A mug and a doughnut count as the same to topology
Topology tracks properties that survive continuous stretching, twisting, and bending—so long as you neither cut nor glue. Under that lens a coffee mug matches a doughnut: both are one-holed shapes you can morph into each other. Exact distances drop out; how parts connect stays.
Homeomorphisms and homotopies are the allowed deformations; invariants under them are topological properties. Dimension separates a line from a surface; compactness separates a line from a circle; connectedness separates one circle from two disjoint ones. Leonhard Euler’s Seven Bridges of Königsberg argument showed you cannot walk every bridge of that town exactly once—connectivity, not lengths, decides. The hairy ball theorem says no continuous nowhere-zero tangent field exists on a sphere: you cannot comb a hairy ball flat without a cowlick, and any hole-free smooth blob shares the fate.
Leibniz dreamed of geometria situs; Euler’s 1736 Königsberg paper and his polyhedron formula V − E + F = 2 (announced to a friend on 14 November 1750) are early theorems. Johann Benedict Listing printed “Topologie” in 1847; Nature’s 1883 English “topology” contrasted qualitative with quantitative geometry. Henri Poincaré’s 1895 Analysis Situs launched homotopy and homology. Maurice Fréchet introduced metric spaces in 1906; Felix Hausdorff coined “topological space” in 1914; Kazimierz Kuratowski’s 1922 generalisation matches today’s definition. Dennis Sullivan won the 2022 Abel Prize for broad contributions across algebraic, geometric, and dynamical topology.
Formally, a topology τ on a set X contains the empty set and X, is closed under arbitrary unions and finite intersections, and the pair (X, τ) is a topological space. Members of τ are open; complements of opens are closed. A map is continuous when preimages of opens are open; a continuous bijection with continuous inverse is a homeomorphism. Cubes match spheres topologically; spheres do not match doughnuts.
Manifolds look locally like Euclidean space: each point of an n-manifold has a neighbourhood homeomorphic to n-dimensional Euclidean space. Lines and circles are one-dimensional manifolds; figure eights are not. Surfaces include the plane, sphere, and torus (embeddable in 3-space) and the Klein bottle and real projective plane (which are not). Algebraic topology seeks algebraic invariants—homotopy groups, homology, cohomology—that classify spaces up to homeomorphism or homotopy equivalence.
Source: Topology
