Talk:Order topology
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[edit] Previous comments
show that IxI in the dictionary order topology is locally connected but not locally path connected.what are the components of this space?
I don't understand IxI. Do you mean the cartesian product of 2 intervals?
MFH 17:30, 8 Apr 2005 (UTC)
[edit] Move section here from "Ordinal number"?
If no one raises an objection in the next few days, I will move the section Ordinal number#Topology and ordinals from that article to this one. My objective is to make that article shorter (and this one longer). JRSpriggs 05:22, 25 September 2006 (UTC)
- Section moved in. JRSpriggs 08:33, 30 September 2006 (UTC)
[edit] Continuous functions from omega-1 to the reals
I think there should be a modification:-
It is also worthy of note that any continuous increasing function from ω1 to R (the real line) is eventually constant
Because if it just has to be continuous I can define it to be 1 at successors of limit ordinals and 0 elsewhere. —The preceding unsigned comment was added by 81.179.130.150 (talk • contribs).
- No. The original formulation, "... any continuous function from ω1 to R (the real line) is eventually constant ...", is true. The function you specified is not continuous. Consider
it is a limit (rather than the successor of a limit) so you give it the value zero. However, it is the limit of the sequence 
and every element in that sequence is mapped to one. Thus the function is discontinuous. JRSpriggs 06:44, 30 October 2006 (UTC)
