 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 rings – 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 spaces generally require allowing certain degeneracies – for example, allowing certain singularities or reducible varieties.