In mathematics, topology is the study of the properties of spaces that are preserved under continuous deformations, such as stretching, twisting, and bending, but not tearing or gluing. Topology began as the study of geometric properties that were invariant under certain transformations. These transformations were later generalized to include continuous changes in shape or size, called homeomorphisms. Today, topology is a broad field with applications in nearly every branch of mathematics. It is also widely used in physics and engineering.
Topology is often divided into two main areas: point-set topology and algebraic topology. Point-set topology deals with the basic properties of space, such as connectedness and compactness. Algebraic topology uses algebraic methods to study more sophisticated properties of space, such as homotopy groups and cohomology groups.
Some of the most famous results in topology include the Brouwer fixed point theorem, the hairy ball theorem, the Jordan curve theorem, and the existence of knots and links. Topologists also study objects called manifolds, which are spaces that locally resemble Euclidean space but may have very different global structure. The most familiar examples of manifolds are surfaces like spheres, tori (surfaces of revolution), and Mobius strips.