[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
the rationals in (0,1) (don't include the minimum). Any isomorphism
countable dense sets extends to an isomorphism between the longish line
> (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
E-MAIL: K.P.Hart at TUDelft.NL PAPER: Faculteit EWI
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