Dimension theory

This article is about dimension theory in topology. For dimension theory in commutative algebra, see dimension theory (algebra).

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

Constructions

Inductive dimension

Main article: 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

Main article: 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.

As a special case, a topological space is zero-dimensional with respect to the covering dimension if every open cover of the space has a refinement consisting of disjoint open sets so that any point in the space is contained in exactly one open set of this refinement.

See also

• Dimension