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 \omega^2\!, 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 \langle (\omega \cdot n) + 1 \, | \, 0 < n < \omega \rangle and every element in that sequence is mapped to one. Thus the function is discontinuous. JRSpriggs 06:44, 30 October 2006 (UTC)
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