Graduation and dimension in locales.

*(English)*Zbl 0555.54020
Aspects of topology, Mem. H. Dowker, Lond. Math. Soc. Lect. Note Ser. 93, 195-210 (1985).

[For the entire collection see Zbl 0546.00024.]

This paper was written in response to a tentative suggestion by the reviewer [Bull. Am. Math. Soc., New Ser. 8, 41-53 (1983; Zbl 0499.54002)] that the theory of locales had reached a state of development at which it was appropriate to start thinking about dimension theory. What has emerged goes far beyond my own modest ambitions. In effect, what is presented here is a new definition of dimension, which the author describes as ”by Lebesgue or Krull, out of Menger-Urysohn”, and which unites most of the best features of these dimension functions on suitable classes of spaces while avoiding their wilder excesses on others. To do this, and to do it not just for spaces but for locales, requires large amounts of technique, and the paper will not provide easy reading for anyone. But it is worth making the effort to master the preliminaries (which occupy rather more than half the paper), since this paper is surely a milestone in the history of dimension theory.

This paper was written in response to a tentative suggestion by the reviewer [Bull. Am. Math. Soc., New Ser. 8, 41-53 (1983; Zbl 0499.54002)] that the theory of locales had reached a state of development at which it was appropriate to start thinking about dimension theory. What has emerged goes far beyond my own modest ambitions. In effect, what is presented here is a new definition of dimension, which the author describes as ”by Lebesgue or Krull, out of Menger-Urysohn”, and which unites most of the best features of these dimension functions on suitable classes of spaces while avoiding their wilder excesses on others. To do this, and to do it not just for spaces but for locales, requires large amounts of technique, and the paper will not provide easy reading for anyone. But it is worth making the effort to master the preliminaries (which occupy rather more than half the paper), since this paper is surely a milestone in the history of dimension theory.

Reviewer: P.T.Johnstone

##### MSC:

54F45 | Dimension theory in general topology |

06D99 | Distributive lattices |

54A05 | Topological spaces and generalizations (closure spaces, etc.) |