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From: Pollux on 4 Jun 2010 14:18
Thanks! Yes I realized that as soon as I had written the post :-(

Pollux
From: Ray Vickson on 4 Jun 2010 18:24
On Jun 4, 2:58 pm, Pollux <frank.ast...(a)gmail.com> wrote:
> Oh, I see! There is an infinity of reals between rationals a and b, but also an infinity of rationals, right? (that must sound trivial)
>
> Yes, I do have a problem of intuition with those things being compatible. I'm sure they are, and I don't care about intuition and its pitfalls. I'm just trying to reconcile "Q being dense in R" and "having an infinity of reals between two rationals" with the correct topological notions.
>
> Pollux

Between two rationals there are infinitely many rationals and
infinitely many reals. Between two reals there are infinitely many
rationals and infinitely many reals.

Nevertheless, the rationals are countable and the reals are not. Note:
the usual ways of listing all the rationals do not correspond to their
order by magnitude, so rational 'a' can come before rational 'b' on
the list even though 'a' comes after 'b' on the number line.

R.G. Vickson
From: Pollux on 4 Jun 2010 14:22
I'm just trying to shelve "intuition" and answer:

1. is Q dense in R: yes,
2. is Q connected: not sure yet,
3. is Q separated: not sure yet,
4. is Q locally compact: no, still working out the proof.

"Intuition" really doesn't help when you consider that Q is dense in R but there are infinitely many more reals than irrationals. It's kind of surprising to me.

Answers to 2 and 3?

Pollux
From: Arturo Magidin on 4 Jun 2010 21:09
On Jun 4, 5:22 pm, Pollux <frank.ast...(a)gmail.com> wrote:
> I'm just trying to shelve "intuition" and answer:
>
> 1. is Q dense in R: yes,
> 2. is Q connected: not sure yet,

No; in fact, Q is "totally disconnected": given any a,b in Q, a=/=b,
there you can express Q as (Q/\A) \/ (Q/\B), where A and B are open
intervals, A/\B is empty, a is in A, and b is in B.

> 3. is Q separated: not sure yet,

See above.

> 4. is Q locally compact: no, still working out the proof.
>
> "Intuition" really doesn't help when you consider that Q is dense in R but there are infinitely many more reals than irrationals. It's kind of surprising to me.
>
> Answers to 2 and 3?

"Yes" and "Yes".

--
Arturo Magidin
From: Pollux on 4 Jun 2010 18:12
Thanks! That really helps.

Pollux
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