Tuples admit a topological interpretation:
How do morphisms come out?
Enriched with the following extra structure, they are covering
systems, in the sense of MacLane and Meordijk (``Sheaves in Geometry
and Logic'', pp. 524 - 527. This extra structure is a partial order
representing inclusion between neighbourhoods. (This can be
represented in various ways: see section 7.)
One can now define a notion of `point'. A point is what is common to a
certain kind of set of neighbourhoods, satisfying two conditions. The
first condition is that the set should be a filter: the set of
neighbourhoods should be directed to the left and closed to the right
(monotone) in .
(This means that for any pair of neighbourhoods
there is another included in them both, and anything that includes a
neighbourhood in the set is also in the set.) The other condition
expresses the idea that a filter contains arbitrarily small
neighbourhoods, and so `identifies a point'. The condition is that
for any tree of basic coverings of any neighbourhood in the filter,
the set of neighbourhoods at the leaves of that tree has non-empty
intersection with the filter.
Warning: the following is probably erroneous.
I am no longer
sure one can identify points with functions
.
For instance, if one
takes A with two elements
,
and as basic coverings, only the
singletons
covering a, and
covering 1, then
a point is determined by the proposition which says it has a as
an element. Hence the collection of points is like the collection of all
propositions, and hence cannot be a set.
One may (at least sometimes) identify points with certain countable sequences
of neighbourhoods, namely those satisfying the following conditions: