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Topology: 2020, SEU Yi Li Topology: 2020, SEU Yi Li SCHOOL OF MATHEMATICS AND SHING-TUNG YAU CENTER,SOUTHEAST UNIVERSITY,NANJING,CHINA 210096 E-mail address: [email protected]; [email protected]; [email protected] Contents Chapter 1. Basic topology 1 1.1. Metric spaces 1 1.2. Topological spaces 11 1.3. Constructions 33 v CHAPTER 1 Basic topology Main references: • Kelley, John L. General topology, Reprint of the 1955 edition [Van Nostrand, Toronto, Ont.], Graduate Texts in Mathe- matics, No. 27, Springer-Verlag, New York-Berlin, 1975. xiv+298 pp. MR0370454 (51# 6681) • Bredon, Glen E. Topology and geometry, Graduate Texts in Mathematics, No. 139, Springer-Verlag, New York, 1993. xiv+557 pp. MR1224675 (94d: 55001) • Lee, John M. Introduction to topological manifolds, Second edi- tion, Graduate Texts in Mathematics, No. 202, Springer, New York, 2011. xviii+433 pp. ISBN: 978-1-4419-7939-1 MR2766102 (2011i: 57001) • Munkres, James R. Topology: a first course, Prentice-Hall, Inc., Englewood Cliffs, N. J., 1975. xvi+413 pp. MR0464128 (57 # 4063) • Burago, Dmitri; Burago, Yuri; Ivanov, Sergei. A course in metric geometry, Graduate Studies in Mathematics, 33, Amer- ican Mathematical Society, Providence, RI, 2001. xiv+415 pp. ISBN: 0-8218-2129-6 MR1835418 (2002e: 53053) 1.1. Metric spaces 1.1.1. Metric spaces. A metric on a set X is a map d : X × X ! R := R [ f¥g satisfying, for any x, y, z 2 X, (a) (positiveness) d(x, y) ≥ 0 and d(x, y) = 0 if and only if x = y; (b) (symmetry) d(x, y) = d(y, x); (c) (triangle inequality) d(x, z) ≤ d(x, y) + d(y, z). A metric space is a pair (X, d) where d is a metric on X. Given a metric space (X, d) we define a relation ∼ on X by x ∼ y () d(x, y) < ¥. Then ∼ is an equivalence relation on X, and the equivalence class [x] of x can be endowed with a natural metric (still denoted by d). EXERCISE 1.1.1. For any x 2 X, show that ([x], d) is a finite metric space. ! A map f : (X, dX) (Y, dY) between metric spaces is called distance-preserving if dY( f (x1), f (x2)) = dX(x1, x2) 1 2 1. BASIC TOPOLOGY 2 for all x1, x2 X. A distance-preserving bijection is called an isometry. A semi-metric on a set X is a map d : X × X ! R := R [ f¥g satisfying, for any x, y, z 2 X, (a) (nonnegativity) d(x, y) ≥ 0; (b) (symmetry) d(x, y) = d(y, x); (c) (triangle inequality) d(x, z) ≤ d(x, y) + d(y, z). A semi-metric space is a pair (X, d) where d is a semi-metric on X. Given a semi-metric space (X, d) we define a relation ∼ on X by x ∼ y () d(x, y) = 0. Then ∼ is an equivalence relation on X. Define Xˆ := X/ ∼= f[x] : x 2 Xg, dˆ([x], [y]) := d(x, y). Then (X/d, d) := (Xˆ , dˆ) is a metric space. EXERCISE 1.1.2. Show that dˆ is well-defined and (X/d, d) is a metric space. EXERCISE 1.1.3. Let X = R2 and define 0 0 0 0 d((x, y), (x , y )) := j(x − x ) + (y − y )j. Show that d is a semi-metric on X. Define f : R2/d ! R by f ([(x, y)]) := x + y. Show that f is an isometry. EXERCISE 1.1.4. Consider the set X of all continuous real-valued functions on [0, 1]. Show that Z 1 d( f , g) := j f (x) − g(x)jdx 0 defines a metric on X. Is this still the case if continuity is weakened to integrabil- ity? EXERCISE 1.1.5. Let (X, dX) and (Y, dY) be two metric spaces. We define q dX×Y((x1, y1), (x2, y2)) := dX(x1, x2) + dY(y1, y2). × Show that (X Y, dX×Y) is a metric space. EXERCISE 1.1.6. As a subspace of R2, the unit circle S1 carries the restricted 2 Euclidean metric from R . We can define another (intrinsic) metric dint by dint(x, y) := the length of the shorter arc between them. 2 p 2 1 Note that dint(x, y) [0, ] for any points x, y S . (a) Show that any circle arc of length less than or equal to p, equipped with the intrinsic metric, is isometric to a straight line segment. (b) The whole circle with the intrinsic metric is not isometric to any subset of R2. 1.1. METRIC SPACES 3 Let (X, d) be a metric space. Set B(x, r) := fy 2 X : d(x, y) < rg, B(x, r) := fy 2 X : d(x, y) ≤ rg. We define the metric topology on X by saying that U ⊂ X is open if and only for any x 2 U there exists e > 0 such that B(x, e) ⊂ U. Let (1.1.1.1) T := fall open subsets in Xg [ fÆ, Xg. The set T satisfies the following (1) U \ V 2 T for any U, V 2 T ; [ 2 T f g ⊂ T (2) i2IUi for any collection Ui i2I ; (3) Æ, X 2 T . In general, we can introduce a topology on any set X.A topology on X is a subset T of 2X satisfying (1) U \ V 2 T for any U, V 2 T ; [ 2 T f g ⊂ T (2) i2IUi for any collection Ui i2I ; (3) Æ, X 2 T . The pair (X, T ) is called the topological space, and any element in T is said to be open; a closed set is a subset of X whose complement in X is open. In particular, we see that any metric space must be a topological space. Given a metric space (X, d). • (X, d) is complete if any Cauchy sequence1 in X has a limit in X. • A diameter of S ⊂ X is defined by (1.1.1.2) diam(S) := sup d(x, y). x,y2S • (X, d) is compact if any sequence in X has a converging subsequence. • ⊂ e f 2 ≤ eg S X is an -net if Se := x X : d(x, S) = infy2X d(x, y) = X. • X is totally bounded if for any e > 0 there exists a finite e-net in X. The following theorem is a basic result. THEOREM 1.1.7. Let (X, d) is a metric space. Then X is compact if and only if X is complete and totally bounded. 1.1.2. Length spaces. Let (X, T ) be a topological space. We say X is Haus- dorff if for any different points x, y 2 X, there exist open sets Ux, Uy ⊂ X such that x 2 Ux, y 2 Uy, and Ux \ Uy = Æ. Clearly that any metric space must be Hausdorff. A path g in X is a continuous2 map g : I ! X. where I ⊂ R is an interval (i.e., connected subset) of R. A length structure on X is a pair (A , L), where A is a collection of paths (called admissible paths) and L : A ! R+ is a map on A (called the length of paths), satisfying 1 f g e 2 A sequence xn n2N in X is said to be Cauchy if for any > 0 there exists an integer n0 N e ≥ such that d(xn, xm) < whenever n, m n0. − 2That is, g 1(U) is open in I for any open set U in X. 4 1. BASIC TOPOLOGY A g ! A gj 2 A (a) ( is closed under restrictions) If : [a, b] X in , then [c,d] for any subinterval [c, d] ⊂ [a, b]; A g ! g (b) ( is closed under concatenations of paths) If 1 : [a, c] X and 2 : ! g g [c, d] X are admissible with 1(c) = 2(c), then the obvious product g g · g ! := 1 2 : [a, d] X is also admissible; (c) (A is closed under reparameterizations) If g : [a, b] ! X is admissible and j : [c, d] ! [a, b] is a homeomorphism, then g ◦ j : [c, d] ! x is also admissible; gj gj gj (d) (Length of paths is additive) L( [a,b]) = L( [a,c]) + L( [c,b]) for any c 2 [a, b] and g : [a, b] ! X in A ; (e) (The length of a piece of a path continuously depends on the piece) If g : [a, b] ! X is a admissible path with L(g) < +¥, then L(g, a, t) := g L( [a,t]) is continuous in t; (f) (L(·) is invariant under reparametrizations) L(g ◦ j) = L(g) for any ad- missible path g and any homeomorphism j as in (c); (g) (Length structures agree with the topology of X) For any x 2 X and open set Ux 3 x, one has inf fL(g) : g(a) = x, g(b) 2 X n U g > 0. g2A x Let (X, T ) is a Hausdorff space with a length structure (A , L). Define (1.1.2.1) d (x, y) := inf fL(g)jg : [a, b] ! X, g(a) = x, g(b) = yg . L g2A ¥ If there is no path between x and y, we set dL(x, y) = + . We say a length struc- A 2 ¥ ture ( , L) is complete if for any x, y X with dL(x, y) < + , there exists a path g 2 A g joining x and y such that dL(x, y) = L( ). EXERCISE 1.1.8. Show that (X, dL) is a metric space, where dL is defined in (1.1.2.1).
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