categories: bases for locally compact spaces

[Paul Taylor <pt@dcs.qmul.ac.uk>]

			Bases for spaces

Has anyone come across this was of seeing locally compact spaces?

We say that a system B_n of VECTORS in a VECTOR SPACE is a BASIS
if any other vector U can be expressed as a sum of scalar multiples
of basic vectors.

Likewise, we say that a system B_n of OPEN SUBSETS of a TOPOLOGICAL
SPACE is a BASIS if any other open set U can be expressed as a "sum"
(disjunction) of basic opens.

How do we find out which basis elements contribute to the sum,
and by what scalar multiple?

By applying the DUAL BASIS  A_n  to the given element U, ie  A_n.U

Then     U  =  sum_n  A_n.U  *  B_n

where
 - "sum" means linear sum, disjunction or existential quantification,
 - "scalars" in the case of topology range over the Sierpinski space,
 - the dot denotes
   - inner product of a dual vector with a vector to yield a scalar,
   - that U is an element of the family A_n, or
   - lambda application,
 - `*'  denotes multiplication by a scalar or a vector,  or conjunction.

In topology, each  A_n  is a Scott-open family of open subsets.

Unfortunately, in general it need not be a filter.  In the case where
it is a filter, it corresponds by the Hofmann--Mislove theorem to a
compact saturated subspace K_n.  Then
    if   A_n.U  is true,  B_n < K_n < U
where < denotes non-strict subset inclusion.   In this case
K_n  provides a basis of compact neighbourhoods.

Anyway, if  A_n.B_n  is true then  B_n  is itself compact open,
and more generally  A_n.U  implies  B_n << U   ("way below").

Thus,  A_n.U  is defined in terms of
  -  the traditional definition of local compactness
     as the existence of a compact subspace between B_n and U
  -  continuous lattices as  B_n << U.

As you might have guessed,  I discovered this during my current work
on domain theory in Abstract Stone Duality,  and have lambda-terms for
A_n and B_n.

Indeed, if such a basis exists for a space X then   X  is a Sigma-split
subspace of  Sigma^N   (in the sense of my recently-announced paper
"Subspaces in Abstract Stone Duality"),  where
   i: X -> Sigma^N          by  x  |-->  lambda n. B_n.x
   I: Sigma^X -> Sigma^2 N  by  U  |-->  lambda psi. some n. A_n.U * psi.n
make  Sigma^X  a retract of  Sigma^2 N.

Conversely, given i and I,we define a basis indexed, not by numbers
themselves, but by lists (k) of numbers, by
   B_k.x  =  all n in k.  (i.x).n
   A_k.U  =  (I.U).(lambda n.n in k)

Paul Taylor
(no academic affiliation)

ASD web page:   http://www.dcs.qmul.ac.uk/~pt/ASD