Computably Based Locally Compact Spaces

Paul Taylor

This is part of the core theory of Abstract Stone Duality for locally compact spaces. It was published in Logical Methods in Computer Science 2 1–70 in 2006 (BibTeX).



Abstract

ASD (Abstract Stone Duality) is a re-axiomatisation of general topology in which the topology on a space is treated, not as an infinitary lattice, but as an exponential object of the same category as the original space, with an associated lambda-calculus. In this paper, this is shown to be equivalent to a notion of computable basis for locally compact sober spaces or locales, involving a family of open subspaces and accompanying family of compact ones. This generalises Smyth’s effectively given domains and Jung’s Strong proximity lattices. Part of the data for a basis is the inclusion relation of compact subspaces within open ones, which is formulated in locale theory as the way-below relation on a continuous lattice. The finitary properties of this relation are characterised here, including the Wilker condition for the cover of a compact space by two open ones. The real line is used as a running example, being closely related to Scott’s domain of intervals. ASD does not use the category of sets, but the full subcategory of overt discrete objects plays this role; it is an arithmetic universe (pretopos with lists). In particular, we use this subcategory to translate computable bases for classical spaces into objects in the ASD calculus.



Acknowledgements

This paper evolved from [], which included roughly Sections ??–?? and ?? of the present version. It was presented at Category Theory and Computer Science 9, in Ottawa on 17 August 2002, and at Domains Workshop 6 in Birmingham a month later. The characterisation of locally compact objects using effective bases (Section ??) 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 [, 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.

BibTeX entry

   @article{TaylorP:comblc,
      author   = {Taylor, Paul},
      title    = {Computably Based Locally Compact Spaces},
      journal  = {Logic Methods in Computer Science},
      year     = 2006,
      volume   = 2, pages = {1--70},
      url      = {PaulTaylor.EU/ASD/comblc},
      amsclass = {54D45, 03D45, 06B35, 54D30, 68N18}}

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Table of contents


Introduction


1. Locally compact sober spaces


2. Locally compact locales


3. Axioms for abstract Stone duality


4. Sets, unions and bases


5. Compact subspaces


6. Effective bases


7. Sigma-split subobjects


8. Every definable object has a basis


9. Basic corollaries


10. Primes and nuclei


11. The way-below relation


12. Domain theory in ASD


13. The lattice basis on ΣN


14. From the basis to the space


15. The points of the new space


16. Morphisms as matrices


17. Relating the classical and term models


References

This document was translated from LATEX by HEVEA.