NetNews Usenet Archive 1992 #23/NN_1992_23.iso/spool/sci/math/13305

Internet Message Format | 1992-10-16 | 1.6 KB

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From: [e-mail] (Jan Willem Nienhuys)
Newsgroups: sci.math
Subject: Re: What a weird group!!!
Message-ID: <[e-mail]>
Date: 16 Oct 92 13:25:24 GMT
References: <[e-mail]>
Sender: [e-mail]
Reply-To: [e-mail]
Organization: Eindhoven University of Technology, The Netherlands
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In article <[e-mail]> [e-mail] (John C. Baez) writes:
>No, I'm not referring to sci.math.  I'm referring to the Bohr
>compactification of the real numbers, something I just learned about.
>This is, of course, due not to Niels Bohr but to his brother Harald.
>
>From an exercise in a book:  The Bohr compactification of the real
>numbers is a compact topological group which contains the real numbers
>as a dense set.  The embedding of the reals into the Bohr group is not a
>homeomorphism.
...
>But what does this group *look like*??  That's what I'd like to know!!

An easier (;-)) way is to consider the reals with discrete topology.
Algebraically, that's an enormous direct sum of copies of Q, and Q of
course is a union of copies of Z, or a sum of Q_p over all p.
Now take the dual (the set of all "continuous" characters).
That is a huge direct product of duals of Q, each a projective limit
of circles;  you may therefore think of (1) an enormous torus (with
a for each element of a Hamel basis of R over Q a coordinate) (2)
many such tori wrapped over each other so that a cross-section looks
like a complicated Cantor discontinuum.

I'm afraid I can't give you a much closer idea.  Hope this helps.
JWN