Topology

Topology is a major area of mathematics that studies properties of space that are preserved under continuous transformations, such as stretching and bending, but not tearing or gluing. It is often referred to as "rubber sheet geometry" due to this focus on properties unchanged by deformation.

Definition

Topology is defined as the mathematical study of the properties that are preserved through deformations, twistings, and stretchings of objects. Tearing, however, is not allowed. A circle is topologically equivalent to an ellipse (into which it can be deformed by stretching) and a sphere is equivalent to an ellipsoid. Similarly, the set of all possible positions of the hour hand on a clock is topologically equivalent to a circle (a one-dimensional closed curve with no intersections that can be embedded in two-dimensional space), the set of all possible positions of the hour and minute hands together is topologically equivalent to the surface of a torus (a two-dimensional a surface that can be embedded in three-dimensional space), and the set of all possible positions of three hands on a clock is topologically equivalent to a three-dimensional object in four-dimensional space.

History

The term topology was introduced by the German mathematician Johann Benedict Listing in the 19th century. However, the concepts underlying topology go back much further. The Seven Bridges of Königsberg problem, solved by Leonhard Euler in 1736, is considered by some to be the first true theorem of topology.

Branches of Topology

Topology can be divided into several subfields.

• General Topology: Also known as point-set topology, focuses on the basic set-theoretic definitions and constructions used in topology.
• Algebraic Topology: Uses tools from abstract algebra to study topological spaces.
• Differential Topology: Focuses on differentiable functions on differentiable manifolds.
• Geometric Topology: Primarily studies manifolds and their embeddings (placements within other manifolds).
• Topological Data Analysis: A recent and rapidly growing field that uses topological techniques to study data.

Applications

Topology has several practical applications in computer graphics, structural biology, DNA sequencing, etc. In these fields, the ability to imagine the structure in three dimensions is a valuable tool, and topology often provides this perspective.

See Also

• Homeomorphism
• Continuous function
• Metric space
• Compact space
• Connected space