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Topology Senior Math Presentation Nate Black Bob Jones University 11/19/07 History • Leonard Euler – Königsberg Bridge Problem Königsberg Bridge Problem J. J. O’Connor, A history of Topology Königsberg Bridge Problem Vertex Degree A 3 B 5 C 3 D 3 C B D A A graph has a path traversing each edge exactly once if exactly two vertices have odd degree. Königsberg Bridge Problem Vertex Degree A 3 B 4 C 3 D 2 C B D A A graph has a path traversing each edge exactly once if exactly two vertices have odd degree. History • Leonard Euler – Königsberg Bridge Problem • August Möbius – Möbius Strip Möbius Strip • A sheet of paper has two sides, a front and a back, and one edge • A möbius strip has one side and one edge Möbius Strip Plus Magazine ~ Imaging Maths – Inside the Klein Bottle History • Leonard Euler – Königsberg Bridge Problem • August Möbius – Möbius Strip • Felix Klein – Klein Bottle Klein Bottle • A sphere has an inside and an outside and no edges • A klein bottle has only an outside and no edges Klein Bottle Plus Magazine ~ Imaging Maths – Inside the Klein Bottle General Topology Overview General Topology Overview • Definition of a topological space • A topological space is a pair of objects, X , , where X is a non-empty set and is a collection of subsets of X , such that the following four properties hold: – 1. X – 2. – 3. If O1 , O2 ,..., On then O1 O2 ... On – 4. If for each I, O then I O General Topology Overview • Terminology – X is called the underlying set – is called the topology on X – All the members of are called open sets • Examples – X x1 , x2 , x3 , x4 , x5 with , x1 , x2 , x1 , x2 , X – Another topology on X , , x1 , X – The real line with open intervals, and in general n General Topology Overview • Branches – Point-Set Topology • Based on sets and subsets • Connectedness • Compactness – Algebraic Topology • Derived from Combinatorial Topology • Models topological entities and relationships as algebraic structures such as groups or a rings General Topology Overview • Definition of a topological subspace • Let Y X , and X , , Y , be topological spaces, then Y is said to be a subspace of X • The elements of O1 , O2 ,..., On are open O sets by definition, if we let O Y , then O , O ,..., O and these sets are said to be relatively open in Y 1 1 2 n 1 General Topology Overview • Definition of a relatively open set • Let , A Y X where Y is a subspace of X and A is a subset of Y . Then A is said to be relatively open in Y if , M X M Y A and M is open in X. • Definition of a relatively closed set • Let , B Y X where Y is a subspace of X and B is a subset of Y . Then B is said to be relatively closed in Y if , N X N Y B and N is closed in X. General Topology Overview • Definition of a neighborhood of a point • Let a X , where X is a topological space. Then a neighborhood of a , denoted N (a) , is a subset of X that contains an open set containing a . • Continuous function property • A continuous function maps open/closed sets in X into open/closed sets in Y Connectedness Connectedness • The general idea that all of the space touches. A point can freely be moved throughout the space to assume the location of any other point. B A Connectedness • General Connectedness – A space that cannot be broken up into several disjoint yet open sets – Consider X x1, x2 where , x1, X x2 x1 B A Connectedness • General Connectedness – A space that cannot be broken up into several disjoint yet open sets • Path Connectedness – A space where any two points in the space are connected by a path that lies entirely within the space – This is different than a convex region where the path must be a straight line Connectedness • Simple Connectedness – A space that is free of “holes” – A space where every ball can be shrunk to a point – A space where every path from a point A to a point B can be deformed into any other path from the point A to the point B General Connectedness • Definition of general connectedness • A topological space X is said to be connected if x X, where x is both open and closed, then x or X . General Connectedness • Example of a disconnected set – X 1,2,3,4 with , 1,2, 3,4, 1,2,3,4 – Let A 1,2, B 3,4 – Then A B X , A B – This implies that B is the complement of A – Since A is open then B is closed – But B is also open since it is an element of – Since B is neither X nor , X is shown to be disconnected General Connectedness • A subset A of a topological space X is said to be connected if a A, where a is both relatively open and relatively closed, then a A , or . Path Connectedness • Definition of a path in X • Let a, b X , f : 0,1 X , where f is a continuous function, and let f 0 a and f 1 b. Then f is called a path in X and f 0,1 , the image of the interval, is a curve in X that connects a to b . 0 f f 1 a b Path Connectedness • Definition of a path connected space • Let x, y X, where X is a topological space. Then X is said to be path connected if there is a path that connects x to y for all x, y. Path Connectedness • Is every path connected space also generally connected? • Let X be a topological space that is path connected. Now suppose that X is disconnected. • Then A X A is both open and closed, and A or X . Path Connectedness • Let a A and b C A. Since X is path connected f : 0,1 X f 0 a, f 1 b. • Consider B t f t A , clearly B since f 0 a A. • In addition, B 0,1 since f 1 b A . • This set B is then either open or closed but not both since 0,1 is connected. • Therefore, A can be open or closed but not both. Path Connected X f 0 a B A f 1 f b Path Connectedness • This is a contradiction, so we conclude that every path connected space is also generally connected. • Proof taken from Mendelson p. 135 Simple Connectedness • Definition of a homotopy • Let f1 , f 2 be paths in X that connect x to y , where x, y X , then f1 is said to be homotopic to f 2 if H : 0,12 X, where H is continuous, such that the following hold true for 0 x, t 1 . H 0, t x H 1, t y H s ,0 f 1 s H s,1 f 2 s Simple Connectedness time Time 0: (0,1) (1,1) Mile marker 0 Mile marker 1 Mile marker 0 Mile marker 1 space (1,0) Time 1: Simple Connectedness time (0,1) f2 (1,1) f1 space (1,0) Simple Connectedness • The function H is called the homotopy connecting f1 to f 2. f1 and f 2 both belong to the same homotopy class. • In a simply connected space any path between two points can be deformed into any other space. • Consider the closed loops, ones in which the starting and ending points are the same. Then they must all be deformable into one another. Simple Connectedness • One such closed path where we leave from a point A and return to it, is to never leave it. • This path is called the constant path and is denoted by e A . Simple Connectedness • Define a simply connected space • Let X be a topological space and x X . Then X is said to be simply connected if for every x there is only one homotopy class of closed paths. Since the constant path is guaranteed to be a closed path for x , the homotopy class must be eA . Simply Connected Entrance Exit Simply Connected Compactness Compactness • Definition of a covering • Let X be a set, A X , and C B I be an indexed subset of A . Then the set C is said to cover A if A c . cC • If only a finite number of sets are needed to cover A , then C is more specifically a finite covering. Compactness • Definition of a compact space • Let X be a topological space, and let C B I be a covering of X . Then if for C D D C and D is finite, then X is said to be compact. Compactness • Example of a space that is not compact • Consider the real line • The set of open intervals n, n 2 n Z is clearly a covering of R . • Removing any one interval leaves an integer value uncovered. • Therefore, no finite subcovering exists. Compactness Remove any interval -5 0 5 2 is no longer covered Compactness • Define locally compact • Let X be a topological space, then X is said to be locally compact if x X N x N x is compact. • Note that every generally compact set is also locally compact since some subset of the finite coverings of the whole set will be a finite covering for some neighborhood of every x in the space. Compactness • Is every closed subset of a compact space compact as well? • Let F be a closed subset of the compact space X . • If U I is an open covering of F , then by adjoining the open set O CF to the open covering of V J we obtain an open covering of X . Compactness • Since X is compact there is a finite subcovering V ,V ,...,V of X . • However, each V is either equal to a U for some I or equal to O . • If O occurs among V ,V ,...,V we may delete it to obtain a finite collection of the U ’s that covers F C O • Proof taken from Mendelson 162-163 1 2 m i 1 2 m Applications Applications • Network Theory • Knot Theory • Genus categorization Applications • Genus = the number of holes in a surface – Formally: the largest number of nonintersecting simple closed curves that can be drawn on the surface without separating it. (Mathworld) 5 holes Matt Black Applications Matt Black Applications 2006 Encyclopædia Britannica Applications • Definition of Fixed Point Theorem • Let f be a continuous function over the interval 0,1, such that f 0 0 and f 1 1 . Then z 0,1 f z z. (1,1) (0,1) z (1,0) Applications • 0,1 is path connected, and since f is continuous, f 0,1 is also path connected. • If v 0,1 , then f v f 0,1 . • Suppose f never crosses the line y x then let a, b be an interval containing some z , where f a a and b f b. Applications • Clearly, there is a path that connects f a to f b since f a, b is a closed interval subset of f 0,1 and is therefore connected. • Since f is continuous we now shrink the width of the interval a, b. • When a z b , the width will be zero and we can substitute z in for a and b . • This implies that f z z f z and, since a path must connect the endpoints, f z z . Applications • Definition of a Conservative Force • The work done in changing an object’s position is said to be path independent. • One significant result of this type of a force is that the work done in moving an object along a closed path is 0. Applications • Let x be a point in a space that is topologically simple, then x X , ex is the homotopy class for all closed paths that include x. • The work done in not moving an object is obviously 0, and since all closed paths containing x are in the same homotopy class, the work done on them is also 0. Applications • We can prove it another way as well • Let us define positive work to be that done on a path heading away from us, and negative work to be done when heading in the opposite direction, toward us. • Then pick any point on the closed path as your starting point and some other point on the path as your ending point. Applications • Move along the straight line connecting those two points, and then move back along the same path. • The total work done during the journey will be 0 since the magnitude of both trips is the same, but the signs are opposite. • Each traversal of the path is homotopically equivalent to traversing one-half of the closed path. Applications 0 total work + work – work Questions Special Thanks • Dr. Knisely for assistance in my paper direction and revision • My brother, Matt Black, for making graphics I couldn’t find online Bibliography • Königsberg city map – http://www-history.mcs.st-and.ac.uk/HistTopics/Topology_in_mathematics.html • Klein Bottle Images and rotating Möbius strip – http://plus.maths.org/issue26/features/mathart/index-gifd.html • Möbius steps and genus diagrams and video – Matthew Black • Coffee Cup Animation – http://www.britannica.com/eb/article-9108691 • Genus definition – http://mathworld.wolfram.com/Topology.html • Mendelson Textbook – Mendelson, Bert. Introduction to Topology. NY: Dover, 1990. • Munkers Textbook – Munkers, James R. Topology. New Delhi: Prentice Hall of India, 2000. • All else, property of Nathanael Black