In mathematics, in the fields of general topology and particularly of algebraic geometry, a generic point P of a topological space X is a point such that every point Q of X is a specialization of P, in the sense of the specialization order (or pre-order). This concept only matters for spaces that are not Hausdorff spaces, because a Hausdorff space with a generic point P can only be the singleton set {P}. The terminology arises from the case of the Zariski topology of algebraic varieties. For example having a generic point is a criterion to be an irreducible set. Mathematics is often defined as the study of topics such as quantity, structure, space, and change. ... In mathematics, general topology or point set topology is that branch of topology which studies elementary properties of topological spaces and structures defined on them. ... Algebraic geometry is a branch of mathematics which, as the name suggests, combines abstract algebra, especially commutative algebra, with geometry. ... Topological spaces are structures that allow one to formalize concepts such as convergence, connectedness and continuity. ... In the branch of mathematics known as topology the specialization (or canonical) preorder defines a preorder on the set of the points of a topological space. ... What is a Pre-Order? A Pre-Order is an item paid for before it is released. ... In topology and related branches of mathematics, a Hausdorff space is a topological space in which points can be separated by neighbourhoods. ... In mathematics, a singleton is a set with exactly one element. ... This article needs to be cleaned up to conform to a higher standard of quality. ... This article is about algebraic varieties. ...

In the foundational approach of André Weil, developed in his Foundations of Algebraic Geometry, generic points played an important role, but were handled in a different manner. For an algebraic variety V over a field K, generic points of V were a whole class of points of V taking values in a universal domain Ω, an algebraically closed field containing K but also an infinite supply of fresh indeterminates. This approach worked, without any need to deal directly with the topology of V (K-Zariski topology, that is), because the specialization could all be discussed at the field level (as in the valuation theory approach to algebraic geometry, popular in the 1930s). André Weil (May 6, 1906 - August 6, 1998) was one of the great mathematicians of the 20th century. ... In abstract algebra, a field is an algebraic structure in which the operations of addition, subtraction, multiplication and division (except division by zero) may be performed, and the same rules hold which are familiar from the arithmetic of ordinary numbers. ... In mathematics, a field F is said to be algebraically closed if every polynomial of degree at least 1, with coefficients in F, has a zero (root) in F (i. ... Model Theory In logic and model theory, a valuation is a map from the set of variables of a first-order language to the universe of some interpretation of that language. ...

This was at a cost of there being a huge collection of equally-generic points. Oscar Zariski, a colleague of Weil's at São Paulo just after World War II, always insisted that generic points should be unique. (This can be put back into topologists' terms: Weil's idea fails to give a Kolmogorov space and Zariski thinks in terms of the Kolmogorov quotient.) Oscar Zariski was one of the most influential mathematicians working in the field of algebraic geometry in the twentieth century. ... The title of this article contains the character ã. Where it is unavailable or not desired, the name may be represented as Sao Paulo. ... World War II was a truly global conflict with many facets: immense human suffering, fierce indoctrination, and the use of new, extremely devastating weapons such as the atomic bomb. ... In topology and related branches of mathematics, the T0 spaces or Kolmogorov spaces form a broad class of well behaved topological spaces. ... In topology and related branches of mathematics, the T0 spaces or Kolmogorov spaces form a broad class of well behaved topological spaces. ...

In the rapid foundational changes of the 1950s Weil's approach became obsolescent. In scheme theory, though, from 1957, generic points returned: this time à la Zariski. For example for R a discrete valuation ring, Spec(R) consists of two points, a generic point (coming from the prime ideal {0}) and a closed point or special point coming from the unique maximal ideal, For morphisms to Spec(R), the fiber above the special point is the special fiber, an important concept for example in reduction modulo p, monodromy theory and other theories about degeneration. The generic fiber, equally, is the fiber above the generic point. Geometry of degeneration is largely then about the passage from generic to special fibers, or in other words how specialization of parameters affects matters. (For a discrete valuation ring the topological space in question is the Sierpinski space of topologists. Other local rings have unique generic and special points, but a more complicated spectrum, since they represent general dimensions. The discrete valuation case is much like the complex unit disk, for these purposes.) In mathematics, a scheme is an important concept connecting the fields of algebraic geometry, commutative algebra and number theory. ... In mathematics, a discrete valuation ring (DVR) is a particular kind of commutative ring that is a local ring, which satisfies conditions that in algebraic geometry come from non-singularity of a point on an algebraic curve. ... In mathematics, a prime ideal is a subset of a ring which shares many important properties of a prime number in the ring of integers. ... In ring theory, a branch of abstract algebra, an ideal is a special subset of a ring which generalizes important properties of integers. ... In topology, Sierpiński space S is the simplest example of a topological space that does not satisfy the T1 axiom. ... In mathematics, more particularly in abstract algebra, local rings are certain rings that are comparatively simple, and serve to describe the local behavior of functions defined on varieties or manifolds. ... A disc of unit radius on a plane is called a unit disc. ...