## Topology

From Wikipedia

**Topology** (Greek Τοπολογία, from τόπος, “place”, and λόγος, “study”) is a major area of mathematics that has emerged through the development of concepts from geometry and set theory, such as those of space, dimension, shape, transformation and others.

Ideas that are now classified as topological were expressed as early as 1736, and toward the end of the 19th century a distinct discipline developed, called in Latin the *geometria situs* (“geometry of place”) or *analysis situs* (Greek-Latin for “picking apart of place”), and later gaining the modern name of topology. In the middle of the 20^{th} century, this was an important growth area within mathematics.

The word *topology* is used both for the mathematical discipline and for a family of sets with certain properties that are used to define a topological space, a basic object of topology. Of particular importance are *homeomorphisms*, which can be defined as continuous functions with a continuous inverse. For instance, the function *y* = *x*^{3} is a homeomorphism of the real line.

Topology includes many subfields. The most basic and traditional division within topology is **point-set topology**, which establishes the foundational aspects of topology and investigates concepts as compactness and connectedness; **algebraic topology**, which generally tries to measure degrees of connectivity using algebraic constructs such as homotopy groups and homology; and **geometric topology**, which primarily studies manifolds and their embeddings (placements) in other manifolds. Some of the most active areas, such as low dimensional topology and graph theory, do not fit neatly in this division.

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.

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Incongruous - 21/11/2009 at 12:50 am |