Generalized topological spaces: not enough opens, so you use sheaves

Steve Vickers
Dept of Pure Maths, Open University

One discovery of sheaf theory has been that sheaves provide an alternative
approach to discussing topology: knowing the sheaves is equivalent to
knowing the topology, and knowing the inverse images of sheaves is
equivalent to knowing a continuous map.

On the face of it this is unappealing, for sheaves are considerably more
complicated than the open subspaces that are normally used. However, the
topos theorists discovered that in some "generalized spaces" there can be
seen structure that is topological in its nature but which can only be
captured using sheaves - there are simply not enough opens.

A fundamental example is the space [set] of sets. By defining its sheaves
suitably, one captures the intuition that a continuous map Y: X -> [set] -
in other words, a continuous set-valued function on X - is exactly a sheaf
over X, Y(x) being the stalk at each point x. The local homeomorphism
property is then seen as a kind of continuity. The opens are insufficient
for this, for there are only enough of them to distinguish between the
empty set and inhabited sets.

The topos theorists also discovered generalized spaces that were
significant in ungeneralized topology but for which the sheaves were easier
to understand than the points - often for constructivist reasons.

Little in my talk will be new. Rather, I shall try to bring out the flavour
of "generalized topology" in some sample generalized spaces, with an
introduction to local homeomorphisms and sheaves. I shall lightly touch on
how generalized spaces can arise from topological groupoids. One
application of this is in finding topological coequalizers: whereas the
coequalizer as ungeneralized space might be trivial, as generalized space
it can retain useful information.

I shall finish with a brief mention of the generalized Priestley duality,
between ordered compact Hausdorf spaces and stably compact spaces. Goubault
has shown the need - in his topological approach to concurrency - to study
"locally" ordered spaces, when dealing with iterative algorithms, and it
seems that any analogue of the Priestley dual would have to be a
generalized space.