Articles that I introduced
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edit- chore division
- Lagrange multipliers
- limit of a function
- mathematical rigor
- Penrose graphical notation
- pursuit-evasion (needs more material on the continuous version)
- spin networks
- Sperner's lemma
Mathematics Organizations
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Topology (Greek topos, "place," and logos, "study") is a branch of mathematics that is an extension of geometry. Topology builds upon set theory in order to investigate the relationship of points in space. In particular, topology captures the notion of proximity without the need for a notion of distance. It can be used to investigate both the fine details of a space (its "local structure") and how a space is put together (its "global structure").
The word topology is used both for the area of study and for a family of sets with certain properties (described below) that are used to define a topological space. The sets in this family are called open sets. Of particular importance in the study of topology are functions or maps, called homeomorphisms, which preserve the "open" property. Informally, these functions can be thought of as those that stretch space without tearing it apart or sticking distinct parts together. Open sets are important because, in an intuitive sense, points are "close together" if they are "usually in the same open set". From this point of view, homeomorphisms are functions which preserve the proximity of points.
When the discipline was first properly founded, toward the end of the 19th century, it was called geometria situs (Latin geometry of place) and analysis situs (Latin analysis of place). From around 1925 to 1975 it was an important growth area within mathematics.
Topology is a large branch of mathematics that includes many subfields. The most basic division within topology is point-set topology, which investigates such concepts as compactness, connectedness, and countability; algebraic topology, which investigates such concepts as homotopy and homology; and geometric topology, which studies manifolds and their embeddings, including knot theory.
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.