The (combinatorial) dimension of a topological space is defined as the supremum of the lengths over all strictly ascending chains of closed irreducible subsets (e.g. Hartshorne). Can it also be defined similarly albeit using strictly descending chains? Are the two notions equivalent? If not, which one is more important and why?
Edited: Also, Atiyah-MacDonald define the Krull dimension of a ring using ascending chains of prime ideals while Matsumura uses descending chains. Any insights?
PS: Merry Christmas :)