Editing Compactification (mathematics) (section)

== Projective space ==
[[Real projective space]] '''RP'''<sup>''n''</sup> is a compactification of Euclidean space '''R'''<sup>''n''</sup>. For each possible "direction" in which points in '''R'''<sup>''n''</sup> can "escape", one new point at infinity is added (but each direction is identified with its opposite). The Alexandroff one-point compactification of '''R''' we constructed in the example above is in fact homeomorphic to '''RP'''<sup>1</sup>. Note however that the [[projective plane]] '''RP'''<sup>2</sup> is ''not'' the one-point compactification of the plane '''R'''<sup>2</sup> since more than one point is added.

[[Complex projective space]] '''CP'''<sup>''n''</sup> is also a compactification of  '''C'''<sup>''n''</sup>; the Alexandroff one-point compactification of the plane '''C''' is (homeomorphic to) the complex projective line '''CP'''<sup>1</sup>, which in turn can be identified with a sphere, the [[Riemann sphere]].

Passing to projective space is a common tool in [[algebraic geometry]] because the added points at infinity lead to simpler formulations of many theorems. For example, any two different lines in '''RP'''<sup>2</sup> intersect in precisely one point, a statement that is not true in '''R'''<sup>2</sup>. More generally, [[Bézout's theorem]], which is fundamental in [[intersection theory]], holds in projective space but not affine space. This distinct behavior of intersections in affine space and projective space is reflected in [[algebraic topology]] in the [[cohomology ring]]s – the cohomology of affine space is trivial, while the cohomology of projective space is non-trivial and reflects the key features of intersection theory (dimension and degree of a subvariety, with intersection being [[Poincaré dual]] to the [[cup product]]).

Compactification of [[moduli space]]s generally require allowing certain degeneracies – for example, allowing certain singularities or reducible varieties. This is notably used in the Deligne–Mumford compactification of the [[Moduli of algebraic curves|moduli space of algebraic curves]].