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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)[reply]

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)[reply]

Section moved in. JRSpriggs 08:33, 30 September 2006 (UTC)[reply]

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)

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

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)[reply]

The example seems to be messed up. I think the author meant the set ${\displaystyle \{0\}\cup \{1/n:n\in \mathbb {N} \}}$.

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)[reply]

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)[reply]

The problem with the case when ${\displaystyle X=\{x\}}$ is a singleton, is that we do not get a base/subbase. We only can choose ${\displaystyle a=b=x}$ and we get ${\displaystyle (-\infty ,b)=(a,b)=(a,\infty )=\emptyset }$. 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)[reply]

Maybe one way to solve it would be to use the sets (a,M] and [m,b) where ${\displaystyle m=\min X}$ and ${\displaystyle M=\max X}$. However, this only works if the set has smallest/largest element. So with this approach, we would not need have singleton as a separate case. But the disadvantage is that the definition would be split into several cases, depending on whether the set $X$ has a smallest/largest element. --Kompik (talk) 05:25, 5 November 2020 (UTC)[reply]

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 ${\displaystyle \{0,1\}}$, 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)[reply]

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)[reply]

The article says The closed sets of a limit ordinal α are just the closed sets in the sense that we have already defined, namely, those that contain a limit ordinal whenever they contain all sufficiently large ordinals below it. but is it correct? For example for ⁠${\displaystyle \alpha =\omega ^{2}}$⁠, the closed set {0, 2, 4, 6, ..., ω} (that is, including all finite even ordinals and ω) is a set that is not closed without ω despite that it does not contain all sufficiently large ordinals below ω. —Quantling (talk | contribs) 16:42, 2 July 2025 (UTC)[reply]

I made an edit. If it needs further work, please edit further or discuss here. Thank you —Quantling (talk | contribs) 18:19, 3 July 2025 (UTC)[reply]