Topology

Topology is a branch of mathematics that studies properties of spaces that are preserved under continuous deformations, such as stretching or bending, but not tearing or gluing. The central idea is to formalize a notion of nearness or continuity without relying on distances. A topology on a set X is a collection T of subsets of X, called open sets, that satisfies three axioms: the empty set and X belong to T; arbitrary unions of members of T belong to T; and finite intersections of members of T belong to T. A topological space is the pair (X, T). A function f from (X, T_X) to (Y, T_Y) is continuous if the preimage of every open set in Y is open in X.

Key concepts include closed sets (complements of opens), interiors, closures, and boundaries of sets; notions of convergence, neighborhood systems, and continuous maps. Important properties such as compactness, connectedness, and separation axioms classify spaces in broad ways. Two spaces are considered the same from a topological viewpoint if they are homeomorphic, meaning there exists a continuous bijection with a continuous inverse.

Examples include the standard Euclidean topology on real spaces, the discrete topology in which all subsets are open, and the trivial/indiscrete topology where only the empty set and the whole space are open. Subspaces inherit topologies from ambient spaces; product topologies arise from product constructions; quotient topologies arise by gluing points together.

Topology has several subfields. Point-set topology studies foundational notions; algebraic topology assigns algebraic invariants, such as homotopy and homology, to classify spaces; differential topology examines smooth structures on manifolds. Applications span analysis, geometry, physics, computer science, and data analysis, where concepts like continuity, shape, and connectivity play central roles.