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Right order and left order are REVERSED. I fixed it and some tool/bot Razorflame reverted them. —Preceding unsigned comment added by 140.247.149.65 ( talk) 21:08, 24 October 2009 (UTC)
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)
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)
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. —Preceding unsigned comment added by 81.179.130.150 ( talk • contribs)
It says: "subspace topology is always finer than the induced order topology;" I think that this is in the wrong direction though; i.e. it's never finer, but could be coarser. I'm not completely sure, but Section 4.1 problem 9 on page 118 of Folland's Real Analysis asks you to prove this; and I'm pretty sure my proof is correct, although it's somewhat messy. I don't trust the example given after it either; but I think it would be better if somebody more acquainted with the subject looked it over rather than me messing with it.
Greeneggsnspam ( talk) 08:28, 14 November 2008 (UTC)
The example seems to be messed up. I think the author meant the set .
Anonymous 12:42, 12 March 2011 (UTC) —Preceding unsigned comment added by 89.27.245.109 ( talk)
Why the definition is restrictive to totally ordered sets? The order topology for partially ordered sets is a nice thing to build interesting counterexamples. Albmont ( talk) 18:32, 30 April 2009 (UTC)
There are multiple sections regarding ordinals, which have some overlap. It would be worthwhile to merge these sections and remove redundancies. -- Jordan Mitchell Barrett ( talk) 05:26, 2 May 2020 (UTC)
The problem with the case when is a singleton, is that we do not get a base/subbase. We only can choose and we get . So this system is not cover of X, which is one of necessary conditions of a base. (And in many sources, this is required also form a subbase.) Although this special case is probably not important enough to be explicitly mentioned. -- Kompik ( talk) 20:09, 4 November 2020 (UTC)
I agree that there is a problem: the topology defined by the rays and the topology defined by the open intervals are different, since, for example, in , the first is the discrete topology and the second is the undiscrete topology. — Preceding unsigned comment added by 2001:861:3743:6E00:47F:FC94:30B1:29FF ( talk) 12:58, 17 December 2021 (UTC)
The section titled An example of a subspace of a linearly ordered space whose topology is not an order topology is utterly opaque. I re-read it several times and keep getting stuck at the sentence Since {-1} is open in Z, there is some point p in M such that the interval (-1, p) is empty. I think this is trying to appeal to some separation axiom, but, well, I can't really tell. This section was added by a single user in 2006, and it is that user's only contribution to wikipedia. I dunno. There must be some easier way to state this example... 67.198.37.16 ( talk) 06:28, 27 November 2023 (UTC)
![]() | This article is rated Start-class on Wikipedia's
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Right order and left order are REVERSED. I fixed it and some tool/bot Razorflame reverted them. —Preceding unsigned comment added by 140.247.149.65 ( talk) 21:08, 24 October 2009 (UTC)
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)
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)
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. —Preceding unsigned comment added by 81.179.130.150 ( talk • contribs)
It says: "subspace topology is always finer than the induced order topology;" I think that this is in the wrong direction though; i.e. it's never finer, but could be coarser. I'm not completely sure, but Section 4.1 problem 9 on page 118 of Folland's Real Analysis asks you to prove this; and I'm pretty sure my proof is correct, although it's somewhat messy. I don't trust the example given after it either; but I think it would be better if somebody more acquainted with the subject looked it over rather than me messing with it.
Greeneggsnspam ( talk) 08:28, 14 November 2008 (UTC)
The example seems to be messed up. I think the author meant the set .
Anonymous 12:42, 12 March 2011 (UTC) —Preceding unsigned comment added by 89.27.245.109 ( talk)
Why the definition is restrictive to totally ordered sets? The order topology for partially ordered sets is a nice thing to build interesting counterexamples. Albmont ( talk) 18:32, 30 April 2009 (UTC)
There are multiple sections regarding ordinals, which have some overlap. It would be worthwhile to merge these sections and remove redundancies. -- Jordan Mitchell Barrett ( talk) 05:26, 2 May 2020 (UTC)
The problem with the case when is a singleton, is that we do not get a base/subbase. We only can choose and we get . So this system is not cover of X, which is one of necessary conditions of a base. (And in many sources, this is required also form a subbase.) Although this special case is probably not important enough to be explicitly mentioned. -- Kompik ( talk) 20:09, 4 November 2020 (UTC)
I agree that there is a problem: the topology defined by the rays and the topology defined by the open intervals are different, since, for example, in , the first is the discrete topology and the second is the undiscrete topology. — Preceding unsigned comment added by 2001:861:3743:6E00:47F:FC94:30B1:29FF ( talk) 12:58, 17 December 2021 (UTC)
The section titled An example of a subspace of a linearly ordered space whose topology is not an order topology is utterly opaque. I re-read it several times and keep getting stuck at the sentence Since {-1} is open in Z, there is some point p in M such that the interval (-1, p) is empty. I think this is trying to appeal to some separation axiom, but, well, I can't really tell. This section was added by a single user in 2006, and it is that user's only contribution to wikipedia. I dunno. There must be some easier way to state this example... 67.198.37.16 ( talk) 06:28, 27 November 2023 (UTC)