Steve Vickers

Dept of Pure Maths, Open University

s.j.vickers@open.ac.uk

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.