In mathematics, the Lebesgue covering dimension of a topological space is defined to be the minimum value of n, such that any open cover has a refinement with no point included in more than n+1 elements.
Here a refinement is a second open cover, of open sets selected from the given open cover. To illustrate the concept, consider open covers of the unit circle, by open arcs. The circle has dimension 1, by this definition, because any such cover can be refined to the stage where a given point x of the circle is contained in at most 2 arcs. That is, whatever arcs we begin with, enough can be discarded so that there are just simple overlaps.
The Lebesgue covering dimension gives the correct answer for the dimension of a finite simplicial complex; this is the Lebesgue covering theorem.
The dimension on any other space will be defined as one greater that the dimension of the object that could be used to completely separate any part of the first space from the rest.
The dimension of a space should be the maximum of its local dimensions where the local dimension is defined as one more than the dimension of the lowest dimensional object with the capacity to separate any neighborhood of the space into two parts.
A topological property of an entity is one that remains invariant under continuous, one-to-one transformations or homeomorphisms.
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