Topologically invariant definition of the dimension of a space
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
topologically invariant way.
Informal discussion.
For ordinary Euclidean spaces, 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 sets.
In general, a topological space "X" can be covered by open sets, in that one can find a collection of open sets such that "X" lies inside of their union. The covering dimension is the smallest number "n" such that for every cover, there is a refinement in which every point in "X" lies in the intersection of no more than "n" + 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 homeomorphisms. 
The general idea is illustrated in the diagrams below, which show a cover and refinements of a circle and a square. 
Formal definition.
The first formal definition of covering dimension was given by Eduard Čech, based on an earlier result of Henri Lebesgue.
A modern definition is as follows. An open cover of a topological space "X" is a family of open sets "U"α such that their union is the whole space, formula_1 "U"α = "X". The order or ply of an open cover formula_2 = {"U"α} is the smallest number "m" (if it exists) for which each point of the space belongs to at most "m" open sets in the cover: in other words Uα1 ∩ ⋅⋅⋅ ∩ Uα"m"+1 = formula_3 for α1, ..., α"m"+1 distinct. A refinement of an open cover formula_2 = {"U"α} is another open cover formula_5 = {"V"β}, such that each "V"β is contained in some "U"α. The covering dimension of a topological space "X" is defined to be the minimum value of "n" such that every finite open cover formula_2 of "X" has an open refinement formula_5 with order "n" + 1. The refinement formula_5 can always be chosen to be finite. Thus, if "n" is finite, Vβ1 ∩ ⋅⋅⋅ ∩ Vβ"n"+2 = formula_3 for β1, ..., β"n"+2 distinct. If no such minimal "n" exists, the space is said to have infinite covering dimension.
As a special case, a non-empty topological space is zero-dimensional with respect to the covering dimension if every open cover of the space has a refinement consisting of 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 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 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 formula_10 has covering dimension "n".
Notes.
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