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The Krull Dimension of Alexandroff T 0 ‐spaces
Author(s) -
WIEDERHOLD PETRA,
WILSON RICHARD G.
Publication year - 1996
Publication title -
annals of the new york academy of sciences
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.712
H-Index - 248
eISSN - 1749-6632
pISSN - 0077-8923
DOI - 10.1111/j.1749-6632.1996.tb49187.x
Subject(s) - dimension theory (algebra) , mathematics , krull dimension , dimension (graph theory) , topological space , lebesgue covering dimension , pure mathematics , space (punctuation) , closed set , discrete mathematics , algebra over a field , computer science , hausdorff dimension , noetherian , operating system
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 for arbitrary Alexandroff spaces. In this paper we will prove that the Alexandroff dimension of a T 0 ‐space is equal to its 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.