Structured frames

Doctoral Thesis


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University of Cape Town

Ehresmann in 1959 first articulated the view that a complete lattice with an appropriate distributivity property deserved to be studied as a generalized topological space in its own right. He called the lattice a local lattice. Here is the distributivity property: x ∧ Vxα = V(x∧xα). A map of local lattices should preserve finite meets and arbitrary joins (and hence top and bottom elements). Dowker and Papert introduced the term frame for a local lattice and extended many results of topology to frame theory. At the 1981 international conference on categorical algebra and topology at Cape Town University a suggestion was made that a study of "uniform frames" (whatever they might be) would be an appropriate and useful start to a project concerned with examining, from a lattice theoretical point of view, the many topological structures which have gained acceptance in the topologist's arsenal of useful tools. It was felt that many of the pre-requisites for such a study had been established, and in fact one of the themes of the conference was the growing role of lattice theory in topology. The suggestion was eagerly accepted, and this thesis is the result.

Bibliography: pages 141-144.