Part 1 of lecture Topology presents the following content: general topology; infinite sets; topological space; continuity; connectedness; convergence; compactness; product of spaces; real functions and spaces of functions; quotient space;...
Lectures notes on TOPOLOGY Huỳnh Quang Vũ π1 Lecture notes on Topology Huỳnh Quang Vũ Version of August 26, 2022 ii This is a set of lecture notes for a series of introductory courses in topology for undergraduate students at the University of Science, Ho Chi Minh City It is written to be delivered by myself, tailored to my students I did not write it with other lecturers or self-study readers in mind In writing these notes I intend that more explanations and discussions will be carried out in class I hope by presenting only the essentials these notes will be more suitable for use in classroom Some details are left for students to fill in or to be discussed in class A sign ✓ in front of a problem notifies the reader that this is a typical problem or an important one (the result is used later) A sign * indicates a relatively more difficult problem Huỳnh Quang Vũ Address: Faculty of Mathematics and Computer Science, University of Science Vietnam National University - Ho Chi Minh City Email: hqvu@hcmus.edu.vn, Web: https://sites.google.com/view/hqvu/ The latest version of this set of notes, together with the source file, are available at https://sites.google.com/view/hqvu/teaching This work is released to Public Domain (CC0) wherever applicable, see http://creativecommons.org/publicdomain/zero/1.0/, otherwise it is licensed under the Creative Commons Attribution 4.0 International License, see http://creativecommons.org/licenses/by/4.0/ Contents I II Introduction General Topology Infinite sets Topological space 14 Continuity 23 Connectedness 31 Convergence 41 Compactness 49 Product of spaces 57 Real functions and Spaces of functions 65 Quotient space 74 Other topics 88 Algebraic Topology 91 10 Structures on topological spaces 93 11 Classification of compact surfaces 101 12 Homotopy 109 13 The fundamental group 113 14 The fundamental group of the circle 120 15 Van Kampen theorem 125 16 Simplicial homology 131 17 Singular homology 138 18 Homology of cell complexes 147 Other topics 149 III Differential Topology 151 19 Smooth manifolds 153 20 Tangent spaces and derivatives 161 21 Regular values 167 22 Critical points of real functions 172 23 Flows 181 24 Boundary 190 iii iv CONTENTS 25 Orientation 197 26 Degrees of maps 205 27 Integration of real functions 211 Guide for further reading 215 Suggestions for some problems 217 Bibliography 223 Index 225 CONTENTS Introduction Topology is a mathematical subject that studies shapes The term comes from the Greek words “topos” (place) and “ology” (study) A set becomes a topological space when each element of the set is given a collection of neighborhoods Operations on topological spaces must be continuous, bringing certain neighborhoods into neighborhoods There is no notion of distance Topology is a part of geometry that does not concern distance Figure 0.1: How to make a closed trip such that every bridge is crossed exactly once? This is the problem “Seven Bridges of Konigsberg”, studied by Leonard Euler in the 18th century It does not depend on the sizes of the bridges Characteristics of topology Operations on topological objects are more relaxed than in geometry: beside moving around (allowed in geometry), stretching or bending are allowed in topology (not allowed in geometry) For example, in topology circles - big or small, anywhere - are same Ellipses and circles are same On the other hand in topology tearing or breaking are not allowed: circles are still different from lines While topological operations are more flexible they still retain some essential properties of spaces Contributions of topology Topology provides basic notions to areas of mathematics where there is a need for a notion of continuity Topology focuses on some essential properties of spaces It can be used in qualitative study It can be useful where metrics or coordinates are not available, not natural, or not necessary Initially developed in the late nineteenth century and early twentieth century to provide basis for abstract mathematical analysis, topology gradually became an influential subject, reaching many achievements in the mid and late twentieth century CONTENTS Topology often does not stand alone: nowadays there are fields such as al- gebraic topology, differential topology, geometric topology, combinatorial topology, quantum topology, Topology often does not solve a problem by itself, but contributes important understanding, settings, and tools Topology features prominently in differential geometry, global analysis, algebraic geometry, theoretical physics Currently we see topology appearing in contemporary research in applied fields such as computation and data analysis Studying topology Emerging later than major branches of mathematics as geometry, algebra, and analysis, it could be said that topology contains concepts, methods, and ideas which are significant advances in mathematical thinking Thus it is very valuable for a student of mathematics to study topology to develop ability to think in new general, abstract, but also concrete manner Notions and results on continuity is now used throughout mathematics The ideas of homology is now used widely in geometry and algebra The concept of manifold has become fundamental in geometry and physics The widespread use of topology implies that studying topology can open students to much wider choices of areas for further works Part I General Topology 76 Figure 9.4: This space is homemomorphic to the cylinder Example Gluing opposite edges of a square gives a torus The torus12 (mặt xuyến) is [0, 1] × [0, 1]/ ∼ where (s, 0) ∼ (s, 1) and (0, t) ∼ (1, t) for all ≤ s, t ≤ Figure 9.5: The torus A of subspace R3 homeomorphic to T can be obtained as the surface of revolution obtained by revolving a circle around a line not intersecting it Suppose that Figure 9.6: The torus embedded in R3 the circle is on the Oyz-plane, the center is on the y-axis and the axis for the rotation is the z-axis Let a be the radius of the circle, b be the distance from the center of the circle to O, (a < b) Let S be the surface of revolution, then the embedding can be given by p [0, 2π ] × [0, 2π ] f / T2 & h S where f (ϕ, θ ) = ((b + a cos θ ) cos ϕ, (b + a cos θ ) sin ϕ, a sin θ ) Thus the torus can be embedded in R3 12 The plural form of the word torus is tori QUOTIENT SPACE 77 z b O a y ϕ θ x Figure 9.7: Equation for this embedded torus: ( x, y, z) = ((b + a cos θ ) cos ϕ, (b + a cos θ ) sin ϕ, a sin θ ), or x + y2 − b + z2 = a2 We can also describe the torus as the quotient of the Euclidean plane by the relation ∀m, n ∈ Z, ( x, y) ∼ ( x + m, y + n), see problem 9.54 and figure 9.8 Figure 9.8: The torus as a quotient of the plane 9.9 Example Gluing a pair of opposite edges of a square in opposite directions gives the Mobius band (dải, lá, mặt Mobius 13 ) More precisely the Mobius band is X = [0, 1] × [0, 1]/ ∼ where (0, t) ∼ (1, − t) for all t 13 Mobius ă or Moebius are other spellings for this name 78 Figure 9.10: The Mobius band Figure 9.11: The Mobius band embedded in R3 The Mobius band could be embedded in R3 It is homeomorphic to a subspace of R3 obtained by rotating a straight segment around the z-axis while also turning that segment “up side down” The embedding can be induced by the map (see Fig 9.12) (s, t) → (( a + t cos(s/2)) cos s, ( a + t cos(s/2)) sin s, t sin(s/2)), with ≤ s ≤ 2π and −1 ≤ t ≤ 1, where a > t a s s/2 Figure 9.12: An embedding of the Mobius band in R3 QUOTIENT SPACE 79 The Mobius band is famous as an example of un-orientable surfaces (mặt không định hướng được) It is also one-sided (một phía) A proof is at 25.4 Example Identifying one pair of opposite edges of a square and the other pair in opposite directions gives a topological space called the Klein bottle More precisely it is [0, 1] × [0, 1]/ ∼ with (0, t) ∼ (1, t) and (s, 0) ∼ (1 − s, 1) Figure 9.13: The Klein bottle It is known that the Klein bottle cannot be embedded in R3 The figure 9.14 does not present an embedded Klein bottle in R3 , instead only an immersed Klein bottle in R3 An immersion (phép nhúng chìm) is a local embedding More concisely, f : X → Y is an immersion if each point in X has a neighborhood U such that f |U : U → f (U ) is a homeomorphism Intuitively, an immersion allows self-intersection (tự cắt) Figure 9.14: The Klein bottle immersed in R3 Example Identifying opposite points on the boundary of a disk (they are called antipodal points) we get a topological space called the projective plane (mặt phẳng xạ ảnh) RP2 The real projective plane cannot be embedded in R3 It can be embedded in R4 , see 9.48 More generally, identifying antipodal boundary points of D n gives us the projective space (không gian xạ ảnh) RPn Example Identifying opposite faces of the cube [0, 1]3 by ( x, y, 0) ∼ ( x, y, 1), ( x, 0, z) ∼ ( x, 1, z), (0, y, z) ∼ (1, y, z) we get a space called the three-dimensional torus It can also be described, similarly to the two-dimensional torus, as the quotient space of R3 by the relation ∀m, n, p ∈ Z, ( x, y, z) ∼ ( x + m, y + n, z + p), see Fig 14 14 14 Figure produced using a computer program by J Weeks, available at http://geometrygames.org 80 Figure 9.15: The three-dimensional torus A person living in this space may be able to see infinitely many copies of her/himself 9.16 Example Let ( Xi )i∈ I be a collection of spaces together with a collection of points xi ∈ Xi The disjoint union of this collection of spaces (see 7.14) quotiented by the relation such that all the points xi are identified, that is, i∈ I Xi /( xi )i∈ I , is denoted by ∨i∈ I ( Xi , xi ), and is called the wedge sum (tổng nêm, chèn) of ( Xi )i∈ I with respect to the points ( xi )i∈ I Figure 9.17: S1 ∨ S1 ∨ S1 ∨ S1 For example a wedge sum of circles S1 ∨ S1 ∨ · · · ∨ S1 (which does not depend on how the identified points are chosen) is called a bouquet of circles (chùm, đóa đường tròn) See 9.30 Cut-and-paste Topology Example We see that the boundary of a Mobius band is a circle Gluing two Mobius bands along their boundaries gives the Klein bottle 15 There is a limerick (humorous poem): A mathematician named Klein Thought the Mobius band was divine Said he, “If you glue 15 see Fig 9.18 The QUOTIENT SPACE 81 process in this example, where parts of spaces are cut and glued, is appropriately called “cut and paste” d d a a c c b b b c b b d d d e a a a d e a e b b a b d a e f f e Figure 9.18: Gluing two Mobius bands along their boundaries gives the Klein bottle Example Gluing a disk to the Mobius band gives the projective plane In other words, deleting a disk from the projective plane gives the Mobius band See Fig 9.19 Problems 9.20 Describe the space [0, 1]/0 ∼ ∼ 9.21 Describe the space that is the quotient of the sphere S2 by its equator S1 9.22 Show that gluing two pairs of endpoints of two line segments gives a circle Precisely, ([0, 1] ∪ [2, 3])/0 ∼ 2, ∼ is homeomorphic to S1 9.23 Show that gluing disks along their boundaries gives a sphere Precisely, (( D2 × {1}) ∪ ( D2 × {2}))/∀ x ∈ S1 , ( x, 1) ∼ ( x, 2) is homeomorphic to S2 9.24 ✓ Show that the torus T is homeomorphic to S1 × S1 9.25 Show that the torus is homogeneous (see 3.32) 9.26 Let I be the rectangle [0, 1] × [0, 2π ] in the Euclidean plane Let ∼ be the equivalence relation on I satisfying all of the requirements below: The edges of two, You’ll get a weird bottle like mine.” 82 c a a a b c a a c a c b a b a c c b c b a c c c a b b b b c b b Figure 9.19: Gluing a disk to the Mobius band gives the projective plane (a) ∀r ∈ [0, 1], (r, 0) ∼ (r, 2π ), (b) ∀θ ∈ [0, 2π ], (0, θ ) ∼ (0, 0) Let f : I → D2 be the polar coordinates on the plane, given by f (r, θ ) = (r cos θ, r sin θ ) Show that f induces a homeomorphism from I/ ∼ onto the disk D2 Can we explain this visually? 9.27 Let I be the rectangle [0, 2π ] × [0, π ] in the Euclidean plane Let ∼ be the equivalence relation on I satisfying all of the requirements below: (a) ∀ϕ ∈ [0, π ], (0, ϕ) ∼ (2π, ϕ), (b) ∀θ ∈ [0, 2π ], (θ, 0) ∼ (0, 0), (c) ∀θ ∈ [0, 2π ], (θ, π ) ∼ (0, π ) Let f : I → S2 be the spherical coordinates on the sphere, given by f (θ, ϕ) = (sin ϕ cos θ, sin ϕ sin θ, cos ϕ) Show that f induces a homeomorphism from I/ ∼ onto S2 Can we explain this visually? 9.28 Show that gluing the boundary circle of a disk together gives a sphere, that is D2 /∂D2 is homeomorphic to S2 , see Fig 9.29 QUOTIENT SPACE 83 Figure 9.29: A map from D2 onto S2 which becomes injective on D2 /∂D2 9.30 Show that S1 ∨ S1 does not depend on how the identified points are chosen, and is homeomorphic to the figure-8, namely, this subspace of the Euclidean plane {( x, y) ∈ R2 | ( x − 1)2 + y2 = 1} ∪ {( x, y) ∈ R2 | ( x + 1)2 + y2 = 1}, see Fig 9.31 Figure 9.31: S1 ∨ S1 is homeomorphic to the figure-8 9.32 ✓ Show that the following space is homeomorphic to the Mobius band, see Fig 9.33 Figure 9.33: Another Mobius band 9.34 ✓ Show that the following two spaces are homeomorphic, one of the two is the Klein bottle, see Fig 9.35 84 b a a a b b a b Figure 9.35: Another Klein bottle 9.36 ✓ What we obtain after we cut a Mobius band along its middle circle? Try it with an experiment Mathematically, to cut a subset S from a space X means to delete S from X, the resulting space is the subspace X \ S In Fig 9.37 the curve CC ′ is deleted B A′ C C′ A B′ B C A Figure 9.37: Cutting a Mobius band along the middle circle 9.38 To obtain a physical Mobius band we can take a long rectangular piece of paper, twist one side once (an angle of 180 degree), then glue to the opposite side What happen if we twist twice? What happen if we twist many times? Do a physical experiment and a computer experiment See Fig 9.39 9.40 ✓ Following the description of the torus a the quotient space of a rectangle, a line with slope 2/3 in the rectangle after the quotient will be a closed curve on the torus that goes around the torus times in one direction and times in another direction, see Fig 9.41 From this it is not difficult to obtain a parametrization of an embedding of that space in R3 as for example ((2 + cos(t/2)) cos(t/3), (2 + cos(t/2)) sin(t/3), sin(t/2)), ≤ t ≤ 12π, see Fig 9.42, from which we can see why this space is often called the trefoil knot, and compare with Fig 9.7 QUOTIENT SPACE 85 Figure 9.39: A rectangle twisted twice then glued Figure 9.41: The trefoil knot on the torus Figure 9.42: The trefoil knot in R3 Show that the trefoil knot is homeomorphic to the circle S1 9.43 In general, the image of a simple closed curve in a Hausdorff topological space X is called a knot (nút) in X Thus a knot is the subspace γ([0, 1]) where γ : [0, 1] → X is a continuous map such that γ(0) = γ(1) and γ|[0,1) is injective, see Fig 9.44 Show that any knot is homeomorphic to the circle 9.45 Show that the projective space RP1 is homeomorphic to S1 9.46 The one-point compactification of the open Mobius band (the Mobius band without the boundary circle) is the projective space RP2 86 Figure 9.44: A knot is the image of a simple closed curve This knot is called the figure-8 knot, from its shape 9.47 * Show that identifying antipodal boundary points of D n is equivalent to identifying antipodal points of Sn In other words, the projective space RPn is homeomorphic to Sn /x ∼ − x 9.48 Consider the projective plane RP2 as the a quotient space of the sphere S2 /x ∼ − x, see 9.47 Show that the map f : → R4 → ( x2 − y2 , xy, yz, zx ) S2 ( x, y, z) induces an imbedding of RP2 into R4 9.49 Show that if X has one of the properties connected, path-connected, compact, then so is the quotient space X/ ∼ 9.50 Show that in order for the quotient space X/ ∼ to be a Hausdorff space, a necessary condition is that each equivalence class [ x ] must be a closed subset of X Is this condition sufficient? 9.51 Suppose ∼ is an equivalence relation on X and f : X → Y is a continuous map satisfying if x1 ∼ x2 then f ( x1 ) = f ( x2 ) Then there exists a unique continuous map g : X/ ∼→ Y such that f = g ◦ p X p f = g◦ p / X/ ∼ " Y g 9.52 Suppose f : X → Y is continuous and onto and ∼ is an equivalence relation on X given by x1 ∼ x2 if and only if f ( x1 ) = f ( x2 ) Show that if f is an open map then f induces a homeomorphism from X/ ∼ onto Y 9.53 On the Euclidean R define x ∼ y if x − y ∈ Z Show that R/ ∼ is homeomorphic to S1 The space R/ ∼ is also described as “The quotient of R by the action of the group Z”, written briefly as R/Z = S1 9.54 On the Euclidean R2 , define ( x1 , y1 ) ∼ ( x2 , y2 ) if ( x1 − x2 , y1 − y2 ) ∈ Z × Z Show that R2 / ∼ is homeomorphic to the torus, briefly R2 /Z2 = T QUOTIENT SPACE 87 9.55 Given a set X and a set Y ⊂ X × X, show that there is an equivalence relation on X that contains Y and is contained in every equivalence relation that contains Y, called the minimal equivalence relation containing Y For example, when we write [0, 1]/0 ∼ we mean the quotient of the set [0, 1] by the minimal equivalence relation on [0, 1] such that ∼ In this case the minimal equivalence relation is clearly {(0, 1), (1, 0), ( x, x ) | x ∈ [0, 1]} 9.56 * A question can be raised: In quotient spaces, if identifications are carried out in steps rather than simultaneously, will the results be different? More precisely, let R1 and R2 be two equivalence relations on a space X and let R be the minimal equivalence relation containing R1 ∪ R2 is also an equivalence relation on space X On the space X/R1 we define an equivalence relation R˜ induced from R2 by: [ x ] R1 ∼ R˜ [y] R1 if ( x ∼ R1 ∪ R2 y) Prove that the map X/( R1 ∪ R2 ) [ x ] R1 ∪ R2 → ( X/R1 )/ R˜ → [[ x ] R1 ] R˜ is a homeomorphism Thus in this sense the results are same 9.57 Show that if H is a normal subgroup of a topological group G (see 7.25) then the quotient group G/H under the quotient topology is a topological group 88 Other topics Below there are several topics for further study and a guide for further reading Strategy for a proof of Tikhonov theorem based on net The proof that we will outline here is based on further developments of the theory of nets and a characterization of compactness in terms of nets Let I and I ′ be directed sets, and let h : I ′ → I be a map such that ∀k ∈ I, ∃k′ ∈ I ′ , (i′ ≥ k′ ⇒ h(i′ ) ≥ k) If n : I → X is a net then n ◦ h is called a subnet of n The notion of subnet is an extension of the notion of subsequence A net ( xi )i∈ I is called eventually in A ⊂ X if there is j ∈ I such that i ≥ j ⇒ xi ∈ A A net n in X is universal if for any subset A of X either n is eventually in A or n is eventually in X \ A Proposition If f : X → Y is continuous and n is a universal net in X then f (n) is a universal net 9.58 Proposition The following statements are equivalent: (a) X is compact (b) Every universal net in X is convergent (c) Every net in X has a convergent subnet The proof of the two propositions above could be found in [Bre93] Then we finish the proof of Tikhonov theorem as follows Proof of Tikhonov theorem Let X = ∏i∈ I Xi where each Xi is compact Suppose that ( x j ) j∈ J is a universal net in X By 7.5 the net ( x j ) is convergent if and only if the projection ( pi ( x j )) is convergent for all i But that is true since ( pi ( x j )) is a universal net in the compact set Xi Metrizability The following is a result on ability for a topology to be given by a metric: 9.59 Theorem (Urysohn metrizability theorem) A regular space with a countable basis is metrizable The proof uses the Urysohn lemma, see [Mun00, p 243] QUOTIENT SPACE 89 The weak topology On a topological vector space the weak topology is the topology generated by continuous linear functionals On a normed space it is the coarsest topology such that all linear functionals which are continuous under the norm are still continuous under the topology This topology has an important role in Functional Analysis and its applications See [H Brezis, Functional analysis, Sobolev spaces and partial differential equations, Springer, 2011, chapter 3], and [Con90, ch 5] Space filling curves 9.60 Theorem There is a continuous curve filling a rectangle on the plane More concisely, there is a continuous map from the interval [0, 1] onto the square [0, 1]2 under the Euclidean topology Such a curve is called a Peano curve It could be constructed as a limit of an iteration of piecewise linear curves [Mun00, p 271] Guide for further reading The book by Kelley [Kel55] has been a common refererence Its presentation is rather abstract: the book contains no figure Munkres’ book [Mun00] is presently a popular textbook The treatment there is more modern than that in Kelley’s book, with many examples, figures and exercises Hocking and Young’s book [HY61] contains many deep and difficult results This book together with Kelley’s and Munkres’ books contain many topics not discussed in our lectures The textbook [Ros99] contains more concrete presentations by focusing in Euclidean spaces The textbook [AF08] contains applications of topology The book by Weeks [Wee02] introduces topology as a study of shapes In the next parts we study properties of spaces using tools from algebra (Algebraic Topology) and analysis (Differential Topology) 90 ... 11 3 14 The fundamental group of the circle 12 0 15 Van Kampen theorem 12 5 16 Simplicial homology 13 1 17 Singular... sin 1x , x > is often called the Topologist’s sine curve This is a classic example of a space which is connected but is not path-connected sin (1/ x) 0.8 0.6 0.4 0.2 -0 .2 -0 .4 -0 .6 -0 .8 -1 0 .1 0.2... diagram: / (1, 2) (1, 1) (1, 3) { { { (2, 1) (2, 2) (2, 3) { { (3, 1) (3, 2) (1, 4) { (4, 1) For detail we can derive the explicit formula for the counting: Z+ × Z+ → Z+ (m, n) → (1 + + · · ·