Some examples of ordered sets that are linear continua include , the set in the dictionary order, and the so-called long line in the dictionary topology. (The third example is a special case of a general result on well-ordered sets and linear continua.)
If is a well-ordered set, then the set is a linear continua in the dictionary order topology.
As a corollary of the preceding , we obtain the result that is in its usual topology, as are the intervals and , where .
If is a linear continuum in the order topology, then every closed interval in is compact.
This is essentially a slightly generalized version of the Heine-Borel Theorem for , and the proof is almost identical. ∎
|Date of creation||2013-03-22 17:17:40|
|Last modified on||2013-03-22 17:17:40|
|Last modified by||azdbacks4234 (14155)|