topology

topology
Name	Topology
Caption	The Möbius strip, a classic non-orientable surface.
Fields	Mathematics
Foundations	Georg Cantor, Felix Hausdorff, Maurice Fréchet
Key people	Henri Poincaré, L. E. J. Brouwer, Kazimierz Kuratowski, John Milnor

topology. Topology is a major area of mathematics concerned with the properties of geometric objects that are preserved under continuous deformations, such as stretching and bending, but not tearing or gluing. It emerged from the study of geometry and set theory, with foundational work by figures like Georg Cantor and Felix Hausdorff. The field provides a unifying language for discussing concepts like continuity, compactness, and connectedness across diverse mathematical contexts.

Overview and basic concepts

The discipline originated from problems in analysis, notably in the work of Augustin-Louis Cauchy and Karl Weierstrass on continuity. The formal definition of a topological space, based on open sets, was established by Felix Hausdorff and Kazimierz Kuratowski. Central notions include the idea of a homeomorphism, which is a bijective continuous map with a continuous inverse, defining when two spaces are considered topologically equivalent. Other fundamental concepts are bases, neighbourhoods, and the closure of a set, which generalize intuitive ideas of proximity without relying on a notion of distance.

Types of topological spaces

There is a vast hierarchy of topological spaces, each defined by specific axioms. Important classes include metric spaces, where topology is derived from a distance function, as studied by Maurice Fréchet. Manifolds, which locally resemble Euclidean space, are central to geometry and physics, with seminal contributions from Bernhard Riemann. Other key types are compact spaces, connected spaces, and Hausdorff spaces, the latter ensuring distinct points have disjoint neighbourhoods. More specialized structures include uniform spaces and topological vector spaces, which blend topological and algebraic properties.

Key topological properties

Topological properties, or invariants, are those preserved by homeomorphisms. Connectedness describes a space that cannot be split into two disjoint open sets, with variations like path connectedness studied by Camille Jordan. Compactness, a generalization of finiteness crucial in analysis, was rigorously defined by Heinrich Eduard Heine and Émile Borel. Separability and the countability axioms involve the size of bases, while metrizability concerns the existence of a compatible metric. Homotopy groups and homology groups, pioneered by Henri Poincaré, are algebraic invariants that classify spaces up to deformation.

Constructions and operations

New spaces are built from old ones using standard constructions. The subspace topology induces a topology on a subset. The product topology, defined on Cartesian products, is fundamental for studying spaces like the Hilbert cube. The quotient topology is formed by gluing points together, creating objects like the real projective plane. Other operations include forming disjoint unions, wedge sums, and adjunction spaces. The Stone–Čech compactification, developed by Marshall Stone and Eduard Čech, is a profound method for embedding a space into a compact Hausdorff space.

Relations to other mathematical disciplines

Topology deeply interconnects with nearly all areas of mathematics. In analysis, it underpins concepts of convergence and continuity in functional analysis, as seen in the work of Stefan Banach. Algebraic topology, founded by Henri Poincaré, uses tools from abstract algebra to solve topological problems. Differential topology, advanced by John Milnor and Stephen Smale, studies smooth manifolds and their mappings. Geometric group theory, influenced by Mikhail Gromov, views groups as metric spaces. Topology also provides essential frameworks for theoretical physics, including string theory and condensed matter physics.

Important theorems and results

The field is marked by profound theorems. The Brouwer fixed-point theorem, proved by L. E. J. Brouwer, states every continuous map from a disk to itself has a fixed point. The Jordan curve theorem, first correctly proved by Oswald Veblen, describes how a simple closed curve divides the plane. The classification of surfaces, completed by Max Dehn and Poul Heegaard, characterizes compact two-dimensional manifolds. The Poincaré conjecture, proven by Grigori Perelman, was a central problem in the Clay Mathematics Institute's Millennium Problems. Other landmark results include the Urysohn lemma, the Tychonoff theorem, and the Alexander duality theorem.

Category:Topology Category:Fields of mathematics