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What are these as topological spaces? A proof I'm reading is referring to these as counter examples and I can't figure out what they mean?

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Each of them has a natural linear order, and they are topologized with the order topology, just as $\Bbb R$ is topologized from its natural order. –  Brian M. Scott Mar 19 '13 at 0:21

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The answer is come from here. It is very interesting. So I copy it for you as an answer.

I find pictures to help. The idea here is that $\omega$ is a limit ordinal and tacking on the ordinal $1$ after it is fundamentally different:

omega

omega+1

The picture for $\omega$ has a curved edge which indicates that it is a limit ordinal opposed to being a successor ordinal. When we tack on $1$ to the right of $\omega$ we have this ordinal $\omega+1$ that contains a limit ordinal which is not something that occurs in $\omega$. This means that $\omega$ and $\omega+1$ can't be isomorphic.

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Wow thank you. One last question, why is $\omega$ a closed subset of $\omega+1$? (That's the last claim of the proof I'm trying to understand). –  user39794 Mar 19 '13 at 0:28
    
Because $\{\omega + 1\}$ is an isolated point. –  Paul Mar 19 '13 at 0:33
    
Note that the question was about $\omega_1$ (and its successor), not $\omega$. –  Andreas Blass Mar 19 '13 at 0:55
    
O, yes, however, it is the same. –  Paul Mar 19 '13 at 2:55

These are ordinal spaces. Ordinals are linearly ordered sets which are also well-ordered, namely every subset has a least element (a generalization of the natural numbers with their ordering).

$\omega_1$ is the linear ordering that every initial segment is countable, but the whole set is uncountable. $\omega_1+1$ is the same order, but now with a maximum which is the unique point that has uncountably many points smaller than itself.

Linear orders generate a very nice topology with the order topology, which is what and how we usually topologize ordinals (I think that any other topology would have been explicitly mentioned).

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Thank you! I wasn't sure what they were called so I couldn't even google them. Thanks again! Also yes it did refer to them in the order topology. –  user39794 Mar 19 '13 at 0:21
    
@AllisonCameron , For more information see, 1. S. Willard General topology p10. and 2. The First Uncountable Ordinal –  M.Sina Mar 19 '13 at 7:10
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In addition to Willard's book, see Andy Miller's notes. Also, a lot of results involving ordinal spaces for countable ordinals--results that are difficult to find elsewhere--can be found in Chapter II.8 (Compact $0$-dimensional spaces) of Zbigniew Semadeni's 1971 book Banach Spaces of Continuous Functions. –  Dave L. Renfro Mar 19 '13 at 14:35
    
@DaveL.Renfro , I search for semandeni'S book but i cant find it in googlebooks or anywhere. If u can please Share a link to read online or download :) Tnx –  M.Sina Mar 20 '13 at 15:36
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@M.Sina: About a week ago I posted some comments about one aspect of Sierpinski's book. Since I'm here, I may as well mention another book of possible interest (and getting even further off topic from the original poster's question): V. Kannan's Ordinal Invariants in Topology (1981). I don't know if this book is freely available on the internet, but you can get a good idea of what it is about here. –  Dave L. Renfro Mar 20 '13 at 18:09

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