[FOM] The "long line".
eilya497 at 013.net
Fri Jun 22 15:38:33 EDT 2007
On Fri, Jun 22, 2007 at 03:44:41PM +1200, 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) ?
> (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?
If the ordinal is countable or equals $\omega_1$, yes. If it is larger,
the point corresponding to $\omega_1$ cannot be automorphically mapped to zero.
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