What is "topology"

Topology \To*pol"o*gy\, n. [Gr. ? place + -logy.] The art of, or method for, assisting the memory by associating the thing or subject to be remembered with some place. [R.]

2. a branch of mathematics which studies the properties of geometrical forms which retain their identity under certain transformations, such as stretching or twisting, which are homeomorphic. See also topologist.

3. configuration, especially in three dimensions; -- used, e. g. of the configurations taken by macromolecules, such as superhelical DNA.

1650s, "study of the locations where plants are found," from topo-, comb. form of Greek topos "place" (see topos) + -logy. Related: Topological.

n. 1 (context mathematics English) A branch of mathematics studying those properties of a geometric figure or solid that are not changed by stretching, bending and similar homeomorphisms. 2 (context mathematics English) A collection ''τ'' of subsets of a set ''X'' such that the empty set and ''X'' are both members of ''τ'' and ''τ'' is closed under arbitrary unions and finite intersections. 3 (context medicine English) The anatomical structure of part of the body. 4 (context computing English) The arrangement of nodes in a communications network. 5 (context technology English) The properties of a particular technological embodiment that are not affected by differences in the physical layout or form of its application. 6 (context topography English) The topographical study of geographic locations or given places in relation to its history. 7 (context dated English) The art of, or method for, assisting the memory by associating the thing or subject to be remembered with some place.

1. n. topographic study of a given place (especially the history of place as indicated by its topography); "Greenland's topology has been shaped by the glaciers of the ice age"

2. the study of anatomy based on regions or divisions of the body and emphasizing the relations between various structures (muscles and nerves and arteries etc.) in that region [syn: regional anatomy, topographic anatomy]

3. the branch of pure mathematics that deals only with the properties of a figure X that hold for every figure into which X can be transformed with a one-to-one correspondence that is continuous in both directions [syn: analysis situs]

4. the configuration of a communication network [syn: network topology]

In mathematics, topology (from the Greek τόπος, place, and λόγος, study) is concerned with the properties of space that are preserved under continuous deformations, such as stretching and bending, but not tearing or gluing. This can be studied by considering a collection of subsets, called open sets, that satisfy certain properties, turning the given set into what is known as a topological space. Important topological properties include connectedness and compactness.

Topology developed as a field of study out of geometry and set theory, through analysis of concepts such as space, dimension, and transformation. Such ideas go back to Gottfried Leibniz, who in the 17th century envisioned the geometria situs (Greek-Latin for "geometry of place") and analysis situs (Greek-Latin for "picking apart of place"). Leonhard Euler's Seven Bridges of Königsberg Problem and Polyhedron Formula are arguably the field's first theorems. The term topology was introduced by Johann Benedict Listing in the 19th century, although it was not until the first decades of the 20th century that the idea of a topological space was developed. By the middle of the 20th century, topology had become a major branch of mathematics.

Topology has many subfields:

• General topology establishes the foundational aspects of topology and investigates properties of topological spaces and concepts inherent to topological spaces. It includes point-set topology, which is the foundational topology used in all other branches (including topics like compactness and connectedness).
• Algebraic topology tries to measure degrees of connectivity using algebraic constructs such as homology and homotopy groups.
• Differential topology is the field dealing with differentiable functions on differentiable manifolds. It is closely related to differential geometry and together they make up the geometric theory of differentiable manifolds.
• Geometric topology primarily studies manifolds and their embeddings (placements) in other manifolds. A particularly active area is low-dimensional topology, which studies manifolds of four or fewer dimensions. This includes knot theory, the study of mathematical knots.

Topology was a peer-reviewed mathematical journal covering topology and geometry. It was established in 1962 and was published by Elsevier. The last issue of Topology appeared in 2009.

Topology is an album by multi-instrumentalist and composer Joe McPhee, recorded in 1981 and first released on the Swiss HatHut label, it was rereleased on CD in 1990.

Topology, the study of surfaces, is a branch of mathematics concerned with spatial properties preserved under bicontinuous deformation (stretching without tearing or gluing); these properties are the topological invariants.

Topology may also refer to:

• Topology, the collection of open sets used to define a topological space
• Topology (journal), a mathematical journal, with an emphasis on subject areas related to topology and geometry
• Spatial effects that cannot be described by topography, i.e., social, economical, spatial, or phenomenological interactions
• The specific orientation of transmembrane proteins.
• Topology (electronics), a configuration of electronic components.
• Network topology, configurations of computer or biological networks.
• Topology (musical ensemble), an Australian post-classical quintet
• Geospatial topology, the study or science of places with applications in earth science, geography, human geography, and geomorphology.
  • In geographic information systems and their data structures, topology and planar enforcement are the storing of a border line between two neighboring areas (and the border point between two connecting lines) only once. Thus, any rounding errors might move the border, but will not lead to gaps or overlaps between the areas.
  • Also in cartography, a topological map is a greatly simplified map that preserves the mathematical topology while sacrificing scale and shape
  • Topology is often confused with the geographic meaning of topography (originally the study of places). The confusion may be a factor in topographies having become confused with terrain or relief, such that they are essentially synonymous.
• In phylogenetics, the branching pattern of a phylogenetic tree.
• TopologiLinux, a Linux distribution

Topology is an indie classical quintet from Australia, formed in 1997. A leading Australian new music ensemble, they perform throughout Australia and abroad and have to date released 12 albums, including one with rock/electronica band Full Fathom Five and one with contemporary ensemble Loops. They were formerly the resident ensemble at the University of Western Sydney. The group works with composers including Tim Brady in Canada, Andrew Poppy, Michael Nyman, and Jeremy Peyton Jones in the UK, and Terry Riley, Steve Reich, Philip Glass, Carl Stone, and Paul Dresher in the US, as well as many Australian composers.

In 2009, Topology won the Outstanding Contribution by an Organisation award at the Australasian Performing Right Association (APRA) Classical Music Awards for their work on the 2008 Brisbane Powerhouse Series.

The topology of an electronic circuit is the form taken by the network of interconnections of the circuit components. Different specific values or ratings of the components are regarded as being the same topology. Topology is not concerned with the physical layout of components in a circuit, nor with their positions on a circuit diagram. It is only concerned with what connections exist between the components. There may be numerous physical layouts and circuit diagrams that all amount to the same topology.

Strictly speaking, replacing a component with one of an entirely different type is still the same topology. In some contexts, however, these can loosely be described as different topologies. For instance, interchanging inductors and capacitors in a low-pass filter results in a high-pass filter. These might be described as high-pass and low-pass topologies even though the network topology is identical. A more correct term for these classes of object (that is, a network where the type of component is specified but not the absolute value) is prototype network.

Electronic network topology is related to mathematical topology, in particular, for networks which contain only two-terminal devices, circuit topology can be viewed as an application of graph theory. In a network analysis of such a circuit from a topological point of view, the network nodes are the vertices of graph theory and the network branches are the edges of graph theory.

Standard graph theory can be extended to deal with active components and multi-terminal devices such as integrated circuits. Graphs can also be used in the analysis of infinite networks.

In chemistry, topology provides a convenient way of describing and predicting the molecular structure within the constraints of three-dimensional (3-D) space. Given the determinants of chemical bonding and the chemical properties of the atoms, topology provides a model for explaining how the atoms ethereal wave functions must fit together. Molecular topology is a part of mathematical chemistry dealing with the algebraic description of chemical compounds so allowing a unique and easy characterization of them.

Topology is insensitive to the details of a scalar field, and can often be determined using simplified calculations. Scalar fields such as electron density, Madelung field, covalent field and the electrostatic potential can be used to model topology.

Each scalar field has its own distinctive topology and each provides different information about the nature of chemical bonding and structure. The analysis of these topologies, when combined with simple electrostatic theory and a few empirical observations, leads to a quantitative model of localized chemical bonding. In the process, the analysis provides insights into the nature of chemical bonding.

Applied topology explains how large molecules reach their final shapes and how biological molecules achieve their activity.

Circuit topology is a topological property of folded linear polymers. This notion has been applied to structural analysis of biomolecules such as proteins and RNAs.