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Topic: Zariski topology

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  PlanetMath: Zariski topology
The Zariski topology is the predominant topology used in the study of algebraic geometry.
Every regular morphism of varieties is continuous in the Zariski topology (but not every continuous map in the Zariski topology is a regular morphism).
This is version 1 of Zariski topology, born on 2002-05-11.
planetmath.org /encyclopedia/ZariskiTopology.html   (150 words)

 This Mathematical Month - April
There were also lectures in the topology of manifolds, singularities, foliations, dynamical systems, algebraic topology, and other subjects.
Zariski received the AMS Cole Prize in Algebra in 1944 and served as president of the AMS from 1969 to 1971.
Read more about Zariski's life and work in the obituary by David Mumford that appeared in the November 1986 issue of the Notices, and the biography The Unreal Life of Oscar Zariski, by Carol Parikh (Academic Press, 1991).
www.ams.org /ams/thismathmonth-apr.html   (713 words)

 Station Information - Oscar Zariski
Oscar Zariski was one of the most influential mathematicians working in the filed of algebraic geometry in the twentieth century.
The Zariski topology, as it was later known, is adequate for biregular geometry, where varieties are mapped by polynomial functions.
Some of his major results, Zariski's main theorem and the Zariski theorem on holomorphic functions, were amongst the results generalized and included in the programme of Alexander Grothendieck that ultimately unified algebraic geometry.
www.stationinformation.com /encyclopedia/o/os/oscar_zariski.html   (544 words)

 Topology - Wikipedia
Topology, in mathematics, is both a structure used to capture the notions of continuity, connectedness and convergence, and the name of the branch of mathematics which studies these.
Some of these terms have been collected together in the Topology Glossary, and the rest of this article assumes that the reader is familiar with them.
In pointless topology one considers instead the lattice of open sets as the basic notion of the theory, while Grothendieck topologies are certain structures defined on arbitrary categories which allow the definition of sheaves on those categories, and with that the definition of quite general cohomology theories.
nostalgia.wikipedia.org /wiki/Topology   (941 words)

 wiki/Oscar Zariski Definition / wiki/Oscar Zariski Research   (Site not responding. Last check: 2007-10-08)
Zariski became professor at Harvard University Harvard University is a private university in Cambridge, Massachusetts, USA and a member of the Ivy League.
Some of his major results, Zariski's main theorem and the Zariski theorem on holomorphic functions, were amongst the results generalized and included in the programme of Alexander GrothendieckAlexander Grothendieck (born March 28, 1928, Berlin) is one of the greatest mathematicians of the 20th century, with major contributions to algebraic geometry, homological algebra, and functional analysis.
Zariski tangent spaceIn algebraic geometry, the Zariski tangent space is a construction that defines a tangent space, at a point P on an algebraic variety V (and more generally).
www.elresearch.com /wiki/Oscar_Zariski   (2680 words)

 Zariski topology
In this topology, named after Oscar Zariski, the closed sets are the sets consisting of the mutual zeros of a finite set of polynomial equations.
This definitions indicates the kind of space that can be given a Zariski topology: for example we define the Zariski topology on a n-dimensional vector space F^n over a field F, using the definition above.
It follows easily that homomorphisms are continuous and so the Zariski topology given to some finite-dimensional vector space doesn't depend on a specific basis chosen.
www.ebroadcast.com.au /lookup/encyclopedia/za/Zariski_topology.html   (135 words)

 Zariski topology -- Facts, Info, and Encyclopedia article   (Site not responding. Last check: 2007-10-08)
In (A science (or group of related sciences) dealing with the logic of quantity and shape and arrangement) mathematics, the Zariski topology is a structure basic to (Click link for more info and facts about algebraic geometry) algebraic geometry, especially since 1950.
The Zariski (The configuration of a communication network) topology is defined by defining the (Click link for more info and facts about closed set) closed sets to be the sets consisting of the mutual zeroes of a set of (A mathematical expression that is the sum of a number of terms) polynomials.
The general case of the Zariski topology is based on the (Click link for more info and facts about affine scheme) affine scheme and (Click link for more info and facts about spectrum of a ring) spectrum of a ring constructions, as local models.
www.absoluteastronomy.com /encyclopedia/Z/Za/Zariski_topology.htm   (455 words)

 Zariski Topology
The Zariski topology is a different kind of topology.
Show that the Zariski topology on the projective line
This confirms once again that the Zariski topology is much coarser than the analytic topology.)
mathcircle.berkeley.edu /BMC3/alg-geom/node3.html   (244 words)

Roughly speaking, a topology is a way of specifying the concept of "nearness"; an open set is "near" each of its points.
The Zariski topology is a purely algebraically defined topology on the spectrum of a ring or an algebraic variety.
Many sets of operators in functional analysis are endowed with topologies that are defined by specifying when a particular sequence of functions converges to the zero function.
www.websters-online-dictionary.org /definition/SUBSPACE+TOPOLOGY   (1143 words)

 Cofinite . Boolean algebra . Zariski topology   (Site not responding. Last check: 2007-10-08)
The cofinite topology on any set X consists of the empty set and all cofinite subsets of X.
In the cofinite topology, the only closed subsets are finite sets, or the whole of X. Then X is automatically compact set compact in this topology, since every open set only omits finitely many points of X. Also, the cofinite topology is the smallest topology satisfying the T1 space T1 axiom; i.e.
One place where this concept occurs naturally is in the context of the Zariski topology.
www.uk.kunsimuna.net /Cofinite   (520 words)

 Topology, Math 5520, Winter 2001
Differential topology studies manifolds, spaces on which it is possible to say not only that something is continuous, but that it is smooth (i.e., differentiable).
It is a very odd topology at first encounter, in comparison with the topologies we are familiar with in the line, the plane, etc. It has proven to be quite a powerful tool, however, playing an important role in the proof of Fermat's last theorem, for example.
For example, the proof using algebraic topology of the Brouwer Fixed Point Theorem, that any continuous function from an n-dimensional ball to itself must have a fixed point, reduces to the simple algebraic fact that the identity function from the integers to the integers is not a constant function.
www.math.wayne.edu /~rrb/classes/m5520w01/Day1.html   (1643 words)

 Topology   (Site not responding. Last check: 2007-10-08)
Topology is the mathematical study of properties of objects which are preserved through deformations, twistings, and stretchings.
Space of all positions of the hour, minute and second hands form a 4-D object that cannot be visualized quite as simply as the former objects since it cannot be placed in our 3-D world, although it can be visualized by other means.
Topology began with the study of curves, surfaces, and other objects in the plane and 3-space.
www.math.sdu.edu.cn /mathency/math/t/t183.htm   (380 words)

 Topological space   (Site not responding. Last check: 2007-10-08)
A purely algebraically defined topology on the spectrum of a ring or an algebraic variety.
Any set with the trivial topology (i.e., only the empty set and the whole space are open, which has the effect of "lumping all points together").
Metric spaces were defined and investigated by Fréchet in 1906, Hausdorff spaces by Felix Hausdorff in 1914 and the current concept of topological space was described by Kuratowski in 1922.
www.termsdefined.net /to/topological-space.html   (1071 words)

 Point set topology   (Site not responding. Last check: 2007-10-08)
Topology may be roughly divided into point-set topology, which considers figures as sets of points having such properties as being open or closed, compact
An introduction to topology, this course concentrates on the two strands of point-set topology and geometric topology.
The basics of topology (often referred to as point set topology) have the merit that they can be usefully applied in a wide variety of contexts.
www.finditeasily.com /q/point-set-topology.html   (989 words)

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