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There exists a well-ordered set $A$ having a largest element $\Omega$ such that the section $S_\Omega$ of $A$ by $\Omega$ is uncountable but every other section is countable.

Can anyone make me understand this theorem?

I can not understand what Mukres has tried to say in his Topology's Book.. I can not visualize what Mukresh has said in the proof. I can not understand why we always get an element $\Omega$ such that the section less than $\Omega$ is countable. What ensures the availability of $\Omega$ in a uncountable well ordered set?

Can anyone make me understand with proper example?

Thank you in Advance.

cmi
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  • How do you visualize very large countable well orders? – Asaf Karagila Dec 28 '18 at 07:45
  • "very large " countable well order means? @AsafKaragila – cmi Dec 28 '18 at 07:47
  • You have answered your own question perfectly in the first sentence of your post. There is only one example (up to isomorphism) and you have described it exactly. – bof Dec 28 '18 at 07:47
  • I am trying to prove that statement..@bof – cmi Dec 28 '18 at 07:49
  • The set $A$ you described is not quite the minimal uncountable well-ordered set, that is the section $S_\Omega$ of $A$. – bof Dec 28 '18 at 07:50
  • I meant I can not visualize how there exists a number $\Omega$ which is greater than only a countable number of elelments.@bof – cmi Dec 28 '18 at 07:52
  • Your question was not about how to prove that statement, was it? – bof Dec 28 '18 at 07:53
  • Is $S_\Omega$ not minimal?@bof – cmi Dec 28 '18 at 07:53
  • Yea...I want to prove the statement. Mukresh proved it . I could not understand it.@bof – cmi Dec 28 '18 at 07:54
  • Yes, $S_\Omega$ is a minimal uncountable well-ordered set. The set $A=S_\Omega\cup{\Omega}$ is not minimal. – bof Dec 28 '18 at 07:55
  • Yea I understood that. But can you tell me about the proof?@bof – cmi Dec 28 '18 at 07:57
  • I meant I can not visualize how there exists a number $\Omega$ which is greater than only a countable number of elelments.@AsafKaragila – cmi Dec 28 '18 at 08:00
  • Why you are being so unkind to my question? I did not find any similarity with that link. Please explain how those questions are similar?@AsafKaragila – cmi Dec 28 '18 at 08:01
  • Please reopen the question or explain how they are similar? You can not abuse your power. Please...I am really in need to understand this.@AsafKaragila – cmi Dec 28 '18 at 08:04
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    Aren't you asking for a proof that uncountable ordinals exist? Why is there "kindness" involved here? It has nothing to do with kindness. Calling Munkres by Mukresh repeatedly is unkind, by the way. – Asaf Karagila Dec 28 '18 at 08:06
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    https://math.stackexchange.com/questions/2745911/what-is-minimal-uncountable-well-ordered-set might be a better duplicate, or perhaps in conjunction with the current one. – Asaf Karagila Dec 28 '18 at 08:27

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