[FOM] The "long line".

K. P. Hart k.p.hart at tudelft.nl
Sat Jun 23 04:42:18 EDT 2007

Bill Taylor wrote:
> IIRC, the long line is obtained by starting with a large ordinal,
> (say aleph_1), and inserting a copy of interval  (0,1)  after each point;
> (and extending the order relation in the obvious way).
> (i) If the starting ordinal is a countable one, is the final result
>     order-isomorphic to  [0,1) ?
Yes, this is so because it has a countable dense subset, which is 
isomorphic to
the rationals in (0,1) (don't include the minimum). Any isomorphism 
between the
countable dense sets extends to an isomorphism between the longish line 
and [0,1).
> (ii) If the long line is preceded by a reversed long line, and the two
>      zero-points identified, is the resulting ordered set automorphic
>      with any other point is mappable to the double-zero?
As long as your ordinal is at most omega_1, yes.
By the above the mid point and any other point can be seen to belong to 
a copy of (-1,1),
which is order-homogeneous.
If alpha > omega_1 then the point at omega_1 cannot be isomorphed to 0 
because it has
character aleph_1 (from the left) whereas 0 has countable character.

K. P. Hart

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