Re: Nice little total order problem.

From: William Elliot (marsh_at_privacy.net)
Date: 03/17/05 

Date: Wed, 16 Mar 2005 23:25:08 -0800

On Wed, 16 Mar 2005 bof@sunflower.com wrote:

> Bill Taylor wrote:
>> Here is a nice little problem that came up in the FoM mailing list.
>> 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.)

Ok.

> 1 of cardinality 1.

Ok.

> 4 of cardinality aleph_0.

Ok.

> 5 of cardinality aleph_1.

Hm.

> A total of 11, or 10 if A has to be nonempty.
>
Ok.