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## Topology

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**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