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

2. 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

3. 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

4. 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

5. 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

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

7. 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

8. 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

9. 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

10. 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

11. 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

12. History of Topology I
Topology is popularly considered to have begun with Leonhard Euler's ( ) 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

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

14. 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.

15. 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.

16. 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 ( ),
August Ferdinand Mobius ( ),
Johann Listing ( ),
Bernhard Riemann ( ),
Felix Klein ( ),
Henri Poincare ( ).
9/22/2018
Sanjay Mishra

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

18. 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 ( ),
Georg Cantor ( ),
Maurice Frechet ( ),
Felix Hausdorff ( ),
Frigyes Riesz ( ),
Hermann Weyl ( )
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

19. Branches of Topology
9/22/2018
Sanjay Mishra

20. 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.
Georg Dolhoff

21. 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.
Georg Dolhoff

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

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

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

25. 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.
Georg Dolhoff

26. 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

27. 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

28. 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

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

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

31. 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

32. Klein Bottle

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

34. 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

35. 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

36. 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

37. 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

38. 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

39. 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

40. 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

41. 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

42. 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

43. 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

44. 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

45. 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,
9/22/2018
Sanjay Mishra

46. 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

47. 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

48. 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

49. 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

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

51. Thank You
9/22/2018
Sanjay Mishra