Hausdorff space. Hausdorff spaces are named after Felix Hausdorff, one of the founders of topology. Hausdorff's original definition of a topological space (in 1914) included the Hausdorff condition as an axiom. Definitions[edit] The points x and y, separated by their respective neighbourhoods U and V. A related, but weaker, notion is that of a preregular space. X is a preregular space if any two topologically distinguishable points can be separated by neighbourhoods.

The relationship between these two conditions is as follows. Equivalences[edit] For a topological space X, the following are equivalent: Examples and counterexamples[edit] Almost all spaces encountered in analysis are Hausdorff; most importantly, the real numbers (under the standard metric topology on real numbers) are a Hausdorff space.

A simple example of a topology that is T1 but is not Hausdorff is the cofinite topology defined on an infinite set. Properties[edit] Let f : X → Y be a continuous function and suppose Y is Hausdorff. Notes[edit] Maximal ideal. In noncommutative ring theory, a maximal right ideal is defined analogously as being a maximal element in the poset of proper right ideals, and similarly, a maximal left ideal is defined to be a maximal element of the poset of proper left ideals. Since a one sided maximal ideal A is not necessarily two-sided, the quotient R/A is not necessarily a ring, but it is a simple module over R. If R has a unique maximal right ideal, then R is known as a local ring, and the maximal right ideal is also the unique maximal left and unique maximal two-sided ideal of the ring, and is in fact the Jacobson radical J(R).

It is possible for a ring to have a unique maximal ideal and yet lack unique maximal one sided ideals: for example, in the ring of 2 by 2 square matrices over a field, the zero ideal is a maximal ideal, but there are many maximal right ideals. Definition[edit] There are other equivalent ways of expressing the definition of maximal one-sided and maximal two-sided ideals.

Examples[edit] Spectral space. In mathematics, a spectral space is a topological space which is homeomorphic to the spectrum of a commutative ring. Definition[edit] Let X be a topological space and let K (X) be the set of all quasi-compact open subsets of X. Then X is said to be spectral if it satisfies all of the following conditions: Equivalent descriptions[edit] Let X be a topological space. X is homeomorphic to a projective limit of finite T0-spaces.X is homeomorphic to the spectrum of a bounded distributive lattice L. Properties[edit] Let X be a spectral space and let K (X) be as before. Spectral maps[edit] A spectral map f: X → Y between spectral spaces X and Y is a continuous map such that the preimage of every open and quasi-compact subset of Y under f is again quasi-compact. References[edit] [edit] Compact space. Kolmogorov space. In topology and related branches of mathematics, a topological space X is a T0 space or Kolmogorov space if for every pair of distinct points of X, at least one of them has an open neighborhood not containing the other.

This condition, called the T0 condition, is one of the separation axioms. Its intuitive meaning is that the points of X are topologically distinguishable. These spaces are named after Andrey Kolmogorov. Definition[edit] A T0 space is a topological space in which every pair of distinct points is topologically distinguishable. That is, for any two different points x and y there is an open set which contains one of these points and not the other. Note that topologically distinguishable points are automatically distinct. Separated ⇒ topologically distinguishable ⇒ distinct The property of being topologically distinguishable is, in general, stronger than being distinct but weaker than being separated. Examples and nonexamples[edit] Spaces which are not T0[edit] Removing T0[edit]

Étale topology. Definitions[edit] For any scheme X, let Ét(X) be the category of all étale morphisms from a scheme to X. This is the analog of the category of open subsets of X (that is, the category whose objects are varieties and whose morphisms are open immersions). Its objects can be informally thought of as étale open subsets of X. The intersection of two objects corresponds to their fibered product over X. Ét(X) is a large category, meaning that its objects do not form a set. An étale presheaf on X is a contravariant functor from Ét(X) to the category of sets. Suppose that X is a Noetherian scheme. Grothendieck originally introduced the machinery of Grothendieck topologies and topoi to define the étale topology.

Local rings in the étale topology[edit] Let X be a scheme with its étale topology, and fix a point x of X. . See also[edit] References[edit] Grothendieck, Alexandre; Dieudonné, Jean (1964). Grothendieck topology. Scheme (mathematics) In mathematics, schemes connect the fields of algebraic geometry, commutative algebra and number theory.

Schemes were introduced by Alexander Grothendieck[when?] So as to broaden the notion of algebraic variety; some consider schemes to be the basic object of study of modern algebraic geometry. Technically, a scheme is a topological space together with commutative rings for all of its open sets, which arises from gluing together spectra (spaces of prime ideals) of commutative rings along their open subsets. There are many ways one can qualify a scheme. According to a basic idea of Grothendieck, conditions should be applied to a morphism of schemes. For detail on the development of scheme theory, which quickly becomes technically demanding, see first glossary of scheme theory.

The algebraic geometers of the Italian school had often used the somewhat foggy concept of "generic point" when proving statements about algebraic varieties. For a scheme , the category of schemes over. Sheaf (mathematics) Cayley plane. In mathematics, the Cayley plane (or octonionic projective plane) P2(O) is a projective plane over the octonions.[1] It was discovered in 1933 by Ruth Moufang, and is named after Arthur Cayley (for his 1845 paper describing the octonions). More precisely, there are two objects called Cayley planes, namely the real and the complex Cayley plane.

The real Cayley plane is the symmetric space F4/Spin(9), where F4 is a compact form of an exceptional Lie group and Spin(9) is the spin group of nine-dimensional Euclidean space (realized in F4). It admits a cell decomposition into three cells, of dimensions 0,8 and 16. [2] The complex Cayley plane is a homogeneous space under a noncompact (adjoint type) form of the group E6 by a parabolic subgroup P1. Properties[edit] In the Cayley plane, lines and points may be defined in a natural way so that it becomes a 2 dimensional projective space, that is, a projective plane. See also[edit] Rosenfeld projective plane Notes[edit] References[edit]

Representation theory. Representation theory is a powerful tool because it reduces problems in abstract algebra to problems in linear algebra, a subject that is well understood.[3] Furthermore, the vector space on which a group (for example) is represented can be infinite-dimensional, and by allowing it to be, for instance, a Hilbert space, methods of analysis can be applied to the theory of groups.[4] Representation theory is also important in physics because, for example, it describes how the symmetry group of a physical system affects the solutions of equations describing that system.[5] A striking feature of representation theory is its pervasiveness in mathematics. There are two sides to this. First, the applications of representation theory are diverse:[6] in addition to its impact on algebra, representation theory: The second aspect is the diversity of approaches to representation theory. The success of representation theory has led to numerous generalizations. Definitions and concepts[edit]

Category theory. A category with objects X, Y, Z and morphisms f, g, g ∘ f, and three identity morphisms (not shown) 1X, 1Y and 1Z. Several terms used in category theory, including the term "morphism", differ from their uses within mathematics itself. In category theory, a "morphism" obeys a set of conditions specific to category theory itself. Thus, care must be taken to understand the context in which statements are made. An abstraction of other mathematical concepts[edit] The most accessible example of a category is the category of sets, where the objects are sets and the arrows are functions from one set to another.

However, the objects of a category need not be sets, and the arrows need not be functions; any way of formalising a mathematical concept such that it meets the basic conditions on the behaviour of objects and arrows is a valid category, and all the results of category theory will apply to it. Category theory has several faces known not just to specialists, but to other mathematicians. Lie group. Inversive ring geometry. Complex projective plane. In mathematics, the complex projective plane, usually denoted P2(C), is the two-dimensional complex projective space. It is a complex manifold described by three complex coordinates where, however, the triples differing by an overall rescaling are identified: That is, these are homogeneous coordinates in the traditional sense of projective geometry. Topology[edit] The Betti numbers of the complex projective plane are The middle dimension 2 is accounted for by the homology class of the complex projective line, or Riemann sphere, lying in the plane. .

Algebraic geometry[edit] In birational geometry, a complex rational surface is any algebraic surface birationally equivalent to the complex projective plane. The group of birational automorphisms of the complex projective plane is the Cremona group. Differential geometry[edit] As a Riemannian manifold, the complex projective plane is a 4-dimensional manifold whose sectional curvature is quarter-pinched. References[edit] See also[edit] Fundamental group. Fundamental groups can be studied using the theory of covering spaces, since a fundamental group coincides with the group of deck transformations of the associated universal covering space. Its abelianisation can be identified with the first homology group of the space. When the topological space is homeomorphic to a simplicial complex, its fundamental group can be described explicitly in terms of generators and relations. Historically, the concept of fundamental group first emerged in the theory of Riemann surfaces, in the work of Bernhard Riemann, Henri Poincaré and Felix Klein, where it describes the monodromy properties of complex functions, as well as providing a complete topological classification of closed surfaces.

Intuition[edit] Start with a space (e.g. a surface), and some point in it, and all the loops both starting and ending at this point — paths that start at this point, wander around and eventually return to the starting point. Definition[edit] or simply π(X, x0). And. Manifold. The surface of the Earth requires (at least) two charts to include every point. Here the globe is decomposed into charts around the North and South Poles. The concept of a manifold is central to many parts of geometry and modern mathematical physics because it allows more complicated structures to be described and understood in terms of the relatively well-understood properties of Euclidean space. Manifolds naturally arise as solution sets of systems of equations and as graphs of functions. Manifolds may have additional features. One important class of manifolds is the class of differentiable manifolds. Motivational examples[edit] Circle[edit] Figure 1: The four charts each map part of the circle to an open interval, and together cover the whole circle.

The top and right charts overlap: their intersection lies in the quarter of the circle where both the x- and the y-coordinates are positive. Figure 2: A circle manifold chart based on slope, covering all but one point of the circle. and. Spectrum of a ring. Zariski topology[edit] to be the set of prime ideals containing I. We can put a topology on Spec(R) by defining the collection of closed sets to be This topology is called the Zariski topology. is a basis for the Zariski topology. Spec(R) is a compact space, but almost never Hausdorff: in fact, the maximal ideals in R are precisely the closed points in this topology.

Sheaves and schemes[edit] Given the space X=Spec(R) with the Zariski topology, the structure sheaf OX is defined on the Df by setting Γ(Df, OX) = Rf, the localization of R at the multiplicative system {1,f,f2,f3,...}. If R is an integral domain, with field of fractions K, then we can describe the ring Γ(U,OX) more concretely as follows. If P is a point in Spec(R), that is, a prime ideal, then the stalk at P equals the localization of R at P, and this is a local ring. Functoriality[edit] Of -1(P) → OP of local rings. Motivation from algebraic geometry[edit] Global Spec[edit] such that for every open affine , the inclusion ). Kevin R. Real projective plane. In mathematics, the real projective plane is an example of a compact non-orientable two-dimensional manifold, that is, a one-sided surface. It cannot be embedded in standard three-dimensional space without intersecting itself.

It has basic applications to geometry, since the common construction of the real projective plane is as the space of lines in R3 passing through the origin. (0, y) ~ (1, 1 − y) for 0 ≤ y ≤ 1 and (x, 0) ~ (1 − x, 1) for 0 ≤ x ≤ 1, as in the leftmost diagram on the right. Examples[edit] Projective geometry is not necessarily concerned with curvature and the real projective plane may be twisted up and placed in the Euclidean plane or 3-space in many different ways.[1] Some of the more important examples are described below. The projective sphere[edit] Consider a sphere, and let the great circles of the sphere be "lines", and let pairs of antipodal points be "points". The projective hemisphere[edit] Boy's surface – an immersion[edit] Roman surface[edit] Hemi polyhedra[edit] . Quaternionic projective space. In mathematics, quaternionic projective space is an extension of the ideas of real projective space and complex projective space, to the case where coordinates lie in the ring of quaternions H.

Quaternionic projective space of dimension n is usually denoted by and is a closed manifold of (real) dimension 4n. It is a homogeneous space for a Lie group action, in more than one way. In coordinates[edit] Its direct construction is as a special case of the projective space over a division algebra. Where the are quaternions, not all zero. In the language of group actions, is the orbit space of by the action of , the multiplicative group of non-zero quaternions. One may also regard as the orbit space of , the group of unit quaternions.[1] The sphere then becomes a principal Sp(1)-bundle over There is also a construction of by means of two-dimensional complex subspaces of , meaning that lies inside a complex Grassmannian.

Projective line[edit] Infinite-dimensional quaternionic projective space[edit] The space . Ringed space. Transformée de Fourier. Analyse non standard. Analyse harmonique (mathématiques)