1 / 64

650 likes | 833 Views

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. C. B. D. A.

Download Presentation
## Topology

**An Image/Link below is provided (as is) to download presentation**
Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author.
Content is provided to you AS IS for your information and personal use only.
Download presentation by click this link.
While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server.
During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

**Topology**Senior Math PresentationNate 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**C B D A A graph has a path traversing each edge exactlyonce if exactly two vertices have odd degree.**Königsberg Bridge Problem**C B D A A graph has a path traversing each edge exactlyonce 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**• 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**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**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 , 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**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 .**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**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 where 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 is said to be connected if , where is both open and closed, then or .**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**General Connectedness**• A subset of a topological space is said to be connected if , where is both relatively open and relatively closed, then .**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**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 .**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 .**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.**Path Connected**0 1**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 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 .**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**Simple Connectedness**time (0,1) (1,1) space (1,0)**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.**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 .**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 .**Simply Connected**Entrance Exit**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.**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.**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.**Compactness**Remove any interval -5 0 5 2 is no longer covered**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.**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 .**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**Applications**• Network Theory • Knot Theory • Genus categorization

More Related