Lebesgue dimension

A dimension defined by means of coverings (cf. Covering (of a set)). It is the most important dimension invariant $\dim X$ of a topological space $X$ and was discovered by H. Lebesgue [1]. He stated the conjecture that $\dim I^n = n$ for the $n$-dimensional cube $I^n$. L.E.J. Brouwer [2] was the first to prove this, as well as the stronger identity: $\dim I^n = \text{Ind}\,I^n = n$. A precise definition of the invariant $\dim X$ (for the class of metric compacta) was given by P.S. Urysohn, who proved that for a space $X$ of this class $$ \dim X = \text{ind}\,X = \text{Ind}\,X $$ (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 $X$ the Lebesgue dimension is defined as the smallest integer $n$ having the property that for any $\epsilon > 0$ there is a finite open $\epsilon$-covering of $X$ that has multiplicity $\le n+1$; an $\epsilon$-covering of a metric space is a covering all elements of which have diameter $< \epsilon$, and the multiplicity of a finite covering of $X$ is the largest integer $k$ such that there is a point of $X$ contained in $k$ elements of the given covering. For an arbitrary normal (in particular, metrizable) space $X$ the Lebesgue dimension is the smallest integer $n$ such that for any finite open covering $\Omega$ of $X$ there is a (finite open) covering $\Lambda$ of multiplicity $n+1$ that refines it. A covering $\Lambda$ is said to be a refinement of a covering $\Omega$ if every element of $\Lambda$ is a subset of at least one element of $\Omega$.

References

[1]	H. Lebesgue, "Sur la non-applicabilité de deux domaines appartenant à des espaces à $n$ et $n+p$ 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.))