Topology

Topology enters engineering almost unnoticed. When we reason about the allowable motions of a mechanism, the connectivity of a structural domain, the existence of a path between two robot configurations, or the global behavior of a nonlinear controller, we are already thinking topologically—even if we do not yet call it that. As modern mechanical engineering confronts increasingly complex geometries, high-dimensional models, and nonlinear behaviors, it becomes essential to understand the qualitative structure of the spaces in which our systems live. This chapter develops the topological ideas that underlie these engineering problems, introducing the concepts, structures, and tools that will serve as a foundation for later applications in mechanics, robotics, control, and numerical analysis.