Lebesgue covering dimension

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. 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. 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 Uα such that their union is the whole space, Uα = X. The order or ply of an open cover = {Uα} 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 ∩ ⋅⋅⋅ ∩ Uαm+1 = for α1, ..., αm+1 distinct. A refinement of an open cover = {Uα} is another open cover = {Vβ}, such that each Vβ is contained in some Uα. The covering dimension of a topological space X is defined to be the minimum value of n such that every finite open cover of X has an open refinement with order n + 1.