. Example 1.19 The Zariski topology on Rn (or on kn for any field k): closed sets are intersections of zero sets of polynomials in n variables. Equivalently: a base for the Zariski topology is given by complements of zero sets of polynomials. Examples 1.20 Other important examples of topological spaces: Sn (the unit sphere in Rn+1), orientable surfaces Σg, nonorientable surfaces Ng, matrix groups,. 1.1.3 Compactness Let X be a topological space. An open cover of X is a collection fUαgα2A of open subsets of X with S X = α2A Uα. For example take X = (0; 2] and Un = (1=n; 2], then fUngn2N is an open cover of X. If we let Y = [0; 2] and U0 = [0; 1=2) then fUngn≥0 is an open cover of Y .
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