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January 10th, 2017, 11:56 PM  #1 
Senior Member Joined: May 2015 From: Arlington, VA Posts: 136 Thanks: 19 Math Focus: Number theory  Separation by randomizing
Can at least two consecutive real numbers remain consecutive (by Brouwer's fixed point theorem) after their real number line is randomized? If not, wouldn't all randomized real numbers approach infinite separation from their original position? 
January 11th, 2017, 03:57 AM  #2 
Math Team Joined: Jan 2015 From: Alabama Posts: 2,210 Thanks: 555 
I cannot make heads or tails out of this! First, what do you mean by "consecutive" real numbers? Second, what do you mean by "their" real number line (I thought there was only one real number line!). Finally, what do you mean by "randomizing" the real number line?

January 11th, 2017, 06:44 AM  #3 
Senior Member Joined: Dec 2012 Posts: 890 Thanks: 20  Two consecutive real numbers ?

January 11th, 2017, 08:53 AM  #4 
Math Team Joined: Dec 2013 From: Colombia Posts: 6,343 Thanks: 2082 Math Focus: Mainly analysis and algebra 
I too have trouble assigning much meaning to terms in the original post, but it is perfectly possible for a randomly selected permutation of an ordered set to have multiple numbers of fixed points. Indeed one possible selection is the identity permutation in which every point is a fixed point. I get the impression that the OP demostrates an at best hazy understanding of real numbers and of randomisation and possibly also of Brouwer which talks about continuous functions not randomisations and says that there always exists at least one fixed point, not how many there can be. 
January 11th, 2017, 09:52 PM  #5  
Senior Member Joined: Dec 2012 Posts: 890 Thanks: 20  Quote:
I think first you've to talk of Rationals than of the function that is the analitic extension whare your problem lay on. Than you can shake as you want, but you can just choose invertible functions to send the points in the domain onto the new codomain. Random will not means that nobody can assure us that two points of the domain can/will be assinged at the same point on the codomain ?  

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randomizing, separating, separation 
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