Compactly generated spaces and locales
by
Martin Escardo
University of Birmingham

The Hofmann-Mislove Theorem (1980) asserts that the first Lawson dual of the topology of a sober topological space is in order-reversing bijection with the compact sets under inclusion. Hofmann and Lawson (1982) gave sufficient conditions on a topological space for its topology being isomorphic to its second Lawson dual. In particular, they concluded that compactly generated Hausdorff spaces have duality. We show that the converse also holds. More precisely, we show that the second Lawson dual of the topology of a Hausdorff space is isomorphic to the lattice of Kelley-open sets. Factoring through the Hofmann-Mislove isomorphism, our isomorphism sends a Kelley open set to the co-Scott open ideal of compact subsets, and its inverse sends such an ideal to its union. These results are obtained by representing topological spaces as locales.

http://www.cs.bham.ac.uk/~mhe/

Date received: July 2, 2001

Copyright © 2001 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # cahy-05.