Editing Lebesgue covering dimension (section)

==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.
|}