Dimension theory

In mathematics, dimension theory is a branch of general topology dealing with dimensional invariants of topological spaces.

Constructions

Inductive dimension

The inductive dimension of a topological space X is either of two values, the small inductive dimension ind(X) or the large inductive dimension Ind(X). These are based on the observation that, in n-dimensional Euclidean space Rⁿ, (n − 1)-dimensional spheres (that is, the boundaries of n-dimensional balls) have dimension n − 1. Therefore it should be possible to define the dimension of a space inductively in terms of the dimensions of the boundaries of suitable open sets.

Lebesgue covering dimension

An open cover of a topological space X is a family of open sets whose union is X. The ply of a cover is the smallest number n (if it exists) such that each point of the space belongs to at most n sets in the cover. A refinement of a cover C is another cover, each of whose sets is a subset of a set in C; its ply may be smaller than, or possibly larger than, the ply of C. The Lebesgue covering dimension of a topological space X is defined to be the minimum value of n, such that every finite open cover C of X has a refinement with ply at most n + 1. If no such minimal n exists, the space is said to be of infinite covering dimension.

Dimension theory (algebra)

In mathematics, dimension theory is a branch of commutative algebra studying the notion of the dimension of a commutative ring, and by extension that of a scheme.

The theory is much simpler for an affine ring; i.e., an integral domain that is a finitely generated algebra over a field. By Noether's normalization lemma, the Krull dimension of such a ring is the transcendence degree over the base field and the theory runs in parallel with the counterpart in algebraic geometry; cf. Dimension of an algebraic variety. The general theory tends to be less geometrical; in particular, very little works/is known for non-noetherian rings. (Kaplansky's commutative rings gives a good account of the non-noetherian case.) Today, a standard approach is essentially that of Bourbaki and EGA, which makes essential use of graded modules and, among other things, emphasizes the role of multiplicities, the generalization of the degree of a projective variety. In this approach, Krull's principal ideal theorem appears as a corollary.