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From: porky_pig_jr on 11 Apr 2010 16:44
On Apr 11, 8:08 am, William Elliot <ma...(a)rdrop.remove.com> wrote:
> It's claimed Q is a dense subset of R.
>
> How can that be?

By the very definition of "dense", sweet heart. By the very
definition.

> It's got more holes in it
> than it has plugs.  Why isn't it called porous?
>

Probably it is, in the subject of plumbing. But "dense" belongs to the
subject of topology.

> On the other hand, isn't R\Q more dense, less porous than Q?
> It's got many fewer holes and far many more plugs than Q.

The way "dense" is defined, a set can't be "more dense" or "less
dense". It's either dense or not.

> In fact R\Q, having more plugs than holes, gives it the
> appearance of having some substance, of being an irrational
> materialist, instead of being mostly not there like Q, the
> rational ghost.

Irrational materialist? Marx and Engels?

>
> Upon magnifying R with the magnifying metric, min{ 1, m|x - y| }
> where m is the magnifying factor, Q appears as a 1/10 inch
> mesh and R\Q as a 1/100 inch mesh.  Yet neither appear dense.
>

With infinite sets, you can magnify them till cows come home, it won't
have any effect on them.

> Thus the quandary:  what is a real dense set?

Uh, dude, if you don't like the standard definition, why don't you
declare that the classical definition of the dense set was not correct
and you have much better definition which will turn the world of
topology upside down. Consider joining the forces with Inverse 19
dudes.

Yours truly,
PPJ.
From: porky_pig_jr on 11 Apr 2010 16:48
On Apr 11, 9:18 am, The Qurqirish Dragon <qurqiri...(a)aol.com> wrote:

> It appears you are confusing measure with the concept of being dense.

Yeah. That's rather amusing, isn't it? From the topological
perspective both Q and R\Q are dense. Yet, say, on [0,1] the
(Lebesgue) measure of R is 0 and the measure of R\Q is 1, so from the
measure perspective, R is practically non-existent.
From: Ross A. Finlayson on 12 Apr 2010 02:46
On Apr 11, 6:14 am, "George Jefferson" <phreon...(a)gmail.com> wrote:
> "William Elliot" <ma...(a)rdrop.remove.com> wrote in message
>
> news:20100411043202.R39951(a)agora.rdrop.com...
>
>
>
> > It's claimed Q is a dense subset of R.
>
> > How can that be?  It's got more holes in it
> > than it has plugs.  Why isn't it called porous?
>
> > On the other hand, isn't R\Q more dense, less porous than Q?
> > It's got many fewer holes and far many more plugs than Q.
> > In fact R\Q, having more plugs than holes, gives it the
> > appearance of having some substance, of being an irrational
> > materialist, instead of being mostly not there like Q, the
> > rational ghost.
>
> > Upon magnifying R with the magnifying metric, min{ 1, m|x - y| }
> > where m is the magnifying factor, Q appears as a 1/10 inch
> > mesh and R\Q as a 1/100 inch mesh.  Yet neither appear dense.
>
> > Thus the quandary:  what is a real dense set?
>
> Dense is a relative term. Saying Q is dense is meaningless unless Q happens
> to be some people that vist sci.math.
>
> Y is dense in X because, relatively speaking, X - Y is small. This doesn't
> work well for infinite sets though and requires a more rigorous definition.
>
> Q is dense in R because the points of Q are densely populated with the
> points of R - Q. Of course, again, one needs a more rigorous definition
> because R - Q seems to be approximately equal to R.
>
> We can see that in the sense that the |X| - |Y| seems to say something about
> how "dense" Y is in X but the it seems to say too much since we feel that Q
> is much more dense than |R| - |Q| = |R| seems to suggest.
>
> But using the concept of set difference(or really the intersection) doesn't
> seem to lead to any useful definition of "denseness" since comparing
> infinite cardinalities does not lead to any useful result.
>
> So what does dense really mean? Take a point in Q and form an open
> neighborhood(or even closed if you want) that contains points in R - Q. The
> neighborhood will always contain other points of Q and R.
>
> In a sense, we cannot separate out Q from R easily. Where we find points in
> R - Q we find points in Q and vice versa. Hence Q is "dense" in R because Q
> is not sparse in R even though there are "many more points" in R - Q than in
> Q. But note the "many more points" is comparing infinite cardinals which
> doesn't make a whole lot of sense for our concept of denseness.
>
> Take Q+. Q+ is not dense in R because there are points in R that have open
> neighborhoods that do not contain points in Q+. Take N. Same argument. Take
> the set of points {1/N}. Same argument.
>
> For a set to be dense in another set every point must have a dense
> neighborhood. A dense neighborhood is what it seems. Not so much that there
> are about equal number of points, again because we can't compare the
> cardinalities and arrive at a meaningful definition, but because of
> "closeness" between points.
>
> Another way to look at it, a more appropriate definition,
>
> A set Y is dense in a set X if the union of all open non-singleton
> neighborhoods for each point in Y is X.
>
> Hence we are generating a cover of Y in X that completely covers X. Hence Y
> is dense in X. We do not use any idea of cardinality between X and Y.
>
> Let p \in Q.
>
> I_p = (p - e, p + e), I_p \in R,
>
> U(I_p, p \in Q) = R
>
> The set of I_p's form a complete open cover of R and therefor we can say Q
> is dense in.
>
> Alternatively we can simply look at the closure of Q which gives us R which
> is the same idea above.
>
> http://en.wikipedia.org/wiki/Topological_closure
>
> Again, the idea is of "closeness" in a topoligical sense and not of
> cardinality(which is what you seem to be using).
>
> Q is not close to R in cardinality but Q's points densely populate R. Think
> of a black 2D plane and use your intuitive concept of denseness. If you used
> a small red dot on each Q then it will cover all of R(Q^2 and R^2 of course)
> and you would see a red plane. If you did this for N you would just see red
> dots at the points (n,n) but almost the whole plane would be black.
>
> For Q^2, no matter how much you zoomed in you or how small the red dots were
> you would still get a red map. This is what "dense" means. We can never get
> some open set in R that just contains points from R - Q even if the
> cardinality of Q is much smaller than that of R.
>
> One more time ;) If you are comparing cardinalities of the sets then your
> approaching it wrong since it only *works* intuitively for finite sets. When
> you get to infinite sets it falls apart and no longer makes sense.

The rationals are dense in the reals.

The rationals then algebraics (roots of polynomials) then along for
example including Mahler's S, T, U irrationals, sees supersets of the
rationals that are dense proper subsets of the reals. Similarly
subsets of the rationals like the ratios to multiples of 2 or 3 are
easily constructed with "less" density than the rationals.

There's obviously there in terms of their density as a relative
quantity that the rationals are somehow "less" dense than something
like the algebraic numbers. That's where it could even be seen that
density chosen as a word to represent the property carries out more of
its natural definition, where the empty set is the least dense set in
the reals, a set with one element is as dense as any other, and proper
supersets are more dense and/or proper subsets less dense. Building
that into the definition to augment it implicitly is explicit. The
rationals being each of the ratios of the integers sees that to
approximate uniform finite partitions is modeled for analog domains
from integer-domained functions. In ratios these are rates. Then in
applications of methods of exhaustion like the integral calculus, the
domain is real-valued. The rationals particularly are generally
useful for that, there are various considerations on sampling rate in
reconsidering the fundamental theorem of calculus to integrate
furthermore, generally in approximating on approximations in
reconstruction.

Modernly that's solved with the measure of all the rationals being
zero. This is in standard measure theory with its standard countable
additivity (geometric reconstruction) of rational patches in the
limit. This is where one might consider there to be a natural
relation between the density of a set within another set and what the
measure of that set would be, compared to the measure of the parent
set. This is where the measure in the continuous case is approximated
by the discrete case. In the discrete case, adding elements that
would have a density (for example apples on the floor of the bin)
adjusts the quantity some finite amount of the measure (or scalar
value) of the density of apples to air. In the continuous case,
particular discrete events are under the Sorites heap or patch, the
frequency resolution level, because they are so small, they could
happen in different events like quantum events, so the measure of the
density doesn't changed in its scale value, except to say that unless
there were enough discrete events to adjust the measure, it wouldn't
change the other way relative to any other. Still, when enough are
added then the values do adjust.

The rationals as ratios and their space leads to wonder about then the
three body system and for (a, b) a ratio and the ratios is (a/b), but
for (a, b, c) it's (a/b, b/c, c/a). As the number of variables
increases then it's more compact their representation when the
mutually correlated variables are combined. Yet still when there's
only one parameter then any number of rate-related variables can be
written in a form that's like a rational number but is instead for
three instead of two values. Then in for example with how orthogonal
functions (of systems of two functions) work on ratios then new
systems of functions (like the n-gonometric compared to sine/cosine,
for example) work on these values. Then looking at rationals n/d and
how they look like parabolas with the endpoints at the points at X and
Y's infinity, with the three, four, and so on, figuring out how those
are dense and in what is an exercise.

Another key thing about the rationals is in the unique representation
of rationals, prime numbers and so on. For example while have halves
then adding quarters and noting that half of those were already
halves, it's particular when a pre-existing partition is subdivided,
while it's the same as the entire partition being divided the first
time, the partition itself already existed.

The rationals are dense in the reals. Then, the rationals' complement
in the reals is of course the irrationals. Topologically, the same
properties apply to the irrationals and rationals (except for example
where zero is a rational and has toplogical properties). While
thatmay be so, standardly there is some preponderance of the
irrationals in that they're uncountable, because the reals are
uncountable so the irrationals couldn't not be. Still, between any
two irrationals as real numbers, the rationals are dense.

Regards,

Ross Finlayson
From: William Elliot on 12 Apr 2010 08:00
On Sun, 11 Apr 2010, porky_pig_jr(a)my-deja.com wrote:
> On Apr 11, 8:08�am, William Elliot <ma...(a)rdrop.remove.com> wrote:
>> It's claimed Q is a dense subset of R.
>>
>> How can that be?
>
> The way "dense" is defined, a set can't be "more dense" or "less
> dense". It's either dense or not.

mu([0,1] /\ Q) = 0 < 1 = mu(R)

Q is almost nowhere; R is almost everywhere.
Which is denser?

>> In fact R\Q, having more plugs than holes, gives it the
>> appearance of having some substance, of being an irrational
>> materialist, instead of being mostly not there like Q, the
>> rational ghost.
>
> Irrational materialist? Marx and Engels?
>
No, capitalists.
From: William Elliot on 12 Apr 2010 08:10
On Sun, 11 Apr 2010, George Jefferson wrote:

> "William Elliot" <marsh(a)rdrop.remove.com> wrote in message

> Y is dense in X because, relatively speaking, X - Y is small. This doesn't
> work well for infinite sets though and requires a more rigorous definition.

D dense subset finite T1 space S iff D = S.
In these cases small means zero.

Let S have the particual point topology { nulset, U | s in U }
where s is the particular point in S.
Then D = {s} is dense and S - D is arbitrarly large.
In these cases small means one point less than the whole space.

----
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