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 April 3rd, 2017, 08:40 AM #51 Senior Member   Joined: Mar 2015 From: New Jersey Posts: 1,215 Thanks: 91 Some insight into previous post can be gained from: Volume paradox -- Banach–Tarski especially the answer to post #17 therein. Last edited by zylo; April 3rd, 2017 at 08:46 AM.
April 3rd, 2017, 09:22 AM   #52
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Joined: Mar 2015
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Posts: 1,215
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Quote:
 Originally Posted by zylo Transformation of (a,b] to [0,pi/2) H [0,$\displaystyle \infty$) by inversion, shrinkage, and displacement: ..........-pi/2|...........0|....(.......|pi/2..........].... H ..........-pi/2|...........0[...........)|pi/2------------ In the same way, without the inversion, (a,b] H (-$\displaystyle \infty$,0]. In general, using inversion, shrinkage/expansion, and displacement: (a,b] H [a,b) H [0,$\displaystyle \infty$) H [K,$\displaystyle \infty$) H (-$\displaystyle \infty$,0] H (-$\displaystyle \infty$,K] (a,b) H (c,d) H (-$\displaystyle \infty$,$\displaystyle \infty$) (a,b) notH [c,d) a$\displaystyle \neq$b, c$\displaystyle \neq$d, a,b,c,d,K finite. New Topic: Completenes Background, Ref Rudin: Questions: Let X and Y denote intervals. 1) Does X H Y and X complete/incomplete imply Y complete/incomplete? Why? 2) Is any open or half=open interval automatically incomplete? f(x)=k on (a,b)? 3) Is [a,$\displaystyle \infty$) complete? See 4). 4) Every convergent sequence in (-$\displaystyle \infty$,$\displaystyle \infty$) converges to a point in (-$\displaystyle \infty$,$\displaystyle \infty$). R is complete. Why is that such a big deal? PS The reals are both open and closed, by definition of open and closed. Google it.
The first part above on homeomorphism is my response to OP.

I gave relevant extracts from Rudin in order to anchor the discussion, and make it relevant to those who don't have Rudin, without any intent to mislead.
I have long since agreed to the more precise statement that 1/n does not converge in (0,1), but this apparently is the only thing you understand or are willing to discuss.

Please feel free to answer questions with any relevant reference to Rudin, or any other specific reference, regardless of what you think my perception of convergence is, which is irrelevant.

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