local compactness in four kinds of topology

local compactness in four kinds of topology

[Paul Taylor]

     Local Compactness and Bases in various formulations of Topology

                www.paultaylor.eu/ASD/loccbv

I would like to ask a specialist question before telling you about
this draft paper.

In the theory of continuous lattices there is a result due to Jimmie
Lawson that says that, if  a<<b  but not  a<<c  then there is a
Scott-open filter that contains  b  but not  c.   The idea is to
iterate the interpolation property:  a<<...<<b2<<b1<<b0=b.  Of
course this requires Dependent Choice.

Does anyone have an example of a topos without Dependent Choice
containing a locale without enough Scott-open filters like this?

-----------------

I have posted a longer version of this announcement on
"Constructive News", addressed to Formal Topologists:

https://groups.google.com/forum/#!topic/constructivenews/i1YWESQPh0Y

A basis for a locally compact space is given by a family of pairs of
subspaces, one open and the other compact.  This paper gives the
complete axiomatisation of the relation on indices that says when a
basic compact subspace is covered by a finite set of basic open ones.
It is a completely rewritten version of one I circulated last year
called "A Concise Presentation for Locally Compact Spaces" and
in particular now also characterises continuous functions.

This work (already in last year's version) is innovative in that
the axiomatisation does not require the basis to be closed under
finite unions and intersections, as had been done in previous
work by Achim Jung and Philipp Sunderhauf and by me.  Therefore
the leading example of a basis - balls in a metric space - is
included.   This step shows a lot of the features of interval
analysis, but generalised from R to locally compact spaces.

An equivalence of categories is proved in Point-Set Topology,
Locale Theory, Formal Topology and Abstract Stone Duality.

It turns out that, even though one can describe the points,
open sets and basic compact sets explicitly, in order to prove
correctness in the classical case it is necessary to go via
Locales and Formal Topology.

Doing this in all four formulations of general topology and their
associated logical foundations provides a setting in which to
examine how they differ in logical strength.  We can say precisely
that a continuous function according to a particular definition
is a certain relation between sets that are definable in the
corresponding logical system.


Paul Taylor




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