Lebesgue dimension

A dimension defined by means of coverings (cf. Covering (of a set)). It is the most important dimension invariant of a topological space and was discovered by H. Lebesgue [1]. He stated the conjecture that for the -dimensional cube . L.E.J. Brouwer [2] was the first to prove this, as well as the stronger identity: . A precise definition of the invariant (for the class of metric compacta) was given by P.S. Urysohn, who proved that for a space of this class

(the Urysohn identity, see Dimension theory). This identity was extended to the class of all separable metric spaces by W. Hurewicz and L.A. Tumarkin in 1925.

For compacta the Lebesgue dimension is defined as the smallest integer having the property that for any there is a finite open -covering of that has multiplicity ; an -covering of a metric space is a covering all elements of which have diameter , and the multiplicity of a finite covering of is the largest integer such that there is a point of contained in elements of the given covering. For an arbitrary normal (in particular, metrizable) space the Lebesgue dimension is the smallest integer such that for any finite open covering of there is a (finite open) covering of multiplicity that refines it. A covering is said to be a refinement of a covering if every element of is a subset of at least one element of .

References

[1]	H. Lebesgue, "Sur la non-applicabilité de deux domaines appartenant à des espaces à et dimensions" Math. Ann. , 70 (1911) pp. 166–168
[2]	L.E.J. Brouwer, "Ueber den natürlichen Dimensionsbegriff" J. Reine Angew. Math. , 142 (1913) pp. 146–152
[3]	P.S. Aleksandrov, B.A. Pasynkov, "Introduction to dimension theory" , Moscow (1973) (In Russian)

Comments

The Lebesgue dimension is also called the covering dimension or Čech–Lebesgue dimension. The multiplicity of a covering is also called the order of the covering.

References

[a1]	R. Engelking, "Dimension theory" , North-Holland & PWN (1978) pp. 19; 50
[a2]	W. Hurevicz, G. Wallman, "Dimension theory" , Princeton Univ. Press (1948) ((Appendix by L.S. Pontryagin and L.G. Shnirel'man in Russian edition.))