Topology

In familiar geometric examples, topology focuses on features that survive continuous changes of shape. A circle can be stretched into an ellipse without being cut, so the two have the same basic topological form; a deformation that tears a surface or joins previously separate points is not allowed. This rubber-sheet picture is useful, but it is only an introduction: topology studies much more than the shapes of objects drawn in ordinary space.

In general topology, a topological space consists of a set together with a specified collection of subsets called open sets. The collection follows rules that make it possible to define ideas such as continuity and neighborhoods without requiring a notion of distance. Mathematicians can then ask whether spaces are connected, compact, or have features such as loops, and whether a continuous, reversible mapping relates one space to another. When such a mapping and its inverse are continuous, the spaces are topologically equivalent, or homeomorphic. These ideas let mathematicians distinguish structural features from measurements such as exact length or angle.

Topology contains several related branches. Point-set, or general, topology develops the language of spaces and continuity; algebraic topology uses algebraic structures to investigate spaces; and differential topology studies smooth spaces and mappings. The concepts are useful in mathematical analysis and geometry, and topological methods also appear in the study of dynamical systems and in mathematical models of physical spaces. The subject provides a common way to reason about continuity and global structure across problems that may otherwise look very different.

Topology grew from questions about geometric form and connectivity. In 1736, Leonhard Euler published a solution to the Seven Bridges of Königsberg problem, showing that a route crossing each of the city's seven bridges exactly once was impossible. By focusing on how locations were joined by bridges rather than on their distances or precise shapes, Euler's approach anticipated topological and graph-theoretic ways of representing structure. The modern subject developed later into several branches that now connect geometry, analysis, and algebra.