Re: Nice little total order problem.

From: Butch Malahide (bof_at_sunflower.com)
Date: 03/17/05 

Date: 16 Mar 2005 23:40:21 -0800

Stephen J. Herschkorn wrote:
> bof@sunflower.com wrote:
>
> >Bill Taylor wrote:
> >
> >
> >>Consider totally ordered sets A, with the property that:
> >>
> >> For all p,q in A s.t. p < q , the open interval (p,q) is
> >> order-isomorphic to the set of
> >>
> >>
> >rationals, Q.
> >
> >
> >>How many distinct such sets A are there? (i.e. No two
> >>
> >>
> >order-isomorphic)
> >
> >1 of cardinality 0. (Not sure if that one counts.)
> >1 of cardinality 1.
> >4 of cardinality aleph_0.
> >5 of cardinality aleph_1.
> >A total of 11, or 10 if A has to be nonempty.
> >
> >
>
> What are the sets of cardinality aleph1?

One of them is a set of order type eta times omega-one. That is, the
set of all ordered pairs (a,r), where a is a countable ordinal and r is
a rational number, ordered by first differences. In other words, a copy
of the rational line is inserted between each countable ordinal and its
successor.

You get a second one by deleting the first element of that one.

You get two more by turning those things around.

The fifth is left as an exercise.

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