Have you ever taken a rubber band and twisted it into different shapes for fun? Whether you were doing it to pass the time in the doctor’s office waiting room or to count how many different shapes you could make, this is actually what we call topology. Taking a shape and transforming it without breaking or having to reattach any corners or lines is exactly what topology is. If you can take that round rubber band and bend it into the shape of a square, then the circle and square are now topologically equivalent.
Topology 101
Topology is a branch of mathematics that studies properties of space that are preserved under transformations, such as stretching, twisting, or bending, but not tearing or gluing. For example, a circle can be stretched into an ellipse without tearing or having to re-attach any lines or corners, making the two shapes topologically the same.
In mathematics, topology investigates concepts such as continuity, convergence, compactness, and connectedness. It focuses on understanding the shape, structure, and spatial relationships of objects, surfaces, and spaces, regardless of measurements like distances or angles. Topology is prominent in many fields outside of mathematics, like physics, computer science, and engineering.
History of Topology
The start of topology is associated with Frenchman Henri Poincaré around the year 1900. Even though many topological ideas had been discussed in mathematics in the previous 150 years, this was a major turning point. Pointcaré’s 1895 publication was titled Analysis Situs, Latin for “analysis of position”.
Many mathematicians contributed to the birth of topology, including Johann Listing, Bernhard Riemann, August Möbius, and Felix Klein. After Pointcaré established the basic context for using algebraic concepts in topology, it was further developed by mathematicians Max Dehn and Poul Heegaard in 1907. Many other mathematicians have worked on developing the types of topology and it is still being studied to this day.
Types of Topology
At a little over 100 years old, topology is still a fairly new mathematical concept. However, through decades of development and research, mathematicians have decided on generally four types of topology.
General or Point Set
In general topology, we mainly focus on understanding the characteristics of spaces at a local level, which often connects closely with analysis. This field broadens the idea of continuity by introducing the notion of topological spaces, where we can explore the behavior of sequences as they approach certain points. Sometimes, we can establish distances within these spaces, leading to what we call metric spaces. However, there are instances where distance doesn’t quite fit the picture.
Combinatorial
Combinatorial topology delves into the overall properties of spaces, pieced together from a network of vertices, edges, and faces. This branch of topology holds the honor of being the oldest, tracing its roots back to the work of Euler. It’s fascinating to note that spaces that are topologically equivalent share a common numerical measure, known as the Euler characteristic.
This characteristic, represented by the formula (V – E + F), where V, E, and F represent the number of vertices, edges, and faces respectively, remains consistent across such spaces. For instance, whether it’s a tetrahedron, a cube, or a sphere, they all share the same Euler characteristic. An interesting fact is that any ‘triangulation‘ of a sphere will always yield an Euler characteristic of 2.
Algebraic
Algebraic topology explores the broader characteristics of spaces, using algebraic tools like groups and rings to tackle questions in topology. It’s a fascinating approach that converts topological puzzles into algebraic ones, often making them easier to comprehend for math students.
For instance, we can associate a group, known as a homology group, with each space. This allows us to distinguish between different spaces – like the torus and the Klein bottle – because they possess distinct homology groups. It is exactly this combination of algebra and topology that offers powerful insights into the structure and nature of spaces.
Differential
Differential topology focuses on spaces where each point has an idea of smoothness attached to it. For instance, while a square and a circle may look similar geometrically, in differential topology, they aren’t considered equivalent because they lack smoothness or differentiability between points.
This field becomes particularly useful when investigating properties of vector fields, such as those seen in magnetic or electric fields. By understanding how smoothness interacts within these spaces, we learn valuable information on their behavior and characteristics.
At Online Math Center
Mathematics is most certainly an important subject to study in school, but also maintains importance in developing skills and understanding in life. Even more recently developed mathematical topics like topology are important as they can be found in other fields like physics, computer science, engineering and more. A child’s fascination with topology may lead them to gain an interest in any of these fields and thus to their future career.
Contact us at OMC today to ensure your child gains a great understanding of mathematical concepts like topology, to further understand how to recognize it in the world. Students can strengthen their skills in foundational topics at OMC through our classes and tutoring. With two difficulty levels per grade, OMC works to get every student as advanced as possible; in mathematics and beyond.
