On the Duality of Compact vs. Open
by
Philipp Sünderhauf
Darmstadt
Coauthors: Achim Jung

A complete metric space can be represented by a dense subset together with the induced metric. The space is recovered by forming the completion of the subset.

A topology on some space can be represented by a subset of opens, i.e. a base, together with the subset relation. The topology is recovered by taking arbitrary unions of base opens.

The present paper is an investigation into representing spaces and topologies by bases. The approach is abstract in the sense that elements of the bases are neither elements nor subsets of the represented space. Thus we follow the localic tradition in topology whose main objects of study are certain lattices, known as frames. Our first goal is to investigate in some detail how a frame can be represented by an abstract basis. The mode of representation is that of continuous domains. Hence we restrict our attention to continuous frames and locally compact spaces, respectively. We show step by step how adding structure to the abstract basis makes the connection to the represented space more well-behaved. This development culminates in what we call \/ -Urysohn lattice bases in Section 4. Apart from the synthetic presentation (and a notably generalization) this material is fairly well known.

For what may seem to be purely aesthetic reasons, we move on to prelocales, whose definition is completely self-dual. As we show in Theorem 5.2, prelocales provide all information about the represented space, its frame of opens and its set of compact saturated subsets in a clean and direct fashion. The duality between compact and open subsets is fully apparent.

Already \/ -Urysohn lattice bases brought us into the realm of compact coherent spaces and it is a natural question whether prelocales-pleasing as they may be-are not too restrictive. Our first main result says that this is not the case: Every compact coherent space can be represented by a prelocale. The construction is not performed by selecting a certain sublattice of opens but by taking pairs consisting of an open and a compact saturated subset as basic tokens. This raises an interesting question about the foundations of (localic) topology.

Constructions on topological spaces are modelled in locale theory by free constructions on frames. Our second main theorem states that these free constructions may be performed at the level of prelocales, reinforcing our thesis that prelocales are not overly specialized.

We conclude the paper by investigating the representation of the canonical quasi-uniformity on compact coherent spaces through prelocales. Once again, we discover that the treatment through prelocales is direct, meaningful, and technically easy to manage.

Date received: April 12, 1996

Copyright © 1996 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 # caaf-82.