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Lecture Notes on General Topology Bit, Spring 2021 LECTURE NOTES ON GENERAL TOPOLOGY BIT, SPRING 2021 DAVID G.L. WANG Contents 1. Introduction 2 1.1. Who cares topology? 2 1.2. Geometry v.s. topology 2 1.3. The origin of topology 3 1.4. Topological equivalence 5 1.5. Surfaces 6 1.6. Abstract spaces 6 1.7. The classification theorem and more 6 2. Topological Spaces 8 2.1. Topological structures 8 2.2. Subspace topology 12 2.3. Point position with respect to a set 14 2.4. Bases of a topology 19 2.5. Metrics & the metric topology 21 3. Continuous Maps & Homeomorphisms 30 3.1. Continuous maps 30 3.2. Covers 35 3.3. Homeomorphisms 37 4. Connectedness 45 4.1. Connected spaces 45 Date: March 2, 2021. 1 4.2. Path-connectedness 53 5. Separation Axioms 57 5.1. Axioms T0, T1, T2, T3 and T4 57 5.2. Hausdor↵spaces 60 5.3. Regular spaces & normal spaces 62 5.4. Countability axioms 65 6. Compactness 68 6.1. Compact spaces 68 6.2. Interaction of compactness with other topological properties 70 7. Product Spaces & Quotient Spaces 75 7.1. Product spaces 75 7.2. Quotient spaces 81 Appendix A. Some elementary inequalities 88 3 1. Introduction 1.1. Who cares topology? The Nobel Prize in Physics 2016 was awarded with one half to David J. Thouless, and the other half to F. Duncan M. Haldane and J. Michael Kosterlitz “for theoretical discoveries of topological phase transitions and topo- logical phases of matter”; see Fig. 1. Figure 1. Topology was the key to the Nobel Laureates’ discoveries, and it explains why electronical conductivity inside thin layers changes in integer steps. Stolen from Popular Science Background of the Nobel Prize in Physics 2016, Page 4(5). Topology, over most of its history, has NOT generally been applied outside of mathematics (with a few interesting exceptions). WHY? TOO abstract? The ancient mathematicians could not even convince of the subject. • It is qualitative, not quantitative? People think of science as a quantitative endeavour. • 1.2. Geometry v.s. topology. Below are some views from Robert MacPherson, a plenary addresser at the ICM in Warsaw in 1983. Geometry (from ancient Greek): geo=earth, metry=measurement. • Topology (from Greek): ⌧´o ⇡oσ=place/position, λ´o γoσ=study/discourse. 4 D.G.L. WANG Figure 2. Screenshot from a video of Robert MacPherson’s talk in Institute for Advanced Study. Topology is “geometry” without measurement. • It is qualitative (as opposed to quantitative) “geometry”. Geometry: The point M is the midpoint of the straight line segment L connecting • A to B. Topology: The point M lies on the curve L connecting A to B. Geometry calls its objects configurations (circles, triangles, etc.) • Topology calls its objects spaces. 1.3. The origin of topology. Here are three stories about the origin of topology. 1.3.1. The seven bridges of K¨onigsberg. The problem was to devise a walk through the city that would cross each of those bridges once and only once; see Fig. 3. The negative resolution by Leonhard Paul Euler (1707–1783) in 1736 laid the founda- tions of graph theory and prefigured the idea of topology. The difficulty Euler faced was the development of a suitable technique of analysis, and of subsequent tests that established this assertion with mathematical rigor. Euler was a Swiss mathematician, physicist, astronomer, logician and engineer who made important and influential discoveries in many branches of mathematics like infinitesimal calculus and graph theory, while also making pioneering contributions to several branches such as topology and analytic number theory. 1.3.2. The four colour theorem. The four colour theorem states that given any separation of a plane into contiguous regions, producing a figure called a map, no more than four colours are required to colour the regions of the map so that no two adjacent regions have the same color. 5 Figure 3. Map of K¨onigsberg in Euler’s time showing the actual layout of the seven bridges, highlighting the river Pregel and the bridges. Stolen from Wiki. The four color theorem was proved in 1976 by Kenneth Appel and Wolfgang Haken. It was the first major theorem to be proved using a computer. 1.3.3. Euler characteristic. χ = v e + f; see Fig. 4. − Figure 4. Stamp of the former German Democratic Republic honouring Euler on the 200th anniversary of his death. Across the centre it shows his polyhedral formula. Stolen from Wiki. We use the terminology polyhedron to indicate a surface rather than a solid. Theorem 1.1 (Euler’s polyhedral formula). Let P be a polyhedron s.t. Any two vertices of P can be connected by a chain of edges. • Any cycle along edges of P which is made up of straight line segments (not necessarily • edges) separates P into 2 pieces. Then the Euler number or Euler characteristic χ =2for P . 1750: first appear in a letter from Euler to Christian Goldbach (1690–1764). • 1860 (around): M¨obiusgave the idea of explaining topological equivalence by thinking • of spaces as being made of rubber.ItworksforconcaveP . 6 D.G.L. WANG David Eppstein collected 20 proofs of Theorem 1.1. • In 3-dimensional space, a Platonic solid is a regular, convex polyhedron. The Euler characteristic of every Platonic solid is 2. In fact, there are only 5 (why?) Platonic solids: the tetrahedron, the cube, the octahedron, the dodecahedron, and the icosahedron; see Fig. 5. Figure 5. The Plotonic solids. Stolen from Wiki. 1.4. Topological equivalence. =homeomorphism;seeSection1.6. p :thicken,stretch,bend,twist,...; :identify,tear,.... ⇥ See Fig. 6. Figure 6. Some surfaces which are not equivalent. Stolen from Haldane’s slides on Dec. 8th, 2016. 7 Theorem 1.2. Topological equivalent polyhedra have the same Euler characteristic. The starting point for modern topology. • Search for unchanged properties of spaces under topological equivalence. • χ =2belongstoS2,ratherthantoparticularpolyhedra define χ for S2. • ! Theorem 1.2:di↵erentcalculations,sameanswer. • 1.5. Surfaces. What exactly do we mean by a “space”? Homeomorphism Continuity. • ! Geometry Boundedness. • ! 1.6. Abstract spaces. The axioms for a topological space appearing for the first time in 1914 in the work of Felix Hausdor↵ (1868–1942). Hausdor↵, a German mathematician, is considered to be one of the founders of modern topology, who contributed significantly to set theory, descriptive set theory, measure theory, function theory, and functional analysis. How has modern definition of a topological space been formed? (1) General enough to allow set of points or functions, and performable constructions like the Cartesian products and the identifying. Enough information to define the continuity of functions between spaces. (2) Cauchy: distances continuity. ! (3) No distance! Continuity neighbourhood axiom. Afunctionf : Em En is continuous if given any x Em and any neighbourhood 1 ! 2 U of f(x), then f − (U)isaneighbourhoodofx. 1.7. The classification theorem and more. Theorem 1.3 (Classification theorem). Any closed surface is homeomorphic to S2 with either a finite number of handles added, or a finite number of M¨obiusstrips added. No two of these surfaces are homeomorphic. Definition 1.4. The S2 with n handles added is called an orientable surface of genus n. Non-orientable surfaces can be defined analogously. Historical notes. The classification of surfaces was initiated and carried through in the orientable case by M¨obiusin a paper which he submitted for consideration for the Grand Prix de Math´ematiques of the Paris Academy of Sciences. He was 71 at the time. The jury did not consider any of the manuscripts received as being worthy of the prize, and M¨obius’ work finally appeared as just another mathematical paper. 8 D.G.L. WANG Decide = or =. ⇠ 6⇠ =: construct a homeomorphism; techniques vary. • ⇠ ⇠=:lookfortopologicalinvariants,e.g.,geometricproperties,numbers,algebraic • systems.6 Examples to show =. 6⇠ E1 = E2:connectedness,h: E1 0 E2 h(0) . • 6⇠ \{ }! \{ } Poincar´e’s construction idea: assign a group to each topological space so that • homeomorphic spaces have isomorphic groups. But group isomorphism does not imply homeomorphism. Here are some theorems that the fundamental groups help prove. Theorem 1.5 (Classification of surfaces). No 2 surfaces in Theorem 1.3 have isomorphic fundamental groups. Theorem 1.6 (Jordan separation theorem). Any simple closed curve in E1 divides E1 into 2 pieces. See Fig. 7. Figure 7. Marie Ennemond Camille Jordan (1838–1922) was a French mathematician, known both for his foundational work in group theory and for his influential Cours d’analyse. The Jordan curve (drawn in black) divides the plane into an “inside” region (light blue) and an “outside” region (pink). Stolen from Wiki. Theorem 1.7 (Brouwer fixed-point theorem). Any continuous function from a disc to itself leaves at least one point fixed. See Fig. 8. Theorem 1.8 (Nielsen-Schreier theorem). A subgroup of a free group is always free. 9 Figure 8. Luitzen Egbertus Jan Brouwer (1881–1966), usually cited as L. E. J. Brouwer but known to his friends as Bertus, was a Dutch mathe- matician and philosopher, who worked in topology, set theory, measure theory and complex analysis. He was the founder of the mathematical philosophy of intuitionism. Stolen from Wiki. 2. Topological Spaces The definition of topological space fits quite well with our intuitive idea of what a space ought to be. Unfortunately it is not terribly convenient to work with. We want an equivalent, more manageable, set of axioms! 2.1. Topological structures. Definition 2.1. Let X be a set and ⌦ 2X . ✓ A(topological)space is a pair (X, ⌦), where the collection ⌦, called a topology or • topological structure on X,satisfiestheaxioms (i) ⌦andX ⌦; ;2 2 (ii) the union of any members of ⌦lies in ⌦; (iii) the intersection of any two members of ⌦lies in ⌦.
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