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The Classification of Continuous Domains
Date: Tue, 7 Nov 89 15:26:06 GMT
Cc: aj%doc.imperial.ac.uk@nsfnet-relay.ac.uk
I have recently finished the classification of continuous domains,
that is, I have characterized all maximal cartesian closed full subcategories
of the category CONT of continuous directed complete partial orders.
The result for dcpo's with bottom is the following:
Theorem: Every cartesian closed full subcategory of CONT_\bot
is contained in the category of continuous L-domains or in the
category of FS-domains.
Continuous L-domains are continuous dcpo's with bottom in which every
principal ideal is a complete lattice.
FS-domains are a new creation: They are those dcpo's with bottom, for
which there is a directed family (f_i) of continuous functions, the supremum
of which yields the identity function and each of which has the following
`finite separation property': There exists a finite set M_i such that for every
element x there is some m in M with x >= m >= f_i(x).
It is not too hard to see that FS-domains are automatically continuous and
form a cartesian closed category.
If one removes the requirement for a least element then each of the classes
splits into two as in the algebraic case.
The insider will naturally ask where the retracts of bifinite domains (these
are the SFP-objects in the countable case) are located. The answer is that
they are FS-domains. However, I do not know, whether there are FS-domains
which are not retracts of bifinites.
FS-domains can also be characterized in the following way. They are Lawson
compact continuous dcpo's for which the function space is again Lawson compact.
and continuous.
Both continuous L-domains and FS-domains appear naturally in Real Analysis.
The former as the set of connected closed subsets of a connected and locally
connected compact space and the latter as the collection of closed discs in
the plane plus the plane itself. (The ordering in both cases is reversed
inclusion.) These examples were found by Klaus Keimel and Jimmie Lawson.
As a by-product of our construction we can give a proof for Mike Smyth's
original result on the maximality of SFP inside \omega-ALG_\bot, which does
not use the Axiom of Choice.
Achim Jung
Imperial College
LONDON