Topology

Topology is a specialized branch of mathematics focused on the properties of geometric objects. These properties remain unchanged during continuous deformations. Such deformations include stretching, twisting, bending, or crumpling an object. However, these transformations must follow strict rules. One cannot tear the object to create new holes. One also cannot glue parts together to close existing holes.

In this field, the exact size or shape of an object matters less than how it is connected. A circle and a square are considered the same because they are both one-dimensional objects that separate a plane into an inside and an outside.

To describe these transformations, mathematicians use specific terms. A homeomorphism is the most basic form of topological equivalence. Two spaces are homeomorphic if one can be deformed into the other without cutting or gluing. A famous illustration of this is the "Topologist's Breakfast." In this example, a coffee mug and a doughnut are considered identical. A pliable torus, which is a doughnut shape, can be reshaped into a mug. One simply creates a dimple that grows into the mug's body while the central hole becomes the handle.

Another concept is homotopy equivalence. This occurs when two objects result from "squishing" a larger object.

Topologists look for properties that are invariant under these deformations. These are called topological properties. One such property is dimension, which distinguishes a line from a surface. Another is compactness, which helps distinguish a line from a circle. Connectedness is also vital, as it allows us to tell a single circle apart from two non-intersecting circles.

Other advanced concepts include orientability and Betti numbers. These help define the complex characteristics of manifolds, which are spaces that look like Euclidean space near every point.

The history of topology involves several centuries of discovery. In the 18th century, Leonhard Euler provided one of the field's first practical applications. He studied the Seven Bridges of Königsberg problem in 1736.

He proved it was impossible to cross all seven bridges exactly once. This result did not depend on bridge length or distance. It only depended on connectivity properties. This discovery helped lead to the development of graph theory. Euler also developed a polyhedron formula involving vertices, edges, and faces.

As the field grew, the formal terminology emerged. Johann Benedict Listing introduced the term "topology" in the 19th century. In 1895, Henri Poincaré published his work on Analysis Situs. This paper introduced the concepts of homotopy and homology, which are parts of algebraic topology. In the early 20th century, the idea of a topological space was fully developed. Maurice Fréchet introduced the metric space in 1906. Later, Felix Hausdorff coined the term "topological space" in 1914.

Kazimierz Kuratowski further generalized these ideas in 1922.

Topology reveals surprising truths about the world. One example is the hairy ball theorem from algebraic topology. This theorem states that one cannot comb the hair flat on a hairy ball without creating a cowlick. Formally, this means there is no nonvanishing continuous tangent vector field on a sphere. This rule is not about the sphere's exact shape. It applies to any smooth blob that lacks holes. Similarly, the Möbius strip is a unique topological object. It is a surface that has only one side and one edge.

Modern topology is a vast discipline with many subfields. General topology, or point-set topology, deals with basic set-theoretic definitions. It uses concepts like open sets, closed sets, and neighborhoods. This serves as the foundation for differential and geometric topology. The field is deeply connected to set theory, which was developed by Georg Cantor. Today, topology remains a vital area of study. Its importance was recognized when Dennis Sullivan won the 2022 Abel Prize for his contributions to its many aspects.