structural:scholar_edition, linguistic:voice_change, structural:summarization
Affine and projective spaces

It is convenient to introduce affine and projective spaces by means of linear spaces, as follows. A n -dimensional linear subspace of a (n +1)-dimensional linear space, being itself a n -dimensional linear space, is not homogeneous; it contains a special point, the origin. Shifting it by a vector external to it, one obtains a n -dimensional affine subspace. It is homogeneous. An affine space need not be included into a linear space, but is isomorphic to an affine subspace of a linear space. All n -dimensional affine spaces over a given field are mutually isomorphic. In the words of John Baez, "an affine space is a vector space that's forgotten its origin". In particular, every linear space is also an affine space.

Given an n -dimensional affine subspace A in a (n +1)-dimensional linear space L, a straight line in A may be defined as the intersection of A with a two-dimensional linear subspace of L that intersects A: in other words, with a plane through the origin that is not parallel to A. More generally, a k -dimensional affine subspace of A is the intersection of A with a (k +1)-dimensional linear subspace of L that intersects A.

Every point of the affine subspace A is the intersection of A with a one-dimensional linear subspace of L. However, some one-dimensional subspaces of L are parallel to A; in some sense, they intersect A at infinity. The set of all one-dimensional linear subspaces of a (n +1)-dimensional linear space is, by definition, a n -dimensional projective space. And the affine subspace A is embedded into the projective space as a proper subset. However, the projective space itself is homogeneous. A straight line in the projective space corresponds to a two-dimensional linear subspace of the (n +1)-dimensional linear space. More generally, a k -dimensional projective subspace of the projective space corresponds to a (k +1)-dimensional linear subspace of the (n +1)-dimensional linear space, and is isomorphic to the k -dimensional projective space.

Defined this way, affine and projective spaces are of algebraic nature; they can be real, complex, and more generally, over any field.

Every real or complex affine or projective space is also a topological space. An affine space is a non-compact manifold; a projective space is a compact manifold. In a real projective space a straight line is homeomorphic to a circle, therefore compact, in contrast to a straight line in a linear of affine space.