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Acknowledgements
Acknowledgements
This paper evolved from [G–],
which included roughly Sections
5–8 and 11 of the present version.
It was presented at
Category Theory and Computer Science9,
in Ottawa on 17 August 2002,
and at Domains Workshop6 in Birmingham a month later.
The characterisation of locally compact objects using effective bases
(Section 6) had been announced on categories
on 30 January 2002.
The earlier version also showed that any object has a filter basis,
and went on to prove Baire’s category theorem, that the intersection
of any sequence of dense open subobjects (of any locally compact overt object)
is dense.
These arguments were adapted from the corresponding ones in the theory
of continuous lattices [GHK++80, Sections I 3.3 and 3.43].
I would like to thank
Andrej Bauer,
Martƒn Escardó,
Peter Johnstone,
Achim Jung,
Jimmie Lawson,
Graham White
and the CTCS and LMCS referees for their comments.
Graham White has given continuing encouragement
throughout the abstract Stone duality project,
besides being an inexhaustible source of mathematical ideas.
This research is now supported by UK EPSRC project GR/S58522,
but this funding was obtained in part on the basis of the work in this paper.
Apart from
5.19,
12.13, 12.14,
15.3, 15.5, 16.15 and 17.6,
the work here was carried out during a period of unemployment,
supported entirely from my own savings.
However, I would have been unable to do this without
the companionship and emotional support of my partner, Richard Symes.