Topologin

Topologin, known in English as 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. It focuses on the notion of nearness and continuity without relying on distances. A topological space consists of a set X together with a collection T of subsets of X, called open sets, which satisfy that the empty set and X are open, and arbitrary unions and finite intersections of open sets are open. The choice of T defines the topology and determines how convergence, continuity, and boundary concepts behave.

Continuity is defined by a preimage condition: a function between topological spaces is continuous if the preimage of every open set is open. This abstract framework unifies many familiar ideas from analysis and geometry. Typical examples include the real numbers with the standard topology, the discrete topology (where every subset is open), and the trivial topology (where only the empty set and X are open). Important constructions include subspace topology, product topology, and quotient topology, which allow building new spaces from existing ones.

Key concepts include open and closed sets, convergence (via nets or sequences), compactness (every open cover has a finite subcover), and connectedness (no separation into disjoint nonempty open sets). Topology classifies spaces up to homeomorphism, a structure-preserving notion of equivalence.

Topologin has several subfields: general or point-set topology studies foundational notions; algebraic topology uses algebraic invariants such as homotopy and homology; differential topology analyzes smooth manifolds; geometric topology concerns spatial structures; topological data analysis applies topological methods to data. Applications span mathematics, physics, computer science, and beyond.