Editing Lebesgue covering dimension (section)

==Relationships to other notions of dimension==

*For a paracompact space {{mvar|''X''}}, the covering dimension can be equivalently defined as the minimum value of {{mvar|''n''}}, such that every open cover <math>\mathfrak A</math> of {{mvar|''X''}} (of any size) has an open refinement <math>\mathfrak B</math> with order {{mvar|''n''}}&thinsp;+&thinsp;1.<ref>Proposition 3.2.2 of {{cite book| url=https://www.maths.ed.ac.uk/~v1ranick/papers/engelking.pdf |mr =0482697 |last= Engelking|first= Ryszard|title= Dimension theory|series= North-Holland Mathematical Library|volume=19|publisher=North-Holland|location=Amsterdam-Oxford-New York|year=1978|isbn= 0-444-85176-3}}</ref> In particular, this holds for all metric spaces.
*'''Lebesgue covering theorem.''' The Lebesgue covering dimension coincides with the [[affine dimension]] of a finite [[simplicial complex]].
*The covering dimension of a [[normal space]] is less than or equal to the large [[inductive dimension]].
*The covering dimension of a [[Paracompact space|paracompact]] [[Hausdorff space|Hausdorff]] space <math>X</math> is greater or equal to its [[cohomological dimension]] (in the sense of [[sheaf (mathematics)|sheaves]]),<ref>Godement 1973, II.5.12, p. 236</ref> that is, one has <math>H^i(X,A) = 0</math> for every sheaf <math>A</math> of abelian groups on <math>X</math> and every <math>i</math> larger than the covering dimension of <math>X</math>.
* In a [[metric space]], one can strengthen the notion of the multiplicity of a cover: a cover has ''{{mvar|r}}-multiplicity'' {{math|''n'' + 1}} if every {{mvar|r}}-ball intersects with at most {{mvar|''n'' + 1}} sets in the cover.  This idea leads to the definitions of the [[asymptotic dimension]] and [[Assouad–Nagata dimension]] of a space: a space with asymptotic dimension {{mvar|n}} is {{mvar|n}}-dimensional "at large scales", and a space with Assouad–Nagata dimension {{mvar|n}} is {{mvar|n}}-dimensional "at every scale".