Topology

1. Topology Senior Math PresentationNate Black Bob Jones University 11/19/07

2. History • Leonard Euler • Königsberg Bridge Problem

3. Königsberg Bridge Problem J. J. O’Connor, A history of Topology

4. Königsberg Bridge Problem C B D A A graph has a path traversing each edge exactlyonce if exactly two vertices have odd degree.

5. Königsberg Bridge Problem C B D A A graph has a path traversing each edge exactlyonce if exactly two vertices have odd degree.

6. History • Leonard Euler • Königsberg Bridge Problem • August Möbius • Möbius Strip

7. 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

8. Möbius Strip Plus Magazine ~ Imaging Maths – Inside the Klein Bottle

9. History • Leonard Euler • Königsberg Bridge Problem • August Möbius • Möbius Strip • Felix Klein • Klein Bottle

10. Klein Bottle • A sphere has an inside and an outside and no edges • A klein bottle has only an outside and no edges

11. Klein Bottle Plus Magazine ~ Imaging Maths – Inside the Klein Bottle

12. General Topology Overview

13. General Topology Overview • Definition of a topological space • A topological space is a pair of objects, , where is a non-empty set and is a collection of subsets of , such that the following four properties hold: • 1. • 2. • 3. If then • 4. If for each then

14. General Topology Overview • Terminology • is called the underlying set • is called the topology on • All the members of are called open sets • Examples • with • Another topology on , • The real line with open intervals, and in general

15. 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

16. General Topology Overview • Definition of a topological subspace • Let , and be topological spaces, then is said to be a subspace of • The elements of are open sets by definition, if we let , then and these sets are said to be relatively open in

17. General Topology Overview • Definition of a relatively open set • Let , where is a subspace of and is a subset of . Then is said to be relatively open in if ,and is open in . • Definition of a relatively closed set • Let , where is a subspace of and is a subset of . Then is said to be relatively closed in if ,and is closed in .

18. General Topology Overview • Definition of a neighborhood of a point • Let , where is a topological space. Then a neighborhood of , denoted , is a subset of that contains an open set containing . • Continuous function property • A continuous function maps open/closed sets in into open/closed sets in

19. Connectedness

20. 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

21. Connectedness • General Connectedness • A space that cannot be broken up into several disjoint yet open sets • Consider where x2 x1 B A

22. 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

23. 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

24. General Connectedness • Definition of general connectedness • A topological space is said to be connected if , where is both open and closed, then or .

25. General Connectedness • Example of a disconnected set • with • Let • Then • This implies that is the complement of • Since is open then is closed • But is also open since it is an element of • Since is neither nor , is shown to be disconnected

26. General Connectedness • A subset of a topological space is said to be connected if , where is both relatively open and relatively closed, then .

27. Path Connectedness • Definition of a path in • Let , , where is a continuous function, and let and . Then is called a path in and , the image of the interval, is a curve in that connects to . 0 1

28. Path Connectedness • Definition of a path connected space • Let , where is a topological space. Then is said to be path connected if there is a path that connects to for all .

29. Path Connectedness • Is every path connected space also generally connected? • Let be a topological space that is path connected. Now suppose that is disconnected. • Then is both open and closed, and or .

30. Path Connectedness • Let and . Since is path connected . • Consider , clearly since . • In addition, since . • This set is then either open or closed but not both since is connected. • Therefore, can be open or closed but not both.

31. Path Connected 0 1

32. Path Connectedness • This is a contradiction, so we conclude that every path connected space is also generally connected. • Proof taken from Mendelson p. 135

33. Simple Connectedness • Definition of a homotopy • Let be paths in that connect to , where , then is said to be homotopic to if , where is continuous, such that the following hold true for .

34. Simple Connectedness time Time 0: (0,1) (1,1) Mile marker 0 Mile marker 1 space (1,0) Time 1: Mile marker 0 Mile marker 1

35. Simple Connectedness time (0,1) (1,1) space (1,0)

36. Simple Connectedness • The function is called the homotopy connecting to . and 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.

37. 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 .

38. Simple Connectedness • Define a simply connected space • Let be a topological space and . Then is said to be simply connected if for every there is only one homotopy class of closed paths. Since the constant path is guaranteed to be a closed path for , the homotopy class must be .

39. Simply Connected Entrance Exit

40. Simply Connected

41. Compactness

42. Compactness • Definition of a covering • Let be a set, , and be an indexed subset of . Then the set is said to cover if . • If only a finite number of sets are needed to cover , then is more specifically a finite covering.

43. Compactness • Definition of a compact space • Let be a topological space, and let be a covering of . Then if for and is finite, then is said to be compact.

44. Compactness • Example of a space that is not compact • Consider the real line • The set of open intervals is clearly a covering of . • Removing any one interval leaves an integer value uncovered. • Therefore, no finite subcovering exists.

45. Compactness Remove any interval -5 0 5 2 is no longer covered

46. Compactness • Define locally compact • Let be a topological space, then is said to be locally compact if 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 in the space.

47. Compactness • Is every closed subset of a compact space compact as well? • Let be a closed subset of the compact space . • If is an open covering of , then by adjoining the open set to the open covering of we obtain an open covering of .

48. Compactness • Since is compact there is a finite subcovering of . • However, each is either equal to a for some or equal to . • If occurs among we may delete it to obtain a finite collection of the ’s that covers • Proof taken from Mendelson 162-163

49. Applications

50. Applications • Network Theory • Knot Theory • Genus categorization