INTRODUCTION

Among many ways of defining or characterizing the dimension of a topological space, three are the inductive (Menger-Urysohn), the covering (Lebesgue), and the chain (Krull). Krull dimension, one less than the maximum length of a chain of completely prime filters of the topology, is spectacularly wrong for the most popular spaces, vanishing for all non-empty Hausdorff spaces; but it seems to be the only dimension of interest for the Zariski spaces of algebraic geometry.

The definition proposed here is by Lebesgue or Krull, out of Menger-Urysohn. The graduated dimension gdim X of a T0 space or locale X is the minimum n such that some sublattice of the topology (respectively, frame) T(X) which is a basis is a directed union of finite topologies of Krull dimension n. The idea of minimizing over bases is from Menger-Urysohn.

A first view of gdim is that it is a modified Krull dimension designed to be reasonable for a wider class of spaces. How reasonable is it? For metrizable spaces or locales X, ind X ≤ gdim X ≤ dim X. There is essentially only one known example for ind ≠ dim in metrizable spaces, Roy's (1968) example, and for it gdim agrees with ind. A second handy class of “geometric” spaces is the compact Hausdorff spaces Y, for which the lineup of familiar dimension functions is dim Y ≤ ind Y ≤ Ind Y (and dim Y = 1, ind Y = 2, Ind Y = 3 is possible (Filippov 1970)).