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Generic Point
In algebraic geometry, a generic point ''P'' of an algebraic variety ''X'' is, roughly speaking, a point at which all generic properties are true, a generic property being a property which is true for almost every point. In classical algebraic geometry, a generic point of an affine or projective algebraic variety of dimension ''d'' is a point such that the field generated by its coordinates has transcendence degree ''d'' over the field generated by the coefficients of the equations of the variety. In scheme theory, the spectrum of an integral domain has a unique generic point, which is the zero ideal. As the closure of this point for the Zariski topology is the whole spectrum, the definition has been extended to general topology, where a generic point of a topological space ''X'' is a point whose closure is ''X''. Definition and motivation A generic point of the topological space ''X'' is a point ''P'' whose closure is all of ''X'', that is, a point that is dense in ''X''. T ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Algebraic Geometry
Algebraic geometry is a branch of mathematics, classically studying zeros of multivariate polynomials. Modern algebraic geometry is based on the use of abstract algebraic techniques, mainly from commutative algebra, for solving geometrical problems about these sets of zeros. The fundamental objects of study in algebraic geometry are algebraic varieties, which are geometric manifestations of solutions of systems of polynomial equations. Examples of the most studied classes of algebraic varieties are: plane algebraic curves, which include lines, circles, parabolas, ellipses, hyperbolas, cubic curves like elliptic curves, and quartic curves like lemniscates and Cassini ovals. A point of the plane belongs to an algebraic curve if its coordinates satisfy a given polynomial equation. Basic questions involve the study of the points of special interest like the singular points, the inflection points and the points at infinity. More advanced questions involve the topology of the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Irreducible Algebraic Set
In algebraic geometry, an irreducible algebraic set or irreducible variety is an algebraic set that cannot be written as the union of two proper algebraic subsets. An irreducible component is an algebraic subset that is irreducible and maximal (for set inclusion) for this property. For example, the set of solutions of the equation is not irreducible, and its irreducible components are the two lines of equations and . It is a fundamental theorem of classical algebraic geometry that every algebraic set may be written in a unique way as a finite union of irreducible components. These concepts can be reformulated in purely topological terms, using the Zariski topology, for which the closed sets are the algebraic subsets: A topological space is '' irreducible'' if it is not the union of two proper closed subsets, and an ''irreducible component'' is a maximal subspace (necessarily closed) that is irreducible for the induced topology. Although these concepts may be considered for ever ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Kolmogorov Space
In topology and related branches of mathematics, a topological space ''X'' is a T0 space or Kolmogorov space (named after Andrey Kolmogorov) if for every pair of distinct points of ''X'', at least one of them has a neighborhood not containing the other. In a T0 space, all points are topologically distinguishable. This condition, called the T0 condition, is the weakest of the separation axioms. Nearly all topological spaces normally studied in mathematics are T0 spaces. In particular, all T1 spaces, i.e., all spaces in which for every pair of distinct points, each has a neighborhood not containing the other, are T0 spaces. This includes all T2 (or Hausdorff) spaces, i.e., all topological spaces in which distinct points have disjoint neighbourhoods. In another direction, every sober space (which may not be T1) is T0; this includes the underlying topological space of any scheme. Given any topological space one can construct a T0 space by identifying topologically indistinguishable ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
World War II
World War II or the Second World War, often abbreviated as WWII or WW2, was a world war that lasted from 1939 to 1945. It involved the vast majority of the world's countries—including all of the great powers—forming two opposing military alliances: the Allies and the Axis powers. World War II was a total war that directly involved more than 100 million personnel from more than 30 countries. The major participants in the war threw their entire economic, industrial, and scientific capabilities behind the war effort, blurring the distinction between civilian and military resources. Aircraft played a major role in the conflict, enabling the strategic bombing of population centres and deploying the only two nuclear weapons ever used in war. World War II was by far the deadliest conflict in human history; it resulted in 70 to 85 million fatalities, mostly among civilians. Tens of millions died due to genocides (including the Holocaust), starvation, ma ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
São Paulo
São Paulo (, ; Portuguese for 'Saint Paul') is the most populous city in Brazil, and is the capital of the state of São Paulo, the most populous and wealthiest Brazilian state, located in the country's Southeast Region. Listed by the GaWC as an alpha global city, São Paulo is the most populous city proper in the Americas, the Western Hemisphere and the Southern Hemisphere, as well as the world's 4th largest city proper by population. Additionally, São Paulo is the largest Portuguese-speaking city in the world. It exerts strong international influences in commerce, finance, arts and entertainment. The city's name honors the Apostle, Saint Paul of Tarsus. The city's metropolitan area, the Greater São Paulo, ranks as the most populous in Brazil and the 12th most populous on Earth. The process of conurbation between the metropolitan areas around the Greater São Paulo (Campinas, Santos, Jundiaí, Sorocaba and São José dos Campos) created the São Paulo Macrometr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
