topologie

Topology is a branch of mathematics that studies properties of spaces that are preserved under continuous deformations, such as stretching or bending, without tearing or gluing. The central object is a topological space, defined as a set X together with a topology T, a collection of subsets of X called open sets, satisfying: the empty set and X are open; arbitrary unions of open sets are open; finite intersections of open sets are open. The open sets determine notions of continuity, convergence and proximity; a function f: X → Y between topological spaces is continuous if the preimage of every open set in Y is open in X.

Beyond open sets, one often uses closed sets, neighborhoods, and convergence of nets or sequences. Typical constructions include subspace topologies (on a subset with the topology induced by X), product topologies (on product spaces) and quotient topologies (induced by a surjective map). Every metric space has an associated topology defined by open balls; conversely, many topologies arise from metrics.

Key properties studied include compactness (every open cover has a finite subcover), connectedness (cannot be divided into two nonempty disjoint open sets), and various separation axioms (T0, T1, Hausdorff). These properties generalize familiar notions from real analysis and geometry. In algebraic topology one studies invariants such as the fundamental group and homology groups, derived from topological spaces to capture their global structure.

Prominent results include the Tychonoff theorem (the product of compact spaces is compact) and the Urysohn lemma in normal spaces; many other results underpin analysis, geometry, and physics. Applications range from analysis and geometry to dynamical systems and data analysis, where topological methods detect shapes and features in data.