thickenings

Thickenings are mathematical constructions that extend a given object by including a surrounding neighborhood or, in algebraic settings, by introducing infinitesimal extensions of that object. They appear in several areas of mathematics under related ideas.

In metric and topological spaces, a subset A of a metric space (X, d) can be thickened to its r-thickening or r-neighborhood A_r = {x in X : dist(x, A) ≤ r}. There is also an open version, A_r^o = {x : dist(x, A) < r}. Thickening a set is equivalent to forming a Minkowski sum with a ball of radius r. As r increases, the thickening contains more points, providing a means to study proximity and continuity of A within X.

In differential geometry, a common form is the tubular neighborhood: for a submanifold M of a smooth manifold X, there exists a neighborhood of M that is diffeomorphic to a neighborhood in the normal bundle NM. This smooth thickening allows one to transfer questions about M to questions about its normal directions and is central to constructions in differential topology and geometry.

In algebraic geometry, thickening has a more algebraic meaning. If X is a scheme and Z ⊆ X is a closed subscheme defined by an ideal sheaf I ⊆ O_X, the n-th thickening Z_n is the subscheme with structure sheaf O_{Z_n} = O_X / I^n. The reduced subscheme is Z. Z_2, for example, represents the first infinitesimal neighborhood of Z in X. Thickenings encode infinitesimal deformations and are fundamental in deformation theory and the study of formal schemes.

Applications span geometric modeling, topology, and algebraic geometry, where thickening serves to formalize neighborhoods, offsets, and infinitesimal extensions.