Department of Mathematics Lovely Professional University Topology Without Tears Sanjay Mishra Department of Mathematics Lovely Professional University Punjab, India 9/22/2018 Sanjay Mishra

Outline Objective of the Session What is Topology History of Topology Different Branches of Topology Basic Concepts of Topology Examples of Topologies in Applications Quiz 9/22/2018 Sanjay Mishra

Objective of the Session Topology is generally considered to be one of the three linchpins of modern abstract mathematics (along with analysis and algebra). In the early history of topology, results were primarily motivated by investigations of real-world problems. Then, after the formal foundation for topology was established in the first part of the twentieth century, the emphasis turned to its abstract development. However, within the past few decades there has been a significant increase in the applications of topology to fields as diverse as economics, engineering, chemistry, medicine, and cosmology. 9/22/2018 Sanjay Mishra

Objective of the Session Recently, topology has also become an important component of applied mathematics, with many mathematicians and scientists employing concepts of topology to model and understand real-world structures and phenomena. Our Goal in this session is: To introduces the traditional approach as an abstract discipline. To demonstrate the utility and signicance of topological ideas in other areas of mathematics, in science, and in engineering. 9/22/2018 Sanjay Mishra

What is Topology (Modern version of Geometry I ) Topology grew out of geometry, expanding on some of the ideas and loosening some of the structures appearing therein. The word topology, literally, means “ the study of position or location” Topology is the study of shapes: including their properties. deformations applied to them. mappings between them. configurations composed of them. 9/22/2018 Sanjay Mishra

What is Topology (Modern version of Geometry II ) “Topology is often described as rubber-sheet geometry” 9/22/2018 Sanjay Mishra

What is Topology (Modern version of Geometry III ) In traditional (Euclidean) geometry, objects such as circles, triangles, planes, and polyhedral are considered rigid, with well-defined distances between points and well-defined angles between edges or faces. We can move things around and flip them over, but you can‘t stretch or bend them. This is called "congruence" in geometry class. Two things are congruent if you can lay one on top of the other in such a way that they exactly match. 9/22/2018 Sanjay Mishra

What is Topology (Modern version of Geometry IV ) But in (Projective geometry) topology, distances and angles are irrelevant. We treat objects as if they are made of rubber, capable of being deformed. We allow continuous transformation on objects as like to be bent, twisted, stretched, shrunk, or otherwise deformed from one to another. But we do not allow the objects to be ripped apart i.e. Discontinuous transformation like cutting, tearing, and puncturing. 9/22/2018 Sanjay Mishra

What is Topology (Modern version of Geometry V ) Here all four shapes that are very different from a Euclidean geometric perspective, but are considered equivalent in topology. Anyone of the four, if made of rubber, can be deformed to each of the others. 9/22/2018 Sanjay Mishra

What is Topology (Modern version of Geometry VII ) Here two objects-the torus and the sphere-that are topologically distinct. We cannot deform a sphere into a torus in any permitted topological manner, and therefore they are not equivalent from a topological perspective. The torus and sphere are not topologically equivalent. 9/22/2018 Sanjay Mishra

What is Topology (Modern version of Geometry VI ) It is often said that a topologist cannot distinguish between a doughnut and a coffee cup. The point is that in topology a coffee cup can be deformed into the shape of a doughnut. These objects are equivalent as far as topology is concerned. A coffee cup and doughnut are topologically equivalent. 9/22/2018 Sanjay Mishra

History of Topology I Topology is popularly considered to have begun with Leonhard Euler's (1707-1783) solution to the famous Konigsberg bridges problem. Let us take a quick look at this problem. In the eighteenth century, the river Pregel flowed through the city of Konigsberg, in Prussia dividing it into four separate regions. There were seven bridges that crossed the river and connected the regions as illustrated in Figure A favoured pastime was to take a bridge-crossing stroll through Konigsberg. People asked, "Can you take a stroll through the city, crossing each bridge exactly once?" Curiously, no one was able to find a way to do so. 9/22/2018 Sanjay Mishra

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

Königsberg Bridge Problem History of Topology III Königsberg Bridge Problem Vertex Degree A 3 B 5 C D C B D A A graph has a path traversing each edge exactly once if exactly two vertices have odd degree.

Königsberg Bridge Problem History of Topology IV Königsberg Bridge Problem Vertex Degree A 3 B 4 C D 2 C B D A A graph has a path traversing each edge exactly once if exactly two vertices have odd degree.

History of Topology V Following the initial work of Euler, a number of prominent mathematicians made valuable contributions to the geometry of position over the next century and a half. These included Carl Friedrich Gauss (1777-1855), August Ferdinand Mobius (1790-1868), Johann Listing (1808-1882), Bernhard Riemann (1826-1866), Felix Klein (1849-1925), Henri Poincare (1854-1912). 9/22/2018 Sanjay Mishra

History of Topology VI Many nineteenth-century efforts in the geometry of position were motivated by applied problems, including James Clerk Maxwell's (1831-1879) and Peter Guthrie Tait's (1831-1901) work on knots (arising from investigations in chemistry), Gustav Kirchoff's (1824-1887) study of electrical networks, Poincare's analysis of celestial mechanics. 9/22/2018 Sanjay Mishra

History of Topology VII In the late nineteenth- and early twentieth-century there were numerous contributions to the growing discipline that would soon become the field of topology. L. E. J. Brouwer (1881-1966), Georg Cantor (1845-1918), Maurice Frechet (1878-1973), Felix Hausdorff (1868-1962), Frigyes Riesz (1880-1956), Hermann Weyl (1885-1955) Hausdorff's introduced an axiomatic foundation for topological spaces and thereby initiated the general study of topology as an abstract mathematics discipline. 9/22/2018 Sanjay Mishra

Branches of Topology 9/22/2018 Sanjay Mishra

Different Branches of Topology The low-level language of topology, which is not really considered a separate "branch" of topology, is known as point-set topology. Algebraic topology (which includes combinatorial topology), Differential topology Low-dimensional topology. 16.1.2008 Georg Dolhoff

Point Set Topology Point-set topology, also called set-theoretic topology or general topology, is the study of the general abstract nature of continuity or "closeness" on spaces. Basic point-set topological notions are ones like continuity, dimension, compactness, and connectedness. 16.1.2008 Georg Dolhoff

Algebraic Topology Study of intrinsic qualitative aspects of spatial objects surfaces, spheres, tori, circles, knots, links, configuration spaces, etc. Includes combinatorial topology 16.1.2008 Georg Dolhoff

Algebraic Topology Rubber-sheet geometry. Study of disconnectivities. Mathematical machinery for studying different kinds of hole structures 16.1.2008 Georg Dolhoff

Differential Topology Study of smooth (differentiable) manifolds. Nonmetrical notions of manifolds (while differential geometry deals with metrical notions of manifold) 16.1.2008 Georg Dolhoff

Low-Dimensional Topology Deals with objects that are two-, three-, or four-dimensional in nature. Low-dimensional topology should be part of differential topology, but General machinery of algebraic and differential topology gives only limited information (particularly noticeable in dimensions three and four) alternative specialized methods have evolved. 16.1.2008 Georg Dolhoff

Basic Concepts of Topology The objects that we study in topology are called topological spaces. These are sets of points on which a notion of proximity between points is established by specifying a collection of subsets called open sets. The line, the circle, the plane, the sphere, the torus, and the Mobius band are all examples of topological spaces. 9/22/2018 Sanjay Mishra

Basic Concepts of Topology The line, the circle, the plane, the sphere, the torus, and the Mobius band are all examples of topological spaces. 9/22/2018 Sanjay Mishra

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

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

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

Klein Bottle Demo on Klein Bottle How can we make Möbius Strip Moebius Highway Through one hole or two Topologie Topology Topology Jordan's Curve Theorem Sphere Inside out Part - I

Klein Bottle

Plus Magazine ~ Imaging Maths – Inside the Klein Bottle The Adventures of the Klein Bottle

Basic Concepts of Topology Manifolds An n-manifold is a topological space that locally resembles n-dimensional Euclidean space. For example, a 1-manifold locally resembles a line, a 2-manifold locally resembles a plane, a 3-manifold locally resembles 3-space, and so on. A 2-manifold is also called a surface. The sphere and torus are examples of surfaces. Each point in a surface lies in an open set that is topologically equivalent to an open set in the plane. 9/22/2018 Sanjay Mishra

Basic Concepts of Topology Manifolds A smooth manifold is a subset of Euclidean space which is locally the graph of a smooth (perhaps vector-valued) function. A more general topological manifold can be described as a topological space that on a small enough scale resembles the Euclidean space of a specific dimension, called the dimension of the manifold. More formally, every point of an n-dimensional manifold has a neighbourhood homeomorphic to an open subset of the n-dimensional space Rn. 9/22/2018 Sanjay Mishra

The sphere (surface of a ball) is a two-dimensional manifold since it can be represented by a collection of two-dimensional maps. 9/22/2018 Sanjay Mishra

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

Examples of Topologies in Applications Digital Topology Digital topology is the study of topological relationships on a digital image display (for example, a computer screen). It plays a role in the field of digital image processing. The digital image display contains a rectangular pixel array. In digital topology, this array is modeled by what is known as the digital plane. 9/22/2018 Sanjay Mishra

Examples of Topologies in Applications Digital Topology An important task in digital image processing is to determine features of an object from a digital image of it. For example, an optical character-recognition program reads a digital image of a character and attempts to determine the character represented by the image so that subsequently it can be used within a word-processing program. Topological properties of the image assist in determining the intended character. The fact that the image encloses two regions helps distinguish the associated character from characters that do not. 9/22/2018 Sanjay Mishra

Examples of Topologies in Applications Digital Topology In the digital line, every odd integer is an open set, and every even integer is a closed set. We view the digital line as a set of open pixels corresponding to the odd integers, along with the set of closed boundaries between the pixels corresponding to the even integers. 9/22/2018 Sanjay Mishra

Examples of Topologies in Applications Digital Topology For each n in Z, define The collection B = {B (n) : n  Z } is a basis for a topology on Z. The resulting topology is called the digital line topology, and we refer to Z with this topology as the digital line. 9/22/2018 Sanjay Mishra

Examples of Topologies in Applications Phenotype Spaces The genotype-phenotype relationship is of fundamental importance in biology. The Genotype is internally coded, inheritable information possessed by all living organisms. The Phenotype is the physical realization of that information. For example, The collection of genes responsible for eye colour in a particular individual is a genotype. The observable eye coloration in the individual is the corresponding phenotype. 9/22/2018 Sanjay Mishra

Examples of Topologies in Applications Phenotype Spaces A model of evolutionary ( gradually development) proximity established by defining a topology on a set of phenotypes. Molecular biologists propose this model as a means for formally defining continuous and discontinuous evolutionary change, providing a mathematical framework for understanding evolutionary processes. 9/22/2018 Sanjay Mishra

Examples of Topologies in Applications An Application to Geographic Information Systems Geographic information systems theory is one area in the field of Spatial Information Science and Engineering, a field that has undergone tremendous growth in recent years. A geographic information system (GIS) is a computer system capable of assembling, storing, manipulating, and displaying geographically referenced data. The data are often used for solving complex planning and management problems. To analyze spatial information, users select data from a GIS by submitting queries. Typical GIS queries incorporate spatial relations to describe constraints about spatial objects to be analyzed or displayed. 9/22/2018 Sanjay Mishra

Examples of Topologies in Applications An Application to Geographic Information Systems It is a field where topology has provided valuable modeling tools. A simple model, originally published in [Ege], that employs topological concepts to define and distinguish relationships between pairs of geographic areas. This model is by no means complete, but it is a start. It has been adopted as a GIS-industry standard for describing the relationships it addresses. [Ege] Egenhofer, E., and Franzosa, R., "Point-Set Topological Spatial Relations" International Journal of Geographical Inforrnation Systems 5 (1991), no. 2,161-174. 9/22/2018 Sanjay Mishra

Examples of Topologies in Applications Automated Guided Vehicles Automated guided vehicles are mobile robots that are used to transport materials from location to location in a manufacturing facility. Part of the challenge in designing and constructing such a facility is properly setting up mobile robot routes so that the robots can move in an efficient and safe manner. The tools and concepts of topology are naturally employed in this planning process. In this section we consider some simple examples. 9/22/2018 Sanjay Mishra

Examples of Topologies in Applications Automated Guided Vehicles We model each robot with a point that moves through a topological space representing the robot routes in the factory. To begin, suppose we have two robots A and B that move on a line represented by R. The configuration space for the two robots is the space 9/22/2018 Sanjay Mishra

Examples of Topologies in Applications Automated Guided Vehicles To prevent robot collisions, we do not allow the two robots to occupy the same point at the same time. That is, we never let The resulting space of permitted configurations is then To distinguish this space from the configuration space C, we call this the safe configuration space for the robots. 9/22/2018 Sanjay Mishra

Examples of Topologies in Applications The Brouwer Fixed Point Theorem Imagine taking two pieces of the same-sized paper and laying one piece on top of the other. Every point on the top sheet of paper is associated with some point right below it on the other sheet. Now take the top sheet of paper and crumple it up into a ball without ripping it. Place the crumpled ball back on top of the bottom sheet of paper. Somewhere on the crumpled ball of paper there is a point that is sitting directly above the same point on the bottom sheet of paper that it sat above before the crumpling took place. 9/22/2018 Sanjay Mishra

Examples of Topologies in Applications The Brouwer Fixed Point Theorem 9/22/2018 Sanjay Mishra

Thank You 9/22/2018 Sanjay Mishra