defined as the intersection of A with a 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 affine subspace of A is the intersection of A with a linear subspace of L that intersects A. Every point of the affine subspace A is the intersection of A with a linear subspace of L. However, some subspaces of L are parallel to A; in some sense, they intersect A at infinity. The set of all linear subspaces of a linear space is, by definition, a 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 linear subspace of the (n+1)-dimensional linear space. More generally, a projective subspace of the projective space corresponds to a linear subspace of the (n+1)-dimensional linear space, and is isomorphic to the 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. Metric and uniform spaces Distances between points are defined in a metric space. Isomorphisms between metric spaces are called isometries. Every metric space is also a topological space. A topological space is called metrizable, if it underlies a metric space. All manifolds are metrizable. In a metric space, we can define bounded sets and Cauchy sequences. A metric space is called complete if all Cauchy sequences converge. Every incomplete space is isometrically embedded, as a dense subset, into a complete space (the completion). Every compact metric space is complete; the real line is non-compact but complete; the open interval (0,1) is incomplete. Every Euclidean space is also a complete metric space. Moreover, all geometric notions immanent to a Euclidean space can be characterized in terms of its metric. For example, the straight segment connecting two given points A and C consists of all points B such that the distance between A and C is equal to the