Topology studies properties that survive stretching, bending, and twisting—but not tearing or gluing. A coffee mug and a donut are topologically the same because both have one hole. A sphere and a cube are equivalent because one can be deformed into the other without cutting.

The field began in the nineteenth century with Euler's work on the Seven Bridges of Königsberg. It grew through Poincaré, Hausdorff, and others. Concepts include open sets, continuity, compactness, and connectedness.

Topology has applications in physics, biology, computer science, and data analysis. It underlies the study of knots, surfaces, and higher-dimensional spaces. The subject is abstract but surprisingly practical, from network design to the shape of the universe.

• Studies properties preserved under deformation
• Coffee mug and donut are topologically equivalent
• Began with Euler's Seven Bridges problem
• Used in physics, biology, and data analysis