SAMS Subject Classification Number: 11Number

Generalized inductive limit topologies (sometimes called mixed topologies) occur surprisingly often in the theory of topolgical vector spaces. In its most general form, such a topology is the finest vector topology coinciding with a given vector topology on all the sets of a family of “bounded” sets (a bornology). In the context of vector lattices, many Mackey topologies are mixed topologies.

In the theory of convergence spaces, there is a similar notion of an inductive limit, or mixed convergence, often referred to as specified sets convergence. There are convergences which are, in general, not derived from a topology that are of this type; an example is order convergence in a vector lattice. We show that a mixed convergence derived from a topological convergence need not itself be topological, and investigate the necessary conditions for a mixed convergence to be topological.

Convergences are usually defined in terms of filters, but there is a recently developed parallel theory in terms of nets which can sometimes be used to simplify arguments.

This is joint work with Michael O’Brien, Vladimir Troitsky and Jan Harm van der Walt.