On the duality of compact vs. open. (English) Zbl 0885.54001
Andima, Susan (ed.) et al., Papers on general topology and applications. Papers presented at the 11th summer conference at the University of Southern Maine, Gorham, ME, USA, August 10–13, 1995. New York, NY: The New York Academy of Sciences. Ann. N. Y. Acad. Sci. 806, 214-230 (1996).
The authors investigate the duality between open sets and compact saturated sets in coherent spaces (which is their term – perhaps not ideally chosen – for locally compact sober spaces in which the compact saturated sets are stable under finite intersections; equivalently, for the spaces of points of arithmetic lattices considered as frames). In order to present this duality with maximum clarity, they introduce a concept of ‘proximity lattice’ which abstracts the finitary part of the notion of an arithmetic lattice. Their main result is a representation theorem stating that the spectra of proximity lattices satisfying two additional conditions are precisely the coherent spaces.
For the entire collection see [Zbl 0879.00049].
For the entire collection see [Zbl 0879.00049].
Reviewer: P.T.Johnstone (Cambridge)
MSC:
| 54A05 | Topological spaces and generalizations (closure spaces, etc.) |
| 06B35 | Continuous lattices and posets, applications |
| 06E15 | Stone spaces (Boolean spaces) and related structures |
| 54D30 | Compactness |
