TOPOLOGY: AN OVERVIEW J. OWUSU ASARE & W. OBENG-DENTEH DEPARTMENT OF MATHEMATICS COLLEGE OF SCIENCE, KNUST- GHANA

Introduction

A Favourite Topological Object

I In our attempt to answer the question of what Topology is, we would briefly journey you through its foundation and formulation. Then you will appreciate why it stands out as one of the most active areas in all of Maths today; playing the role as one of the 3 main areas in pure Maths(together with Algebra and Analysis) and further acting as an indispensable component of Applied Mathematics.In our world today, it is almost impossible to understand many real world structures without its use [1]. Remember how you employed the use of quantitative relationships in high school when you studied figures like triangles, angles and circles in geometry? Well, Topology grew out of geometry by expanding on some of the ideas of geometry and the loosening of some of the rigid geometrical structures you disliked. It earned the name rubber-sheet geometry as a result. So your childhood muse about deforming scaled figures may hold in Topology, Yes! If you do not touch the qualitative properties of that figure. By that, we mean the properties of the figure that will remain untouched after all deformations.

Credit: Alon

Figure 3: The torus

History

Note

I In the eighteenth century, a river called Pregel flowed through the city of Konigsberg, dividing it into exactly four separate regions. Interestingly, seven bridges crossed the river Pregel and connected the four regions of the city as shown in Figure 1. The natural curiosity of people led to the question of whether it was possible to take an entire stroll through the city by crossing each bridge exactly once? Leonhard Euler(1707-1783), solved this long pending problem about the famous seven bridges in Konigsberg, Kaliningrad(Russia). Since he was the first one to discuss geometry without measurement, he is considered by many as the father of Topology. But aside him are notable mentions like Descartes, Poincare, Listing, Riemann, Cantor, Riesz, Frechet etc. To a topologist, a coffee cup and a doughnut are not distinguishable! But you would be amazed to find out what a topologist can do [1], [2], [3]. For more on history see [4].

I I I I I I I I I Credit: [4]

Figure 1: The Konigsberg bridges

Amit (Quora)

I

The objects we study in Topology are called Topological spaces. Simply put, a space is a set with an added structure. We may choose to define Topological spaces in terms of open or closed sets. A subset A of a topological space X is open if the set X − A is closed. A subset A of a topological space X is closed if the set X − A is open. Open sets are considered as “compliments” of closed sets instead of “opposites”. A set may be open, closed, half-open or half-closed. When the set is half-open or half-closed, it’s called clopen. Some of the properties of sets that we are usually interested in are its interior, exterior, closure, boundary and derived sets. To deeply appreciate Topology, some key concepts to look out for are metric spaces, continuity, connectedness and compactness.

Applications of Topology

Credit: Business

Insider

Figure 2: coffee cup and a doughnut

Definitions 1).Topology is the mathematical discipline concerned with giving precise definitions for the concept of spatial structure, comparing the various definitions of spatial structure that have been or may be given, and investigating relations between properties that can be induced into a topological system [2]. 2). Topology is the study of shapes, including their properties, deformations applied to them, mappings between them and configurations composed of them [1]. 3). Topology is the study of all properties of a space that are invariant under 1:1 bi-continuous mappings [2].∗ 4). Let X be a set. A topology T on X is a collection of subsets of X, each called an open set, such that; •∅ and X are open sets; •The intersection of finitely many open sets is an open set; •The union of any collection of open sets is an open set. The set X together with a topology T on X is called a topological space [1]. We shall now proceed to show you one of the most favourite topological objects known. ∗The use of (1) and (2), is usually preferred by many to the use of (3) since no prior knowledge base is required by the reader.

I I I I I I I I I I I I I

Topology has many profound applications some of which extend to; Embedded systems and Network systems Epidemiology Data Analysis and Population modelling Computer vision Manifolds and Cosmology Gauge transformations in Physics Braids, Knots and DNA Medical imaging such as CT Scan Fixed Points and Economics Geographic information systems Motion planning in robotics Computer Science Phenotype Spaces in Evolutionary Biology etc.

Conclusion This poster may not be enough to even contain a fraction of the endless applications of Topology let alone Mathematics. So the next time you wonder about what Topologists or Mathematicians can do, remind yourself of their endless viabilities and careers. It is rather unfortunate that we end our journey here! However, if your interest has been sparked, do continue to explore further. Who knows, you may become a contributor to this indispensable form of Mathematics someday. References [1] Adams C, Franzosa R. Introduction To Topology, Pure and Applied Pearson Prentice Hall;2007. [2] Wolfgang J. Thron Topological Structures Holt, Rinehart and Winston Inc;1966 [3] James Munkres. Topology, Upper Saddle River, N.J . Prentice Hall;2000 [4] O’Connor, JJ and Robertson, EF. Topology in Mathematics, accessed at http://www-history.mcs.st-andrews.ac.uk/HistTopics/Topology in mathematics.html

Introduction

A Favourite Topological Object

I In our attempt to answer the question of what Topology is, we would briefly journey you through its foundation and formulation. Then you will appreciate why it stands out as one of the most active areas in all of Maths today; playing the role as one of the 3 main areas in pure Maths(together with Algebra and Analysis) and further acting as an indispensable component of Applied Mathematics.In our world today, it is almost impossible to understand many real world structures without its use [1]. Remember how you employed the use of quantitative relationships in high school when you studied figures like triangles, angles and circles in geometry? Well, Topology grew out of geometry by expanding on some of the ideas of geometry and the loosening of some of the rigid geometrical structures you disliked. It earned the name rubber-sheet geometry as a result. So your childhood muse about deforming scaled figures may hold in Topology, Yes! If you do not touch the qualitative properties of that figure. By that, we mean the properties of the figure that will remain untouched after all deformations.

Credit: Alon

Figure 3: The torus

History

Note

I In the eighteenth century, a river called Pregel flowed through the city of Konigsberg, dividing it into exactly four separate regions. Interestingly, seven bridges crossed the river Pregel and connected the four regions of the city as shown in Figure 1. The natural curiosity of people led to the question of whether it was possible to take an entire stroll through the city by crossing each bridge exactly once? Leonhard Euler(1707-1783), solved this long pending problem about the famous seven bridges in Konigsberg, Kaliningrad(Russia). Since he was the first one to discuss geometry without measurement, he is considered by many as the father of Topology. But aside him are notable mentions like Descartes, Poincare, Listing, Riemann, Cantor, Riesz, Frechet etc. To a topologist, a coffee cup and a doughnut are not distinguishable! But you would be amazed to find out what a topologist can do [1], [2], [3]. For more on history see [4].

I I I I I I I I I Credit: [4]

Figure 1: The Konigsberg bridges

Amit (Quora)

I

The objects we study in Topology are called Topological spaces. Simply put, a space is a set with an added structure. We may choose to define Topological spaces in terms of open or closed sets. A subset A of a topological space X is open if the set X − A is closed. A subset A of a topological space X is closed if the set X − A is open. Open sets are considered as “compliments” of closed sets instead of “opposites”. A set may be open, closed, half-open or half-closed. When the set is half-open or half-closed, it’s called clopen. Some of the properties of sets that we are usually interested in are its interior, exterior, closure, boundary and derived sets. To deeply appreciate Topology, some key concepts to look out for are metric spaces, continuity, connectedness and compactness.

Applications of Topology

Credit: Business

Insider

Figure 2: coffee cup and a doughnut

Definitions 1).Topology is the mathematical discipline concerned with giving precise definitions for the concept of spatial structure, comparing the various definitions of spatial structure that have been or may be given, and investigating relations between properties that can be induced into a topological system [2]. 2). Topology is the study of shapes, including their properties, deformations applied to them, mappings between them and configurations composed of them [1]. 3). Topology is the study of all properties of a space that are invariant under 1:1 bi-continuous mappings [2].∗ 4). Let X be a set. A topology T on X is a collection of subsets of X, each called an open set, such that; •∅ and X are open sets; •The intersection of finitely many open sets is an open set; •The union of any collection of open sets is an open set. The set X together with a topology T on X is called a topological space [1]. We shall now proceed to show you one of the most favourite topological objects known. ∗The use of (1) and (2), is usually preferred by many to the use of (3) since no prior knowledge base is required by the reader.

I I I I I I I I I I I I I

Topology has many profound applications some of which extend to; Embedded systems and Network systems Epidemiology Data Analysis and Population modelling Computer vision Manifolds and Cosmology Gauge transformations in Physics Braids, Knots and DNA Medical imaging such as CT Scan Fixed Points and Economics Geographic information systems Motion planning in robotics Computer Science Phenotype Spaces in Evolutionary Biology etc.

Conclusion This poster may not be enough to even contain a fraction of the endless applications of Topology let alone Mathematics. So the next time you wonder about what Topologists or Mathematicians can do, remind yourself of their endless viabilities and careers. It is rather unfortunate that we end our journey here! However, if your interest has been sparked, do continue to explore further. Who knows, you may become a contributor to this indispensable form of Mathematics someday. References [1] Adams C, Franzosa R. Introduction To Topology, Pure and Applied Pearson Prentice Hall;2007. [2] Wolfgang J. Thron Topological Structures Holt, Rinehart and Winston Inc;1966 [3] James Munkres. Topology, Upper Saddle River, N.J . Prentice Hall;2000 [4] O’Connor, JJ and Robertson, EF. Topology in Mathematics, accessed at http://www-history.mcs.st-andrews.ac.uk/HistTopics/Topology in mathematics.html