Mimimal uncountable well ordered set, every section of which is countable

topsquark

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Aug 27, 2012
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I was just reading an old Topology text and it is interesting how the method of presentation is rather different from the more modern texts.

I ran into an interesting comment on the definition of the ordinal numbers. It used \(\displaystyle \Omega\) to define the ordinals and goes on to define the cardinals in a similar fashion. I must say I rather liked that presentation better then in my other texts.

This brings up a question that I've had for a while and I've managed to ask it on every new site I've gotten on. You all have some advanced members and questions so perhaps I should ask here, too: Is there an explicit representation of \(\displaystyle \Omega\)? I've never found one and it drives me a bit crazy because I think I should be able to construct it, since the concept of it is fairly straightforward. But I can't.

Thanks!

-Dan
 

pka

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Jan 29, 2005
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Dan, can you give some idea how far back the text you found go. Here is my situation. I spent three & half years studying point set topology (Moore spaces) with one of Moore's students( R L Moore if the god-farther of American topology) If you doubt that just take a look at his PhD students.
Here is what Keith Devlin has to say about MOORE
 

topsquark

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Aug 27, 2012
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The text is "General Topolgy" by Willard (1970). Looks like it's a first edition. It's younger than I had thought. Based on the wear of the cover I thought it went back to the 50's. Still it is a bit different than I'm used to.

-Dan
 
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