Editing Lebesgue covering dimension

{{short description|Topologically invariant definition of the dimension of a space}}
{{more footnotes|date=April 2018}}
In [[mathematics]], the '''Lebesgue covering dimension''' or '''topological dimension''' of a [[topological space]] is one of several different ways of defining the [[dimension]] of the space in a
[[topological invariant|topologically invariant]] way.<ref name="Lebesgue">{{cite journal|url=http://matwbn.icm.edu.pl/ksiazki/fm/fm2/fm2130.pdf|title=Sur les correspondances entre les points de deux espaces|volume= 2 |year= 1921|first=Henri|last= Lebesgue| author-link= Henri Lebesgue|journal=
[[Fundamenta Mathematicae]]| pages= 256–285|doi= 10.4064/fm-2-1-256-285|lang=fr}}</ref><ref name="Duda">{{cite journal|title=The origins of the concept of dimension|volume=42|year= 1979|first=R.|last= Duda|journal=Colloquium Mathematicum
|pages= 95–110|doi=10.4064/cm-42-1-95-110|url=https://www.impan.pl/en/publishing-house/journals-and-series/colloquium-mathematicum/all/42/1/102445/the-origins-of-the-concept-of-dimension|mr=0567548|doi-access=free}}</ref>

==Informal discussion==
For ordinary [[Euclidean space]]s, the Lebesgue covering dimension is just the ordinary Euclidean dimension: zero for points, one for lines, two for planes, and so on. However, not all topological spaces have this kind of "obvious" [[dimension]], and so a precise definition is needed in such cases. The definition proceeds by examining what happens when the space is covered by [[open set]]s.

In general, a topological space ''X'' can be [[open cover|covered by open sets]], in that one can find a collection of open sets such that ''X'' lies inside of their [[union (set theory)|union]]. The covering dimension is the smallest number ''n'' such that for every cover, there is a [[refinement (topology)|refinement]] in which every point in ''X'' lies in the [[intersection (set theory)|intersection]] of no more than ''n''&thinsp;+&thinsp;1 covering sets. This is the gist of the formal definition below. The goal of the definition is to provide a number (an [[integer]]) that describes the space, and does not change as the space is continuously deformed; that is, a number that is invariant under [[homeomorphism]]s. 

The general idea is illustrated in the diagrams below, which show a cover and refinements of a circle and a square. 
{| |-
| [[Image:Refinement of the cover of a circle.svg|thumb|Refinement of the cover of a circle]]
|The first image shows a refinement (on the bottom) of a colored cover (on the top) of a black circular line. Note how in the refinement, no point on the circle is contained in more than two sets, and also how the sets link to one another to form a "chain".
|-
|| [[Image:Refinement on a planar shape.svg|thumb|Refinement of the cover of a square]]
|The top half of the second image shows a cover (colored) of a planar shape (dark), where all of the shape's points are contained in anywhere from one to all four of the cover's sets. The bottom illustrates that any attempt to refine said cover such that no point would be contained in more than ''two'' sets—ultimately fails at the intersection of set borders. Thus, a planar shape is not "webby": it cannot be covered with "chains", per se. Instead, it proves to be ''thicker'' in some sense. More rigorously put, its topological dimension must be greater than 1.
|}

==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.

==Examples==
The empty set has covering dimension −1: for any open cover of the empty set, each point of the empty set is not contained in any element of the cover, so the order of any open cover is 0.

Any given open cover of the [[unit circle]] will have a refinement consisting of a collection of [[open (topology)|open]] arcs. The circle has dimension one, by this definition, because any such cover can be further refined to the stage where a given point ''x'' of the circle is contained in ''at most'' two open arcs. That is, whatever collection of arcs we begin with, some can be discarded or shrunk, such that the remainder still covers the circle but with simple overlaps.

Similarly, any open cover of the [[unit disk]] in the two-dimensional [[plane (mathematics)|plane]] can be refined so that any point of the disk is contained in no more than three open sets, while two are in general not sufficient. The covering dimension of the disk is thus two.

More generally, the ''n''-dimensional [[Euclidean space]] <math>\mathbb{E}^n</math> has covering dimension ''n''.

==Properties==
*[[Homeomorphic]] spaces have the same covering dimension. That is, the covering dimension is a [[topological invariant]].
*The covering dimension of a normal space ''X'' is <math>\le n</math> if and only if for any [[closed subset]] ''A'' of ''X'', if <math> f:A\rightarrow S^n </math> is continuous, then there is an extension of <math> f </math> to <math> g:X\rightarrow S^n </math>.  Here, <math> S^n </math> is the [[n-sphere|''n''-dimensional sphere]].
* '''Ostrand's theorem on covering dimension.'''  If {{mvar|X}} is a normal topological space and <math>\mathfrak A</math> = {{{mvar|''U''}}<sub>{{mvar|α}}</sub>} is a locally finite cover of {{mvar|''X''}} of order ≤ {{mvar|''n''}} + 1, then, for each 1 ≤ {{mvar|''i''}} ≤ {{mvar|''n''}} + 1, there exists a family of pairwise disjoint open sets <math>\mathfrak B</math><sub>{{mvar|''i''}}</sub> = {{{mvar|''V''}}<sub>{{mvar|''i''}},{{mvar|α}}</sub>} shrinking <math>\mathfrak A</math>, i.e. {{mvar|''V''}}<sub>{{mvar|''i''}},{{mvar|α}}</sub> ⊆ {{mvar|''U''}}<sub>{{mvar|α}}</sub>, and together covering {{mvar|''X''}}.{{sfn|Ostrand|1971}}

==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".

==See also==
* [[Carathéodory's extension theorem]]
* [[Geometric set cover problem]]
* [[Dimension theory]]
* [[Metacompact space]]
* [[Point-finite collection]]

== Notes ==
{{Reflist|30em}}

== References ==
*{{cite book|mr=2356043|last=Edgar|first= Gerald A.|title=Measure, topology, and fractal geometry|edition=Second| series= Undergraduate Texts in Mathematics| publisher= [[Springer-Verlag]]| year= 2008|isbn= 978-0-387-74748-4|chapter=Topological Dimension|pages=85–114}}
* {{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}}
* {{cite book | last1=Godement | first1=Roger | author1-link = Roger Godement | title=Topologie algébrique et théorie des faisceaux | publisher=Hermann | location=Paris | mr= 0102797 | year=1958|lang=fr | series= 
Publications de l'Institut de Mathématique de l'Université de Strasbourg| volume= III}}
* {{cite book|mr =0006493 |last1= Hurewicz|first1= Witold|last2= Wallman|first2= Henry|title= Dimension Theory | series =Princeton Mathematical Series| volume= 4 | publisher= [[Princeton University Press]]| year= 1941}}
* {{cite book | last=Munkres |first=James R. |author-link=James Munkres|title=Topology |year=2000 |edition=2nd |publisher=Prentice-Hall |isbn=0-13-181629-2|mr=3728284 }}
* {{cite journal| mr= 0288741 |last=Ostrand|first= Phillip A.|title = Covering dimension in general spaces| journal= General Topology and Appl.| volume= 1 |year=1971|issue= 3|pages= 209–221|doi=10.1016/0016-660X(71)90093-6 }}

==Further reading==

===Historical===
* [[Karl Menger]], ''General Spaces and Cartesian Spaces'', (1926) Communications to the Amsterdam Academy of Sciences. English translation reprinted in ''Classics on Fractals'', Gerald A.Edgar, editor, Addison-Wesley (1993) {{isbn|0-201-58701-7}}
* [[Karl Menger]], ''Dimensionstheorie'', (1928) B.G Teubner Publishers, Leipzig.
=== Modern ===
* {{cite book|first=Alan R.|last =Pears |title=Dimension Theory of General Spaces|year=1975|publisher= [[Cambridge University Press]] |isbn=0-521-20515-8 | url = https://archive.org/details/dimensiontheoryo0000pear |mr=0394604}}
* V. V. Fedorchuk, ''The Fundamentals of Dimension Theory'', appearing in ''Encyclopaedia of Mathematical Sciences, Volume 17, General Topology I'', (1993) A. V. Arkhangel'skii and [[L. S. Pontryagin]] (Eds.), Springer-Verlag, Berlin {{isbn|3-540-18178-4}}.

==External links==
* {{springer|title=Lebesgue dimension|id=p/l057830}}

{{Dimension topics}}

[[Category:Dimension theory]]