Topology

Lore & Background

The ideas underlying topology go back to Gottfried Wilhelm Leibniz in the 17th century, who envisioned geometria situs and analysis situs. Leonhard Euler's Seven Bridges of Königsberg problem and polyhedron formula are arguably the field's first theorems. The term topology was introduced by Johann Benedict Listing in the 19th century, though the idea of a topological space was not developed until the first decades of the 20th century.

Reader's Guide

Topology's significance lies in its ability to solve geometric problems that depend not on exact shape but on how objects are put together. For example, Euler demonstrated that crossing each of the seven bridges of Königsberg exactly once was impossible, a result depending only on connectivity properties. Similarly, the hairy ball theorem states that one cannot comb the hair flat on a hairy ball without creating a cowlick, applying to any smooth blob without holes. These insights led to the formalization of homeomorphism and homotopy equivalence. The development of topology in the 20th century included Maurice Fréchet's introduction of metric spaces in 1906, Felix Hausdorff's coining of 'topological space' in 1914, and Kazimierz Kuratowski's generalization in 1922. Modern topology depends strongly on set theory developed by Georg Cantor. The 2022 Abel Prize was awarded to Dennis Sullivan for his contributions to topology in its broadest sense.

Did You Know?

• The term 'topology' comes from the Greek words τόπος ('place, location') and λόγος ('study').
• Leonhard Euler's 1736 paper on the Seven Bridges of Königsberg is regarded as one of the first practical applications of topology.
• The surface of a coffee mug and the surface of a doughnut (a torus) are homeomorphic; a pliable torus can be reshaped into a mug by creating a dimple and enlarging it while shrinking the central hole into the handle.
• The 2022 Abel Prize was awarded to Dennis Sullivan for his groundbreaking contributions to topology.