What is the difference between surface boundary and point-set boundary?

[Electric Jaguar]

I'm having trouble understanding how to find the boundary in the sense of boundary of a surface, as opposed to boundary in the sense of boundary of a point-set. From what my professor said, it seems as though for surface boundaries, you just stare at the surface and figure out its boundary, but there's got to be a formalized method for doing this.

For example, take the upper hemisphere of a sphere in 3 space. Its boundary according the the point-set definition of boundary is the entire surface; put another way, if the surface is thought of as a subset of R3, it has no interior.

But the "surface boundary" of the hemisphere is the circle defining its base, according to my professor. How did he reach this conclusion?

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[matt grime]

Try asking your professor, you'd be surprised how keen they are to explain things. and it would mean we wouldn't have to second guess their interpretation - for what it's worth, probably nothing, the boundary is the closure less the interior, at least that's one view, but I don't know, and nor will any other poster, what level you are taking this course.

Q. What's the boundary of the rationals inside the reals?


To explain one interpretation of the example you state, the upper hemisphere is locally a 2-d object, except at the boundary where it is some part of the upper half plane, and in this view the boundary is 'the edge', that is the circle. imagine deforming it so it was the disk, then the boundary would be the circle.

 

[Electric Jaguar]

Originally posted by matt grime
but I don't know, and nor will any other poster, what level you are taking this course.

Right now I'm reviewing for a midterm and specifically was looking at Stoke's Theorem.

Q. What's the boundary of the rationals inside the reals?

The boundary of rational numbers inside the set of real numbers is the set of rational numbers itself.

I suppose I'll better understand the idea after I've taken a course on topology. But that is a long ways away...

 

[matt grime]

OK, so we know what you want this for:

the boundary can safely be called the edge. It will be 1 dimension less than th interior. The upper hemisphere, is locally 2-d, the boundary is 1-d

it's perhaps best to give loads of examples so you see the pattern.

Let, D be the disk in R^2 - the points (x,y) with X^2+y^2 <= 1

the boundary is the circle, called S^1 usually.

The disk in R^n is the set of n-tuples (x,y,..,z) with x^2+y^2+...+z^2 <= 1, the boundary is the n-1 sphere S^(n-1), the set of points with x^2+y^2+...+z^2 = 1,


The boundary of the interval [a,b] is the set withw two points, {a,b}.


The boundary of the points (x,y) in R^2 with 0<=x,y<=1 is... the edge of the unit square. got the idea?

Stokes theorem says that, under certain conditions you can reduce a question about integration over some object to one about integration over its edge.

So go with the boundary being the naive (and sufficient here) notion of edge.

What about the boundary of S^1 above? Ans, it doesn't have one.

If it's not clear still we can try another more rigorous way of doing it.






I do'nt want to explain what it is for pathological examples like Q<R, because that is not relevant here in the slightest. And the boundary of a point set isn't just the point set

And midterm isn't a phrase used much in England, and which year anyway? (for god sake don't say sophomore! that isn't used except in the odd archaic place in England, and in episodes of American TV programs).

 

[Electric Jaguar]

Thanks, I understand the idea much better now.

At the University of Washington, the year is divided up into quarters, with one course taking up a quarter. Midterms are tests about halfway (usually) through the course. I'm currently a 2nd year undergraduate student (since you don't like the term sophomore, which is what we call it here).

 

[selfAdjoint]

And midterm isn't a phrase used much in England, and which year anyway? (for god sake don't say sophomore! that isn't used except in the odd archaic place in England, and in episodes of American TV programs).

In the US we play baseball, not cricket, we call your football soccer, and our four year colleges have freshmen, sophmores, juniors, and seniors, and frequently give midterm exams. Other lands other customs. Get used to it.

 

[matt grime]

Originally posted by selfAdjoint
And midterm isn't a phrase used much in England, and which year anyway? (for god sake don't say sophomore! that isn't used except in the odd archaic place in England, and in episodes of American TV programs).

In the US we play baseball, not cricket, we call your football soccer, and our four year colleges have freshmen, sophmores, juniors, and seniors, and frequently give midterm exams. Other lands other customs. Get used to it.

As some one who has lived and worked in the US (at an American University in fact teaching maths) all the phrases are well understood. Sorry it came across badly, but just writing 'for a midterm' doesn't tell me which year that person is in and what we can assume of their experience. It was tongue in cheek. As it is sophomre IS an archaic English term; lots of the English in America preserves the usages of this country from the time of the pilgrim fathers (in back of rather than behind). The oddities are that fortnight doesn't survive in the US, and that the English think that spelling with -ize is new fangled, cos it's American, when actually it is old fashioned and the preserve of those of us educated at Oxbridge, it would appear.


edit: just realized that it doesn't say where my location is on the side panel

 

[HallsofIvy]

Actually, it came across as
"For God's sake, everyone be civilized and at least pretend you are from England!"

 

[matt grime]

You mean there are people not from England? Strange idea. You're all speaking English, though. Are you sure you're not English? Look outside and check, if the weather's awful, you're paid crap and it costs nearly 5 bucks for a gallon of gas (note the choice of words) it must be England. (The first person to call me a Brit will, will... well, when I find out where you live, then we'll see!).

It's all a revenge of course from teaching evaluations, written by people wearing basebal caps indoors and back to front, that called my level of English inadequate.