topologiaan

Topology is a branch of mathematics that studies properties of space that are preserved under continuous deformations, such as stretching or bending, but not tearing or gluing. The central object is a topological space, defined as a set together with a topology: a collection of open subsets satisfying union and finite intersection axioms. A function between topological spaces is continuous if the preimage of every open set is open; two spaces are considered the same from a topological viewpoint if they are homeomorphic, meaning there is a bicontinuous bijection between them.

Topologists study various constructions and notions that generalize metric ideas without relying on distance. These include bases and subbases for generating topologies, the product topology on Cartesian products, and quotient topologies arising from equivalence relations. Important properties include compactness, which generalizes finiteness in many senses; connectedness and path-connectedness; and separability. Specializations include metric topology, where a distance function induces a topology, and more expansive frameworks found in algebraic topology, differential topology, and geometric topology.

Historically, modern topology formed in the early 20th century through the work of Cantor, Fréchet, Hausdorff, Brouwer, and others, with later synthesis via set theory and category theory. It provides tools and concepts used across mathematics and the sciences, including analysis, geometry, and data analysis, where methods such as persistent homology in topological data analysis help uncover shape-related features in data.