Topology is a branch of mathematics that deals with selected properties of collections of related physical or abstract elements, specifically, those properties that endure when the collection undergoes distortion (as long as it remains intact). A lump of clay, for example, may be regarded as a collection of physical points that can be deformed (say, into a ball or a long, thin rod) without changing topologically.
The influence of topology extends to almost every other branch of mathematics and to disciplines that formerly were not considered susceptible to mathematical treatment. For example, aspects of topology are closely related to symbolic logic; topology bears upon the design of mechanical devices, geographic maps, distribution networks, and systems for planning and controlling complex activities.
An offshoot of geometry, topology originated during the 19th century. A simple (but not entirely accurate) definition has often been used: topology is the study of those properties that an object retains under deformation--specifically, bending, stretching and squeezing, but not breaking or tearing. Thus, a triangle is topologically equivalent to a circle but not to a straight line segment. Similarly, a solid cube made of modeling clay could be deformed into a ball by kneading. It could not, however, be molded into a solid torus (ring) unless a hole were bored through it or two surfaces were joined together. A solid cube is therefore not topologically equivalent to a finger ring.
More precisely, if there are given two geometric objects or sets of points and if some two-way transformation (or operation or mapping) takes each point p of either set into one and only one point p' of the other and if the transformation is continuous in the sense (which can be made precise) that points close to p become points close to p', then the transformation is called a homeomorphism and the two sets are said to be topologically equivalent. Topology is, then, the study of properties that remain invariant under homeomorphisms.
Excerpt from the Encyclopedia Britannica without permission.