The pseudocomplement in pointfree topology.
by
Anneliese Schauerte
University of Cape Town

Frame (or locale) theory is a way of approaching topological spaces that makes the collection of open sets the basic notion under consideration; the underlying set is of lesser importance. Since finite intersections and arbitrary unions of open sets are open, the open sets of a topological space form a complete lattice, in which finite intersections distribute over arbitrary unions. Such a lattice is called a frame. This simple change of viewpoint has many interesting consequences. It is well known that the Tychonoff Product Theorem (``any product of compact spaces is compact") is equivalent to the Axiom of Choice. Not so in frame theory - the corresponding result is true without any choice assumptions. Often, when frames and spaces differ, the situation in the former is better - for example, coproducts of regular frames preserve the Lindelof property; product of regular spaces do not. These ideas can also be approached from a different direction, beginning with uniform structures. Any uniform space has a natural topology underlying it; any quasi-uniform space (drop the symmetry axiom) has two natural underlying topologies. These bitopological spaces have their frame counterparts also; they are called biframes, and will be the particular topic of this talk. The pseudocomplement is a useful tool (arising in notions like regularity, booleanness and rimcompactness), which illustrates the similarities and the differences between frames and biframes.

Date received: May 23, 2002