# Dimension theory

In mathematics, **dimension theory** is a branch of general topology dealing with dimensional invariants of topological spaces.^{[1]}^{[2]}^{[3]}^{[4]}

## 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 *R*^{n}, (*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.

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

## References

- ↑ Katětov, Miroslav; Simon, Petr (1997), "Origins of dimension theory",
*Handbook of the history of general topology, Vol. 1*, Kluwer Acad. Publ., Dordrecht, pp. 113–134, MR 1617557. - ↑ Hurewicz, Witold; Wallman, Henry (1941),
*Dimension Theory*, Princeton Mathematical Series, v. 4, Princeton, N. J.: Princeton University Press, MR 0006493. - ↑ Nadler, Sam B., Jr. (2002),
*Dimension theory: an introduction with exercises*, Aportaciones Matemáticas: Textos [Mathematical Contributions: Texts],**18**, Sociedad Matemática Mexicana, México, ISBN 970-32-0026-5, MR 1925171. - ↑ Lipscomb, Stephen Leon (2009),
*Fractals and universal spaces in dimension theory*, Springer Monographs in Mathematics, Springer, New York, doi:10.1007/978-0-387-85494-6, ISBN 978-0-387-85493-9, MR 2460244.