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From: Pollux on 4 Jun 2010 12:55
Hi-

I'm only a beginner in topology, so please no flames... Here is my question: if almost all reals are not rationals, should it be the case that for any two rationals, there is an infinity of reals "between" them? But Q is dense in R, so I can always find a rational as close to any real as I want, there are no "gaps". What's wrong with my reasoning?

Also, if I can always find a rational as close as I want to a real, isn't Q somehow in bijection with R? Again, there must be something wrong with my naive ideas here.

Again, I'm sure the argument is wrong, I'm just trying to reconcile it with the right notions of density/connectedness/separation as found in topology.

Thanks!

Pollux
From: A N Niel on 4 Jun 2010 17:28
In article
<1728676549.280893.1275684950560.JavaMail.root(a)gallium.mathforum.org>,
Pollux <frank.astier(a)gmail.com> wrote:

> Hi-
>
> I'm only a beginner in topology, so please no flames... Here is my question:
> if almost all reals are not rationals, should it be the case that for any two
> rationals, there is an infinity of reals "between" them?

This is a non-sequitur. The "if" neither implies nor denies the
"should". It *is* the case that there is an infinity of reals between
any two rationals. But this is not because (nor in spite of) the fact
that almost all reals are not rationals.

> But Q is dense in R,
> so I can always find a rational as close to any real as I want, there are no
> "gaps".

This is correct.

> What's wrong with my reasoning?

So far, you have nothing wrong. You are making correct assertions.

>
> Also, if I can always find a rational as close as I want to a real, isn't Q
> somehow in bijection with R?

Not according to the definition of "bijection". There is no use for
"somehow" in mathematical reasoning.

> Again, there must be something wrong with my
> naive ideas here.

The only thing wrong is that you think these things conflict with each
other.

>
> Again, I'm sure the argument is wrong, I'm just trying to reconcile it with
> the right notions of density/connectedness/separation as found in topology.
>
> Thanks!
>
> Pollux
From: Pollux on 4 Jun 2010 13:54
Thanks.

But, and again, this is going to sound stupid I'm sure, if there is an infinity of reals between any two rationals, let's say a and b, does it mean there is a real x in [a,b], and there is an epsilon such that no rational is inside ]x-eps, x+eps[?

Pollux
From: Pollux on 4 Jun 2010 13:58
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
From: Arturo Magidin on 4 Jun 2010 18:05
On Jun 4, 4:54 pm, Pollux <frank.ast...(a)gmail.com> wrote:
> Thanks.
>
> But, and again, this is going to sound stupid I'm sure, if there is an infinity of reals between any two rationals, let's say a and b, >does it mean there is a real x in [a,b],

Yes.

> and there is an epsilon such that no rational is inside ]x-eps, x+eps[?

No.

Say you are taking a and b, a<b, both rational. Yes: there is real x
in [a,b]; even an *irrational* x (which I guess is what you meant).
But why should there be an e>0 such that there is no rational in (x-e,x
+e)? It's not even true even if you restrict yourself to rationals
and forget about all those pesky irrationals! Given any two distinct
rationals, there is always a rational strictly between them. So you
can find x_1 in (a,b) which is rational; and you can find x_2 in
(a,x_1) and x_3 in (x_1,b) which are both rationals. And you can find
x_4 in (a,x_2), and x_5 in (x_2,x_1), and x_6 in (x_1,x_3), and x_7 in
(x_3,b) which are all rational. Etc.

Since there is no "next" rational after a, there is no reason why you
would have that epsilon.

--
Arturo Magidin
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