Welcome to Jürgen Braungardt's Website.

Topology

Definition

Topology is a mathematical sub-discipline that studies the properties of objects and spaces. Topology is the modern version of geometry; it was used first in 1847 by the German mathematician Johann Benedict Listing. The shapes of objects can change through twisting or stretching them, and topologists ask: what are the object’s properties that remain intact? In these deformations, tearing is not allowed. From a topological point of view, therefore, a circle is equivalent to an ellipse (into which it can be deformed by stretching) and a donut (also called a “torus”, a two-dimensional a surface embedded in three-dimensional space) is equivalent to a coffee cup.

Topology began with the study of curves, surfaces, and other objects in two- and three-dimensional space. One of the central ideas in topology is that spatial objects like circles and spheres can be treated as objects in their own right, and knowledge of objects is independent of how they are embedded in space.

Topology focuses on the the inherent connectivity of objects while ignoring their detailed form. Because of this abstraction from the detailed form, it is possible to define the “objects” of topology as “topological spaces”. If two objects have the same topological properties, they are said to be “homeomorphic.”

Moebius Strip

A good example for a topological object is the Moebius strip. It is a one-sided surface that can be constructed by connecting the ends of a rectangular strip after giving one of the ends a one-half twist. This space has interesting properties, such as having only one side and remaining in one piece when split down the middle. The properties of the strip were discovered independently and almost simultaneously by two German mathematicians, August Ferdinand Moebius and Johann Benedict Listing, in 1858. The picture above is an example for it.