# 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:

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

Field With One Element - Computations
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Projective Space - Generalizations
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Projective Linear Group
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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 AndrÃ© 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 premiÃ¨re to Hate and Pain;
And Space urbanely keeping us apart.
Philip Larkin (1922–1986)