Topology

A branch of mathematics that deals with selected (topological) properties of collections of related physical or abstract elements and specifically those that endure without rupture when the collection undergoes distortion. A topological property is therefore any property of a structure which is invariant or unchanged by such deformation. A topological space can be thought of as a set from which has been eliminated all structure irrelevant to the continuity of functions defined on it.

1. The broad field of topology includes domains such as: the homology and cohomology theory of complexes, and of more general spaces; dimension theory; the theory of differentiable and Riemannian manifolds and of Lie groups; the theory of continuous curves; the theory of Banach and Hilbert spaces and their operators, and of Banach algebras; and abstract harmonic analysis on locally compact groups.

2. Applications of topology have been found in a wide range of disciplines, particularly those concerned with networks or relationships of some kind. Since topology is not limited to quantitative problems, it may contribute to such fields as the social sciences, previously not considered susceptible to mathematical treatment.