Infiniti Q45
 

If you type in infinity into Google you get the result:

Results 1 - 10 of about 43,600,000 for infinity

This is a disappointing reality of the finite nature of humankind, in my humble opinion.

However, I can't be bothered to click through to the last of the 43,600,000 references, or maybe there is a surprise at the end of this bounded universe?

And how annoying that Google thinks I want to buy a luxury car!!!!

[QUOTE=robert44444uk;108222]And how annoying that Google thinks I want to buy a luxury car!!!![/QUOTE]

Googling "Infiniti" returns 26,100,000 hits. I'm not sure if that should make us better about humankind or not. If not, perhaps we could sue the manufacturer for false advertising.

Does not click!
 

[QUOTE=robert44444uk;108222]If you type in infinity into Google you get the result:

Results 1 - 10 of about 43,600,000 for infinity

This is a disappointing reality of the finite nature of humankind, in my humble opinion.

However, I can't be bothered to click through to the last of the 43,600,000 references, or maybe there is a surprise at the end of this bounded universe?

And how annoying that Google thinks I want to buy a luxury car!!!![/QUOTE]

:smile:

I dont see why you should guage the infinitude of human kind to a finite quantity of references of infinity. It just does not click logically.

Of course you can question my statement as I believe in eternal life- that it is eternal. I cant prove it scientifically though. Its a belief in Faith of the Word.

The universe is not bounded. Would you call a sphere bounded ? Where is the boundary ? And thats only in 3 dimensions. 4 dimensions will be even more interesting I can assure you.

Mally :coffee:

[QUOTE=robert44444uk;108222]However, I can't be bothered to click through to the last of the 43,600,000 references, or maybe there is a surprise at the end of this bounded universe?[/QUOTE]

Just as well. Google will show you only the first 1000 anyway :wink:

[QUOTE=mfgoode;108238]The universe is not bounded. Would you call a sphere bounded ? Where is the boundary ?[/QUOTE]
Boundedness and finiteness are independent quantities. Objects can exist which can have either, both or neither of these characteristics.

In one dimension, circle is finite but unbounded. A line open at both ends is infinite and unbounded. A line segment with one end point is infinite and bounded. A line segment with two end points is finite and bounded.

As you point out, a sphere is unbounded. It is, however, finite.

Observational evidence suggests that the universe is probably bounded and infinite. The boundary for which the best evidence exists is the Big Bang, where *all* spacetime forms a singularity analogous to the end of a line segment. Recent measurements of the expansion of spacetime and theoretical models of black holes suggests that the future is unbounded with no long-lasting singularities.


Paul

One dimension.
 

[QUOTE=xilman;108277]
In one dimension, circle is finite but unbounded.
Paul[/QUOTE]

:surprised:

Could you please explain and clarify why this is so?

Mally :coffee:

[QUOTE=mfgoode;108328]Could you please explain and clarify why this is so?[/QUOTE]

You find yourself on a circular walkway. Staying on the walkway, start walking. Stop when you reach the end.

More mathematically: a circle is the boundary of an open ball in 2-D. (i.e. an Open disc). The open ball has finite volume (area), hence its closure is also finite. As you can see from the subtext under my username, the boundary (that's the outer of the 2 deltas) of a boundary (that's the "delta omega") is the empty set.

(At least in general; the topological caveats on this are beyond the scope of this discussion, but are out there on the web, for the interested or merely mathematically masochistic.)

[QUOTE=mfgoode;108328]:surprised:

Could you please explain and clarify why this is so?

Mally :coffee:[/QUOTE]Ernst has given a reasonably rigorous explanation. Here's a more picturesque explanation.

Pick a circle large enough to walk around, then step on to it with a pot of paint and a paint brush in your hand. Walk around the circle, backwards by preference or you'll get your feet messy, and paint the circle where you've been. After a little while the entire circle is painted, using a finite amount of paint. The circle is therefore finite (to be more precise, it has finite length) but you have not yet come across a boundary, nor will you ever do so because after going around once without finding a boundary you are in exactly the same situation as when you started (except, possibly, by having paint on your feet).

The analogous situation in two dimensions would have you paint the entire surface of a sphere with a finite amount of paint without ever finding an edge to the sphere.


Paul

Dimensions.
 

:smile:
Thank you both Ernst and Paul for giving me a down to earth clarification on my query.

Whereas I understand that a circle is boundless, I am confused in which least dimension it can exist. I was always under the impression that a line requires just one dimension (linear) with one direction of freedom for it to be drawn. The second direction of freedom would put it into the second dimension.

I have checked my 4 dictionaries of maths and surprisingly find there is no strict geometrical definitions for lines and circles (and discs). They say there are only intuitive definitions, for the straight line at least.

Then again a circle can be traced with one direction of freedom but its a planar curve and so can only be generated where there is a plane and that's 2dimensions A sphere can only exist in 3 minimum dimensions I consider only 3dimensions.

If we consider it algebraically in The equations for a straight we have the exponent as 1, (y=mx + c). I take that to be one dimension. The circle eqn is x^2 + y^2 = r^2 and so I take that to be 2 Dim.

Similarly the eqn of the sphere has an exponent of 3, hence it is in 3 dim. The area of the surface of the sphere is in 2 dim. though I would say to generate it 3 dim is required. If we take a semi circle (2 dim) and rotate it we require the 3rd dim to form a sphere.

In both your posts I find that you have gone down one dim less. i.e 'circle' 1 dim.. 'open ball' 2 dim. This is what I find very confusing so please clarify and let me know where I have gone wrong, unless 0 is a dimension in which case the dim.1 will be the 2nd. dim and so on.

A point was made about open and closed discs. We have adequately thrashed this out in a former thread and I wont to labour on it here.

For the record an open disc is the set of points such that x^2 + y^2 < r. That is the points are all inside except on the circumference. For the closed disc the points are also on the circumference i.e. x^2 +y^2 <= r^2.

Thank you once again for bearing up with me,

Mally :coffee:

[QUOTE=mfgoode;108361]Similarly the eqn of the sphere has an exponent of 3, hence it is in 3 dim.[/QUOTE]
You must be reading a different book from the one that I do.

For a sphere, I have [TEX]x^2+y^2+z^2 = R^2[/TEX] with the equality replaced by < or ≤, as needed, depending on whether you are speaking of the shell or the volume enclosed, etc.

It is not the exponent, but the number of degrees of freeedom that relates to the dimentionality.

[QUOTE=mfgoode;108361]:If we consider it algebraically in The equations for a straight we have the exponent as 1, (y=mx + c). I take that to be one dimension. The circle eqn is x^2 + y^2 = r^2 and so I take that to be 2 Dim.[/QUOTE]As I thought, you are confusing dimensionaility with exponents in the govenrning equation in Cartesian coordinates.

A circle in radial coordinates with origin at the center of the circle is r=constant. A linear equation and an extremely simple one. However this may also be misleading as, once again, you may be tempted to confuse dimensionality with an exponent in a particular coordinate system.

When a mathematician speaks of the dimensionality of a geometric object (at least for relatively simple ones, we'll ignore pathological cases such as fractals and disconnected sets of points for the moment), what is meant is the minimum number of quantities required to specify any point on the object. In the case of a circle, you need only one number --- the distance from a specific point (the place where you started painting) to the point in question. A circle is thus one-dimensional. That's not to say you can't use more numbers if you wish --- your x and y coordinates for instance --- but the important point is that you don't [i]need[/i] to.

To specify a location on a sphere you [i]need[/i] two numbers ---- one is not sufficient. Conventionally the numbers are the latitude and longitude, but there are many other reasonable choices --- the (x,y) coordinates of the location on a Mercator projection, for instance.


Paul

One can also regard the unit circle as the set of all points in [0, 2pi], provided that we identify 0 and 2pi as the same point.

[quote=robert44444uk;108222]If you type in infinity into Google you get the result:
Results 1 - 10 of about 43,600,000 for infinity
This is a disappointing reality of the finite nature of humankind, in my humble opinion.
[/quote]
Well, maybe merely the finite nature of computer storage media connected to internet ?
Also, note that you have to end the question in "=" in order to force mathematical evaluation in case of ambiguity.
[quote]However, I can't be bothered to click through to the last of the 43,600,000 references, or maybe there is a surprise at the end of this bounded universe? [/quote]
The "last" just refers to some humanly arbitrarily chosen partial(!) preorder relation.
Also, surprises are usually hidden somewhere (of course not exactly) in the middle and not necessarily at one of the extremities : even if it's the farest away, it would be easier to find, thus contradicting to several basic principles of statistical thermodynamics and computer science (e.g. Murphy's law).

[quote=jinydu;108413]One can also regard the unit circle as the set of all points in [0, 2pi], provided that we identify 0 and 2pi as the same point.[/quote]
[0,2pi] is the same than [0,1] etc. Anyway all of this refers to the same manifold.

[QUOTE=xilman][I]In one dimension, circle is finite but unbounded. [/I]
[/QUOTE]
I think you mix up "unbounded" and boundary-less.

[QUOTE]More mathematically: a circle is the boundary of an open ball in 2-D. (i.e. an Open disc). The open ball has finite volume (area), hence its closure is also finite. As you can see from the subtext under my username, the boundary (that's the outer of the 2 deltas) of a boundary (that's the "delta omega") is the empty set.[/QUOTE]

A bound is not at all the same than a boundary.
The first has to do with POsets, the second with topology.

Finally, I'd rather suggest that the boundedness of the universe remains an open question so far.

[QUOTE=m_f_h;108415]
I think you mix up "unbounded" and boundary-less.[/QUOTE]
I don't think so. I was using the terms in the sense used earlier in the thread for continuity and to avoid introducing another possible cause of confusion.

Further, the word "unbounded" is used in precisely this same sense in cosmology, which leads me on to:

[QUOTE=m_f_h;108415]
Finally, I'd rather suggest that the boundedness of the universe remains an open question so far.[/QUOTE]

As I said, the best available observational evidence [i]suggests[/i] a boundary at the big bang and no boundary in the future. Surely, that's (a) an accurate summary of present knowledge and (b) acknowledges that the definitive answer is not yet known.


Paul

Balls!
 

[QUOTE=Wacky;108363]You must be reading a different book from the one that I do.

For a sphere, I have [TEX]x^2+y^2+z^2 = R^2[/TEX] with the equality replaced by < or ≤, as needed, depending on whether you are speaking of the shell or the volume enclosed, etc.

It is not the exponent, but the number of degrees of freeedom that relates to the dimentionality.[/QUOTE]

:smile:

Thank you both Richard and Paul for giving me a keen insight into dimensions and exponents.

I was thinking of the areas, such as length (0 area but length one dimension)

Area of circle *pi*r^2. Area of sphere 4*pi*r^2. Area of square a^2

And volume of cube a^3, Volume of ball 4/3*pi*r^3.

There seemed to be some relation between exponents and dimensions at least where areas are concerned

The next logical question : what would be the volume of a 4D cube (tesseract). Would it include the exponent 4 in its formula? From the net I get it is of R^4.

Why I ask is that in my opinion after 3D solids we go onto 4 dimensions. Here our quest branches in to different geometries not necessarily Euclidean. This is a lacuna I would like to plug.

In browsing the net I find that ewmayer has also explained it well and used the correct terminology.

["Unfortunately, geometers and topologists adopt incompatible conventions for the meaning of "-sphere," with geometers referring to the number of coordinates in the underlying space ("thus a two-dimensional sphere is a circle," Coxeter 1973, p. 125) and topologists referring to the dimension of the surface itself ("the -dimensional sphere is defined to be the set of all points in satisfying ," Hocking and Young 1988, p. 17; "the -sphere is ," Maunder 1996, p. 21). As a result, geometers call the surface of the usual sphere the 3-sphere, while topologists refer to it as the 2-sphere and denote it ".

"Regardless of the choice of convention for indexing the number of dimensions of a sphere, the term "sphere" refers to the surface only, so the usual sphere is a two-dimensional surface. The colloquial practice of using the term "sphere" to refer to the interior of a sphere is therefore discouraged, with the interior of the sphere (i.e., the "solid sphere") being more properly termed a "ball." ]

Regards the equation Richard gave of the sphere this is where the < or = comes in for the surface and the ball. The same is for circles which is the difference between the circle itself and the disc.

Milli Gratia, Ernst, Paul and Richard.

Mally :coffee:

unbounded / boundary less
 

well, maybe it's a matter of definition.
It's surprising how even (relatively reasonable, I'd say) mathematicians can disagree about such basic questions:
[quote=xilman] [I][I]In one dimension, circle is finite but unbounded. [/I][/I][/quote]
To me a circle is the set of points having a given distance from a chosen origin, which is (as I see it) usually infinite, but bounded.

So it's exactly the opposite of your statement... :unsure:

I think Paul meant "finite but unbounded" in the sense of "having finite extent but no boundary", i.e. finite length but no endpoint in the case of a circle.

[quote=ewmayer;108518]I think Paul meant "finite but unbounded" in the sense of "having finite extent but no boundary", i.e. finite length but no endpoint in the case of a circle.[/quote]
:shock: d-d-did he ? :unsure:

PS: hey, since any of my profound philosophical discoveries are immediately flooded by trivialities, I'll simply re-edit it hereafter. --
It's surprising how even (relatively reasonable, I'd say) mathematicians can make seemingly contradictory statements about things that one would never suspect to have an ambiguous definition.
[quote=xilman][I] [I][I]In one dimension, circle is finite but unbounded. [/I][/I][/I][/quote]
[quote=m_f_h]To me a circle is the set of points having a given distance from a chosen origin, which is (as I see it) usually infinite, but bounded.
[/quote]
These statements seem to be manifestly contradictory, while both of them can be considered to be true...:cool:
(Reminds me of some other "fuzzy truth" thread...)

Philosophy!
 

[QUOTE=m_f_h;108532]:shock: d-d-did he ? :unsure:

PS: hey, since any of my profound philosophical discoveries are immediately flooded by trivialities, I'll simply re-edit it hereafter. --
It's surprising how even (relatively reasonable, I'd say) mathematicians can make seemingly contradictory statements about things that one would never suspect to have an ambiguous definition.


These statements seem to be manifestly contradictory, while both of them can be considered to be true...:cool:
(Reminds me of some other "fuzzy truth" thread...)[/QUOTE]

:smile:

Naturally m_f_h !

As I see it linearity is the dimension of a straight line i.e. it only has length but no area. Lets call it one dimensional. Its equation is X=0 or Y=0 in one dimension; y=mx + c in two D as it lies in between the X and the Y axis but is still linear!

One cannot visualise a circle in one dimension. At most we can study its projection from its two dimensions on to the visualised straight line and that will be a straight line.

This straight line can be finite with two end points (bounded) or one point, and an unreachable second end and we call it unbounded, which means it cannot be measured along the visualised line. Also it if it has no end points no matter how far you go it is also unbounded in the one dimension.

A bounded straight line is the length of part of a circle with an infinite radius
if you want to bring a circle into the definition. We cannot visualise an infinite circle only parts of it and these are straight lines.

To draw a str. line you have one direction of freedom say the X axis. For a circle you require two axis both the X and Y axis and so on as it has area pi.r^2

A finite circle has finite area but no limits to its end as one simply goes 'round and round the mullberry bush' in the words of an old poem

If you can visualise a ball in the third dimension, as ewmayer puts it then it(the circle) is the circumference formed by the intersection of the ball or sphere by a plane This is a true circumference and not the disk which is inside the ball.

Please note that a ball is the full content whereas a sphere is only the sur face.

No fuzzy logic there!

m_f_h you said something of boundary and boundedness ? Kindly clarify the difference if you care.

Mally :coffee:

[QUOTE=m_f_h;108514]To me a circle is the set of points having a given distance from a chosen origin, which is (as I see it) usually infinite, but bounded.

So it's exactly the opposite of your statement... :unsure:[/QUOTE]

The set of points is infinite, but the resulting curve is not.

Human infinity
 

Just taking a view on where we are... the direction of the thread is a little random, and that is a good thing.... but..... I am particularly interested in the concept that, we as humans, cannot deal with infinity except in a conceptual (mathematical) fashion, and that relatively small numbers, 4.36*10^7 for example, are big, big numbers for everyone.

Take Bill Gates' wealth.... quoted in cents....even he got bored with this small number and created a useful foundation for the rest of the world to benefit from. (And I am not a big fan of the Gatester, but I would hope to do the same thing as he, should such luck befall me)

Out here in Bangladesh, they have the lovely numbering system, lacs and crores. A lac is 10^5 and a crore 10^7, but nothing above is commonly used, as it is beyond everyday occurrence. Steal a crore taka from the people and you will make the front page headlines here, as many politicians and businessmen have found to their surprise recently. But a taka is a small amount, and a rickshaw driver will laugh if you offer him only 12 taka for a ride. (don't worry, I pay more!!!) But have a look at [url]http://en.wikipedia.org/wiki/Indian_numbering_system[/url] for those inflationary days still to come.

As a human, I am always amazed at infinity. I remember I had conceptual problems not very long ago in realising that the set of factors of the integers 2^n-1, n from 1 to infinity, (an infinitessimally small group of integers) contained all of the prime numbers. Hilbert's hotel and all of that.

In any case there seems to me to be a smaller infinity, that defined by human endeavour, and that is quite a small quantity in the scale of things. So we will pride ourselves on finding small primes of the order 2^n-1 where n is 30 million or so, and think it has taken all of the last 4.5 billion years to find this. How the infinity god (small g) must be laughing at our puny efforts!!!!

Infinity!
 

:smile:

Further to my post 20 I give below some more axioms. Take it or leave it!

1) Infinity can only be represented as a point. I call it the Daleth point (3rd letter of the Hebrew alphabet) .

2) All else outside the Daleth is finite.

3) The infinity point Daleth contains all and is in all.

4) All lengths proceeding from it contract, so Zero cannot be reached no matter how far from the Daleth. Hence the universe has no boundary which is infinitely far and is expanding. No question of a Lorentz contraction here!

5) Matter can only react to consciousness at the speed of the limiting velocity of light and is proportional to its distance from the light source or Consciousness. .

6)Hence we say that Alpha Centaurus is 4.3 light years away.
That's the speed of a physical signal and the time it takes at the speed of 'c'. I think of Sirius and am transported instantly yet the signal from me will take 4.3 light years for it to react.

7) Consciousness is known by different names be it Prana, Akash, Tao etc.

These are some of the basics I formulated at 20 yrs and still am convinced of its truth. I still have to perfect my theory and am working on it.

Mally :coffee:

[quote]boundary and boundedness ? Kindly clarify the difference if you care.[/quote]
Basic definitions:
1) The boundary of a set in a topological space is its closure minus its interior: dA=[A] \ A°.
2) A subset of a topological vector space is bounded iff it is absorbed by any neighborhood of zero.
Alternatively,
2') A subset of a metric space is bounded if its diameter is finite.
Some more blabla:
ad 1):
The closure of A is the set of all points which are the limit of some sequence contained in A.
The interior of A, is the set A° of points in A that have a neighborhood which is contained in A. Removing the interior A° is usually(?) equivalent to taking the intersection with the closure of the complement of A.
Then the boundary is also the intersection of the closure of A with the closure of its complement. Going to complements, this is equivalent to say that the boundary of A is the complement of the union of A's interior and A's complement's interior.
Clearly, the boundary of a boundary is the boundary itself: dd=d. This is called idempotence.
EWMayer said the boundary of a boundary is empty (dd=0). This clearly is another concept of boundary. Not that I don't like cohomology, or exterior calculus... but I'd say this is much more abstract idea of a boundary, and probably not the first one Mr Everyday is thinking of when hearing "boundary".
It is left to the reader to decide if she considers a sphere (i.e. the boundary of a /solid/ ball), to be /without/ boundary, or to be it's own boundary. (Think of a, say, basket ball (or any inflatable toy of your preference, to leave more room for your fantasy...), and call its boundary the region where the material it is made of is in contact with air...)

ad 2):
"absorbed" means that by multiplying it with some big number, any (yet so small) neighborhood of zero can be made big enough to contain the set A. Or, since we're in a field, by multiplying
with a small enough number, the set A can be shrunk as to fit into that ("yet so small") neighborhood of zero.
Most topological vector spaces you can think of are in fact normed (or normable) spaces;
in that case you can take the neighborhood to be the (solid) unit ball. (Finally, it's radius does not matter.)
Well, there are less trivial topol.vector spaces, and, even more interesting, topological modules. There, it turns out that the good definition is : A is bounded if for any neighborhood V of the zero vector there's a neighborhood w of the number zero such that wA is contained in V...
ad 2'): The diameter of A is of course the supremum of all distance d(x,y) between points of A.
It turns out that for normed spaces, definitions 2) and 2') are equivalent. Fortunately...
Now (as if this was not enough...), let me end on a new thought:
According to basic common sense, a circle (set of points of given distance from its center, thus bounded) does not have a beginning or an end. Thus, it is (literally) endless, i.e., (literally!) infinite!
([I]regardless[/I] of the question if it has a boundary or not, and [I]regardless[/I] of the cardinality of its points...) - I think this also replies adequately to
[quote=ewmayer]The set of points is infinite, but the resulting curve is not.[/quote]
Oh no, this made me think of another thing: you probably knew that, while balls are never empty (guys...we speak about math!), a sphere may well be empty... (I mean, an empty set: no points on it!)
But there are funny (ultra)metric spaces, in which any closed ball Bc(0,r) = {x | d(x,0) <= r } is an open set, and the collection of all open balls Bo(x,r) = { y | d(x,y) < r } of same radius, centered in any point x of the former, forms a partition of the former! Ain't that great?!

[quote=mfgoode;109509]Further to my post 20 I give below some more axioms. (...)[/quote]
You seem to forget that there are different infinities.
There are already names for them : aleph[0], aleph[1],...
it is well known that there are much more real numbers than rational numbers, but as much rational numbers as positive integers.

Also, the boundary of the universe is not infinitely far away.
(In addition to the current folklore of saying that the radius of the universe cannot be bigger than ((age of the universe)/(speed of light))
= 10^10 light years, roughly, consider the following provocative reasoning:
The mass of the universe is finite, and since anything is quantized,
the diameter is finite.
no?

Boundless universe!
 

[QUOTE=m_f_h;109525]You seem to forget that there are different infinities.
There are already names for them : aleph[0], aleph[1],...
it is well known that there are much more real numbers than rational numbers, but as much rational numbers as positive integers. [/QUOTE]

Good reasoning m_f_h ! But can these cardinal numbers Aleph_null .... be represented on the number line? If so then the Daleth point encompasses all.

[QUOTE = m_f_h] Also, the boundary of the universe is not infinitely far away.
(In addition to the current folklore of saying that the radius of the universe cannot be bigger than ((age of the universe)/(speed of light))
= 10^10 light years, roughly, consider the following provocative reasoning:
The mass of the universe is finite, and since anything is quantized,
the diameter is finite.
no?[/QUOTE]

Well I meant the geometrical hypothetical universe. Not the physical one.
The physical universe is expanding into the boundless geometrical one I would say as from infinity Zero cannot be reached. Putting the cart before the horse it would mean that infinity cannot be reached from zero.

I think you mean the diameter of the universe = age of universe x c ?

Please bear in mind in the physical world the Daleth Point has infinite energy!

Mally :coffee:

[QUOTE=m_f_h;109524]Clearly, the boundary of a boundary is the boundary itself: dd=d. This is called idempotence.
EWMayer said the boundary of a boundary is empty (dd=0). This clearly is another concept of boundary.[/QUOTE]

I wonder if you may be confusing "boundary" with "closure" here. The 2nd [url=http://en.wikipedia.org/wiki/Kuratowski_closure_axioms]closure axiom[/url] states that the closure of a closure is the closure itself.

Now as you have pointed out, the boundary dS of a set S satisfies dS = closure(S) - interior(S).

Now the boundary dS of S is itself a set, so we ask, what is the closure of dS? Since dS is contained in the closure of S, closure(dS) can add nothing to closure(S), thus dS is identical to its own closure. But that implies that dS is "all interior," hence has the empty set as its boundary. QED.

[quote=ewmayer;109536]I wonder if you may be confusing "boundary" with "closure" here.[/quote]
clearly not, since I use both in the same phrase
[quote]closure(dS) can add nothing to closure(S), thus dS is identical to its own closure. But that implies that dS is "all interior,"[/quote]
wrong: e.g. a segment in usual space is closed, but with empty interior and identical with its boundary.
in fact, a boundary is "usually always" with empty interior (as common sense suggests), except for weird cases
[quote=wikipedia]For any set [I]S[/I], ∂[I]S[/I] ⊇ ∂∂[I]S[/I], with equality holding if and only if the boundary of [I]S[/I] has no interior points. This is always true if [I]S[/I] is either closed or open. Since the boundary of any set is closed, ∂∂[I]S[/I] = ∂∂∂[I]S[/I] for any set [I]S[/I]. The boundary operator thus satisfies a weakened kind of [URL="http://en.wikipedia.org/wiki/Idempotence"][COLOR=#0000ff]idempotence[/COLOR][/URL]. In particular, the boundary of the boundary of a set will usually be nonempty.[/quote]

Well, then perhaps our definitions are different in some fundamental way. I am thinking e.g. of geometrical manifolds, for instance this simple classic case:

1) Start with S := open ball in R^3 or radius r, get its closure, which adds the set of points at distance = r from the origin, i.e. dS.

2) The boundary dS is just the surface of the sphere, which is a closed 2-manifold which is identical to its own closure, i.e. has no boundary.

To paint a cartoon: Imagine yourself being a 2-D being living on the surface of the sphere. Where is the boundary of your world?

I think I may have figured out where the disparity in my and m_f_h's takes on this lies. Again using the above example of the 3-sphere S, whose boundary dS is a 2-D closed surface.

Now, when we consider the boundary of the boundary of S, d(dS), in some sense the result depends on your point of view:

1) If you are still viewing things 3-dimensionally, then dS is (by definition) "all boundary and no interior" and so will be equivalent to its own boundary. (I believe this is m_f_h's point of view).

2) If you instead consider dS as a geometrical object in its own right when asking about d[sup]2[/sup]S, then S (i.e. the thing dS bounds) is irrelevant to that argument, because one is now working in the 2-D world that is dS. Its boundary will be an object of geometric/topological dimension 1, which happens however to be null. In this approach, dS is "all interior," because every point in dS has only neighbors also in dS, where we now use the 2-D definition of neighborhood which is appropriate to the surface dS.

m_f_h, does that resolve the disparity for you?