Editing Lebesgue covering dimension (section)

==Formal definition==
[[File:Radford-stretcher-bond.jpeg|thumb|upright=1|[[Henri Lebesgue]] used closed "bricks" to study covering dimension in 1921.{{sfn|Lebesgue|1921}}]] 
The first formal definition of covering dimension was given by [[Eduard Čech]], based on an earlier result of [[Henri Lebesgue]].<ref>{{citation|title=Collected Works of Witold Hurewicz|volume=4|series=American Mathematical Society, Collected works series|editor-first=Krystyna|editor-last=Kuperberg|editor-link=Krystyna Kuperberg|publisher=American Mathematical Society|year=1995|isbn=9780821800119|at=p.&nbsp;xxiii, footnote&nbsp;3|url=https://books.google.com/books?id=6EICfJrepKQC&pg=PR23|quote=Lebesgue's discovery led later to the introduction by E. Čech of the covering dimension}}.</ref>

A modern definition is as follows. An [[open cover]] of a topological space {{mvar|''X''}} is a family of [[open set]]s {{mvar|''U''}}<sub>{{mvar|α}}</sub> such that their union is the whole space, <math>\cup_\alpha</math> {{mvar|''U''}}<sub>{{mvar|α}}</sub> = {{mvar|''X''}}. The '''order''' or '''ply''' of an open cover <math>\mathfrak A</math> = {{{mvar|''U''}}<sub>{{mvar|α}}</sub>} is the smallest number {{mvar|''m''}} (if it exists) for which each point of the space belongs to at most {{mvar|''m''}} open sets in the cover: in other words {{mvar|U}}<sub>{{mvar|α}}<sub>1</sub></sub> ∩ ⋅⋅⋅ ∩ {{mvar|U}}<sub>{{mvar|α}}<sub>{{mvar|''m''}}+1</sub></sub> = <math>\emptyset</math> for {{mvar|α}}<sub>1</sub>, ..., {{mvar|α}}<sub>{{mvar|''m''}}+1</sub> distinct. A [[refinement (topology)|refinement]] of an open cover <math>\mathfrak A</math> = {{{mvar|''U''}}<sub>{{mvar|α}}</sub>} is another open cover <math>\mathfrak B</math> = {{{mvar|''V''}}<sub>{{mvar|β}}</sub>}, such that each {{mvar|''V''}}<sub>{{mvar|β}}</sub> is contained in some {{mvar|''U''}}<sub>{{mvar|α}}</sub>. The '''covering dimension''' of a topological space {{mvar|''X''}} is defined to be the minimum value of {{mvar|''n''}} such that every finite open cover <math>\mathfrak A</math> of ''X'' has an open refinement <math>\mathfrak B</math> with order {{mvar|''n''}}&thinsp;+&thinsp;1. The refinement <math>\mathfrak B</math> can always be chosen to be finite.<ref>Proposition 1.6.9 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> Thus, if {{mvar|''n''}} is finite, {{mvar|V}}<sub>{{mvar|β}}<sub>1</sub></sub> ∩ ⋅⋅⋅ ∩ {{mvar|V}}<sub>{{mvar|β}}<sub>{{mvar|''n''}}+2</sub></sub> = <math>\emptyset</math> for {{mvar|β}}<sub>1</sub>, ..., {{mvar|β}}<sub>{{mvar|''n''}}+2</sub> distinct. If no such minimal {{mvar|''n''}} exists, the space is said to have infinite covering dimension.

As a special case, a non-empty topological space is [[zero-dimensional space|zero-dimensional]] with respect to the covering dimension if every open cover of the space has a refinement consisting of [[disjoint set|disjoint]] open sets, meaning any point in the space is contained in exactly one open set of this refinement.