View source for 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 [[#References|[1]]]. He stated the conjecture that $\dim I^n = n$ for the $n$-dimensional cube $I^n$. L.E.J. Brouwer [[#References|[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====
<table>
<TR><TD valign="top">[1]</TD> <TD valign="top">  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</TD></TR>
<TR><TD valign="top">[2]</TD> <TD valign="top">  L.E.J. Brouwer,   "Ueber den natürlichen Dimensionsbegriff"  ''J. Reine Angew. Math.'' , '''142'''  (1913)  pp. 146–152</TD></TR>
<TR><TD valign="top">[3]</TD> <TD valign="top">  P.S. Aleksandrov,   B.A. Pasynkov,   "Introduction to dimension theory" , Moscow  (1973)  (In Russian)</TD></TR>
</table>



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

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