Abstract:
It is well known that, whilst the closed subsets of a topological
space may be regarded as the formal duals of its open subsets, when
one considers continuous mappings between spaces the correct dual of
the class of open maps is (not the class of closed maps, but) the
class of `relatively compact maps', commonly known as proper maps.
In the late 1940's Alfred Tarski, in collaboration with J.C.C.
McKinsey, showed that much of general topology could be reduced to
algebra by the use of a notion of `formal space' based on lattices
of closed sets; this was an important forerunner of the modern
theory of locales (or frames), although workers in locale theory
traditionally take open-set lattices as the primitive notion. In
1994, Japie Vermeulen in a ground-breaking paper `rediscovered the
closed-set lattices', and showed that one could use them as the
basis for a completely formal duality between open and proper maps
of locales, whereby results proved for one class could be easily
translated into results about the other. More recently, Vermeulen
collaborated with Ieke Moerdijk in writing an AMS Memoir, published
just three months before his untimely death in February 2001, which
investigated the theory of proper maps of toposes, and showed that
all the major results of the Joyal--Tierney theory of open maps have
their `proper' duals. The proofs, however, are not as yet formally
dual; in general it is significantly harder to prove the results
about proper maps. In this talk, after describing the key results on
open and proper maps of locales and toposes, I shall offer some
speculations on what one might take as `the closed-set lattice of a
topos', in the hope of setting up a formal duality at this level.