Basic Geometry and Topology Stephan Stolz September 7, 2015 Contents 1 Pointset Topology 1 1.1 Metric spaces . .1 1.2 Topological spaces . .5 1.3 Constructions with topological spaces . .6 1.3.1 Subspace topology . .6 1.3.2 Product topology . .7 1.3.3 Quotient topology. .8 1.4 Properties of topological spaces . 12 1.4.1 Hausdorff spaces . 13 1.4.2 Compact spaces . 13 1.4.3 Connected spaces . 15 1 Pointset Topology 1.1 Metric spaces We recall that a map f : Rm ! Rn between Euclidean spaces is continuous if and only if 8x 2 X 8 > 0 9δ > 0 8y 2 X d(x; y) < δ ) d(f(x); f(y)) < , (1.1) where p 2 2 d(x; y) = jjx − yjj = (x1 − y1) + ··· + (xn − yn) 2 R≥0 is the Euclidean distance between two points x; y in Rn. Example 1.2. (Examples of continuous maps.) 2 1. The addition map a: R ! R, x = (x1; x2) 7! x1 + x2; 2 2. The multiplication map m: R ! R, x = (x1; x2) 7! x1x2; The proofs that these maps are continuous are simple estimates that you probably remember from calculus. Since the continuity of all the maps we'll look at in these notes is proved by expressing them in terms of the maps a and m, we include the proofs of continuity of a and m for completeness. 1 1 POINTSET TOPOLOGY 2 2 Proof. To prove that the addition map a is continuous, suppose x = (x1; x2) 2 R and > 0 2 are given. We claim that for δ := /2 and y = (y1; y2) 2 R with d(x; y) < δ we have d(a(x); a(y)) < and hence a is a continuous function. To prove the claim, we note that p 2 2 d(x; y) = jx1 − y1j + jx2 − y2j and hence jx1 − y1j ≤ d(x; y), jx1 − y1j ≤ d(x; y). It follows that d(a(x); a(y)) = ja(x) − a(y)j = jx1 + x2 − y1 − y2j ≤ jx1 − y1j + jx2 − y2j ≤ 2d(x; y) < 2δ = . To prove that the multiplication map m is continuous, we claim that for δ := minf1; /(jx1j + jx2j + 1)g 2 and y = (y1; y2) 2 R with d(x; y) < δ we have d(m(x); m(y)) < and hence m is a continuous function. The claim follows from the following estimates: d(m(y); m(x)) = jy1y2 − x1x2j = jy1y2 − x1y2 + x1y2 − x1x2j ≤ jy1y2 − x1y2j + jx1y2 − x1x2j = jy1 − x1jjy2j + jx1jjy2 − x2j ≤ d(x; y)(jy2j + jx1j) ≤ d(x; y)(jx2j + jy2 − x2j + jx1j) ≤ d(x; y)(jx1j + jx2j + 1) < δ(jx1j + jx2j + 1) ≤ n n Lemma 1.3. The function d: R × R ! R≥0 has the following properties: 1. d(x; y) = 0 if and only if x = y; 2. d(x; y) = d(y; x) (symmetry); 3. d(x; y) ≤ d(x; z) + d(z; y) (triangle inequality) Definition 1.4. A metric space is a set X equipped with a map d: X × X ! R≥0 with properties (1)-(3) above. A map f : X ! Y between metric spaces X, Y is continuous if condition (1.1) is satisfied. an isometry if d(f(x); f(y)) = d(x; y) for all x; y 2 X; Two metric spaces X, Y are homeomorphic (resp. isometric) if there are continuous maps (resp. isometries) f : X ! Y and g : Y ! X which are inverses of each other. Example 1.5. An important class of examples of metric spaces are subsets of Rn. Here are particular examples we will be talking about during the semester: 1 POINTSET TOPOLOGY 3 n n n n n 1. The n-disk D := fx 2 R j jxj ≤ 1g ⊂ R , and Dr := fx 2 R j jxj ≤ rg, the n-disk of radius r > 0. The dilation map n n D −! Dr x 7! rx n n is a homeomorphism between D and Dr with inverse given by multiplication by 1=r. However, these two metric spaces are not isometric for r 6= 1. To see this, define the diameter diam(X) of a metric space X by diam(X) := supfd(x; y) j x; y 2 Xg 2 R≥0 [ f1g: n For example, diam(Dr ) = 2r. It is easy to see that if two metric spaces X, Y are n n isometric, then their diameters agree. In particular, the disks Dr and Dr0 are not isometric unless r = r0. 2. The n-sphere Sn := fx 2 Rn+1 j jxj = 1g ⊂ Rn+1. 3. The torus T = fv 2 R3 j d(v; C) = rg for 0 < r < 1. Here 2 2 3 C = f(x; y; 0) j x + y = 1g ⊂ R is the unit circle in the xy-plane, and d(v; C) = infw2C d(v; w) is the distance between v and C. 4. The general linear group n n GLn(R) = fvector space isomorphisms f : R ! R g n ! f(v1; : : : ; vn) j vi 2 R ; det(v1; : : : ; vn) 6= 0g n n n2 = finvertible n × n-matricesg ⊂ R × · · · × R = R | {z } n Here we think of (v1; : : : ; vn) as an n × n-matrix with column vectors vi, and the bijection is the usual one in linear algebra that sends a linear map f : Rn ! Rn to the matrix (f(e1); : : : ; f(en)) whose column vectors are the images of the standard basis n elements ei 2 R . 5. The special linear group n n2 SLn(R) = f(v1; : : : ; vn) j vi 2 R ; det(v1; : : : ; vn) = 1g ⊂ R 6. The orthogonal group n n O(n) = flinear isometries f : R ! R g n n2 = f(v1; : : : ; vn) j vi 2 R ; vi's are orthonormalg ⊂ R n We recall that a collection of vectors vi 2 R is orthonormal if jvij = 1 for all i, and vi is perpendicular to vj for i 6= j. 1 POINTSET TOPOLOGY 4 7. The special orthogonal group n2 SO(n) = f(v1; : : : ; vn) 2 O(n) j det(v1; : : : ; vn) = 1g ⊂ R 8. The Stiefel manifold n k n Vk(R ) = flinear isometries f : R ! R g n kn = f(v1; : : : ; vk) j vi 2 R ; vi's are orthonormalg ⊂ R Example 1.6. The following maps between metric spaces are continuous. While it is pos- sible to prove their continuity using the definition of continuity, it will be much simpler to prove their continuity by `building' these maps using compositions and products from the continuous maps a and m of Example 1.2. We will do this below in Lemma 1.22. 1. Every polynomial function f : Rn ! R is continuous. We recall that a polynomial function is of the form f(x ; : : : ; x ) = P a xi1 ····· xin for a 2 . 1 n i1;:::;in i1;:::;in 1 n i1;:::;in R n2 2. Let Mn×n(R) = R be the set of n × n matrices. Then the map Mn×n(R) × Mn×n(R) −! Mn×n(R)(A; B) 7! AB given by matrix multiplication is continuous. Here we use the fact that a map to the n2 product Mn×n(R) = R = R × · · · × R is continuous if and only if each component map is continuous (see Lemma 1.21), and each matrix entry of AB is a polynomial and hence a continuous function of the matrix entries of A and B. Restricting to the invertible matrices GLn(R) ⊂ Mn×n(R), we see that the multiplication map GLn(R) × GLn(R) −! GLn(R) is continuous. The same holds for the subgroups SO(n) ⊂ O(n) ⊂ GLn(R). −1 3. The map GLn(R) ! GLn(R), A 7! A is continuous (this is a homework problem). The same statement follows for the subgroups of GLn(R). n p 2 2 The Euclidean metric on R given by d(x; y) = (x1 − y1) + ··· + (xn − yn) for x; y 2 Rn is not the only reasonable metric on Rn. Another metric on Rn is given by n X d1(x; y) = jxi − yij: (1.7) i=1 The question arises whether it can happen that a map f : Rn ! Rn is continuous with respect to one of these metrics, but not with respect to the other. To see that this doen't happen, it is useful to characterize continuity of a map f : X ! Y between metric spaces X, Y in a way that involves the metrics on X and Y less directly than Definition 1.4 does. This alternative characterization will be based on the following notion of \open subsets" of a metric space. 1 POINTSET TOPOLOGY 5 Definition 1.8. Let X be a metric space. A subset U ⊂ X is open if for every point x 2 U there is some > 0 such that B(x) ⊂ U. Here B(x) = fy 2 X j d(y; x) < g is the ball of radius around x. To illustrate this, lets look at examples of subsets of Rn equipped with the Euclidean n n n n metric. The subset Dr = fv 2 R j jjvjj ≤ rg ⊂ R is not open, since for for a point v 2 Dr n with jjvjj = r any open ball B(v) with center v will contain points not in Dr . By contrast, n the subset Br(0) ⊂ R is open, since for any x 2 Br(0) the ball Bδ(x) of radius δ = r − jjxjj is contained in Br(0), since for y 2 Bδ(x) by the triangle inequality we have d(y; 0) ≤ d(y; x) + d(x; 0) < δ + jjxjj = (r − jjxjj) + jjxjj = r: Lemma 1.9.