Projective Space

A projective space S can be defined axiomatically as a set P (the set of points), together with a set L of subsets of P (the set of lines), satisfying these axioms :

  • Each two distinct points p and q are in exactly one line.
  • Veblen's axiom: If a, b, c, d are distinct points and the lines through ab and cd meet, then so do the lines through ac and bd.
  • Any line has at least 3 points on it.

The last axiom eliminates reducible cases that can be written as a disjoint union of projective spaces together with 2-point lines joining any two points in distinct projective spaces. More abstractly, it can be defined as an incidence structure (P,L,I) consisting of a set P of points, a set L of lines, and an incidence relation I stating which points lie on which lines.

A subspace of the projective space is a subset X, such that any line containing two points of X is a subset of X (that is, completely contained in X). The full space and the empty space are always subspaces.

The geometric dimension of the space is said to be n if that is the largest number for which there is a strictly ascending chain of subspaces of this form:

Read more about Projective SpaceMorphisms, Dual Projective Space, Generalizations

Other articles related to "projective space, projective spaces, space, spaces, projective":

Field With One Element - Computations
... structures on a set are analogous to structures on a projective space, and can be computed in the same way Sets are projective spaces The number of elements of, the (n ... of the q-integer into a sum of powers of q corresponds to the Schubert cell decomposition of projective space ... of a set can be considered a filtered set, as a flag is a filtered vector space for instance, the permutation (0, 1, 2) corresponds to the filtration ...
Projective Space - Generalizations
... dimension The projective space, being the "space" of all one-dimensional linear subspaces of a given vector space V is generalized to Grassmannian manifold, which is parametrizing ... sequence of subspaces More generally flag manifold is the space of flags, i.e ... other subvarieties Even more generally, moduli spaces parametrize objects such as elliptic curves of a given kind ...
Projective Linear Group
... In mathematics, especially in the group theoretic area of algebra, the projective linear group (also known as the projective general linear group or PGL) is the induced ... Explicitly, the projective linear group is the quotient group PGL(V) = GL(V)/Z(V) where GL(V) is the general linear group of V and Z(V) is the subgroup of all nonzero scalar transformations of V these are ... The projective special linear group, PSL, is defined analogously, as the induced action of the special linear group on the associated projective space ...
Abstract Variety
... in the definition of manifold independent of any ambient space (Hassler Whitney, in the 1930s) by some years, the first notions being those of Oscar Zariski and André Weil in the 1940s ... gave a first acceptable definition of algebraic variety that stood outside projective space ... affine varieties can be completed, by embedding them in projective space ...
Lie Sphere Geometry
... spheres) of infinite radius and that points in the plane (or space) should be regarded as circles (or spheres) of zero radius ... The space of circles in the plane (or spheres in space), including points and lines (or planes) turns out to be a manifold known as the Lie quadric (a quadric hypersurface in projective space) ... To handle this, curves in the plane and surfaces in space are studied using their contact lifts, which are determined by their tangent spaces ...

Famous quotes containing the word space:

    Time in his little cinema of the heart
    Giving a première to Hate and Pain;
    And Space urbanely keeping us apart.
    Philip Larkin (1922–1986)