The Krull dimension of Alexandroff T-0-spaces

Alexandroff T-0-spaces have been studied as topological models of the supports of digital images and as discrete models of continuous spaces in theoretical physics. In [17] we have defined (topologically) the Alexandroff dimension far arbitrary Alexandroff spaces. In this paper we will prove that the Alexandroff dimension of a T-0-space is equal to ifs Krull dimension, which is defined in terms of the lattice of closed sets of the space and which was first studied in [16]. Since the category of Alexandroff T-0-spaces is known to be isomorphic to the category of posets (see [3, Theorem 3.5]), the results could be formulated in this latter category as well.