In commutative algebra this mathematical concept is defined for rings as the supremum of the number of strict inclusions in a chain of prime ideals and is named after Krull. The invariance of domain theorem shows that this is a homeomorphism invariant for Euclidean spaces. For subsets of Euclidean space this concept has a generalization to fractional values given by finding the critical value alpha at which the exterior alpha Hausdorff measure does not vanish or equal infinity, which is used to 
In commutative algebra this mathematical concept is defined for rings as the supremum of the number of strict inclusions in a chain of prime ideals and is named after Krull. The invariance of domain theorem shows that this is a homeomorphism invariant for Euclidean spaces. For subsets of Euclidean space this concept has a generalization to fractional values given by finding the critical value alpha at which the exterior alpha Hausdorff measure does not vanish or equal infinity, which is used to define this for fractals. For connected manifolds this integer is defined in reference to what Euclidean space the manifold is locally homeomorphic to. For vector spaces this is defined to be the cardinality of any basis. For 10 points name this concept, which for the Euclidean space R^n [read "R-N"] is equal to the number of coordinates n.
In commutative algebra this mathematical concept is defined for rings as the supremum of the number of strict inclusions in a chain of prime ideals and is named after Krull.
{dimension}