Topology

Topology (from the Greek τόπος, “place”, and λόγος, “study”) is a major area of mathematics concerned with the most basic properties of space, such as connectedness. More precisely, topology studies properties that are preserved under continuous deformations, including stretching and bending, but not tearing or gluing. The exact mathematical definition is given below. Topology developed as a field of study out of geometry and set theory, through analysis of such concepts as space, dimension, and transformation.

Ideas that are now classified as topological were expressed as early as 1736. Toward the end of the 19th century, a distinct discipline developed, which was referred to in Latin as the geometria situs (“geometry of place”) or analysis situs (Greek-Latin for “picking apart of place”). This later acquired the modern name of topology. By the middle of the 20th century, topology had become an important area of study within mathematics.

Topology has many subfields. Point-set topology establishes the foundational aspects of topology and investigates concepts inherent to topological spaces (basic examples include compactness and connectedness). Algebraic topology tries to measure degrees of connectivity using algebraic constructs such as homology and homotopy groups. Geometric topology primarily studies manifolds and their embeddings (placements) in other manifolds. A particularly active area in geometric topology is low dimensional topology, which studies manifold of 4 or fewer dimensions. This includes knot theory, the study of mathematical knots.

See also: topology glossary for definitions of some of the terms used in topology and topological space for a more technical treatment of the subject.