Infinity

Originally posted by adam warlock
I don't think I understood you. The surface of a sphere does have a boundary. And a sphere does have a boundary too. I'm using regular topological definitions.

Is this a case of the same words being used in different contexts type of confusion? 😕

In spaces like R^3, yes, but I thought you meant in isolation (i.e. in the space formed by itself). In R^3 it's straightforward that it is bounded and has a boundary so in no way is it boundless.

So, at best, I can interpret your claim of a sphere being boundless as not having a boundary in isolation but if you go to it as a subset of a larger space then I don't see how you can in any meaningful way say it is boundless.

Originally posted by Palynka
In spaces like R^3, yes, but I thought you meant in isolation (i.e. in the space formed by itself). In R^3 it's straightforward that it is bounded and has a boundary so in no way is it boundless.

So, at best, I can interpret your claim of a sphere being boundless as not having a boundary in isolation but if you go to it as a subset of a larger space then I don't see how you can in any meaningful way say it is boundless.

Boundless in the sense that you can walk all over it and never reach an end.

It is hardly my claim: I'm sure you can find plenty of books saying that a sphere is boundless and finite and that [a,b] is infinite and bounded as examples of sets that demonstrate that even though boundless (no limits) and infinite are related terms but aren't equivalent.

http://www.bartleby.com/173/31.html

Let us consider now a second two-dimensional existence, but this time on a spherical surface instead of on a plane. The flat beings with their measuring-rods and other objects fit exactly on this surface and they are unable to leave it. Their whole universe of observation extends exclusively over the surface of the sphere. Are these beings able to regard the geometry of their universe as being plane geometry and their rods withal as the realisation of “distance”? They cannot do this. For if they attempt to realise a straight line, they will obtain a curve, which we “three-dimensional beings” designate as a great circle, i.e. a self-contained line of definite finite length, which can be measured up by means of a measuring-rod. Similarly, this universe has a finite area, that can be compared with the area of asquare constructed with rods. The great charm resulting from this consideration lies in the recognition of the fact that the universe of these beings is finite and yet has no limits.

Notice that I didn't use the terms boundary at first I just said boundless.

Originally posted by adam warlock
Boundless in the sense that you can walk all over it and never reach an end.

Can you define this mathematic property precisely or point me to a precise definition of it?

The way Einstein is talking about it there seems to me about boundaries not about boundedness.

Edit- It's also hardly my claim that a sphere is bounded.

Originally posted by adam warlock
Topologically speaking the surface of a sphere is closed and not open. So in what sense do you say that that it is continuous and open?

This is not a matter of you being convinced. It's a matter of precise mathematical definitions.

I was thinking in terms of set theory. The set of points on a sphere is open - I think, but it may also be closed. It may even depend on the co-ordinate system and whether we are treating it as 2D or 3D. The set is also continuous. I am fairly sure that the set is bounded.

Originally posted by adam warlock
Boundless in the sense that you can walk all over it and never reach an end.

I think the correct mathematical term is continuous.

Originally posted by twhitehead
I think the correct mathematical term is continuous.

Even though I understand waht you're trying to say The term continuous is only applied to mappings and not to sets.

But even if we apply continuous to sets I think you'd say that X=[a,b] is continuous and it certainly isn't boundless.
If you walk over X you're going to find its limits. On the other hand if you walk over the surface of a sphere you won't find any limits.

Originally posted by twhitehead
I was thinking in terms of set theory. The set of points on a sphere is open - I think, but it may also be closed. It may even depend on the co-ordinate system and whether we are treating it as 2D or 3D. The set is also continuous. I am fairly sure that the set is bounded.

I was talking about the surface of a sphere and not about the sphere. But even if you are talking about a sphere it is closed.

A sphere is given by the condition x^2+y^2+z^2 <= R^2 (let me you know if you disagree). The set of the limit points of a sphere (which is the sphere itself) is contained in the sphere (trivially). Thus, by definition, a sphere is closed.

The same is true for the surface of the sphere.

Originally posted by adam warlock
Even though I understand waht you're trying to say The term continuous is only applied to mappings and not to sets.

Maybe the term I was looking for was complete

I haven't done set theory for over 15 years 🙂

Originally posted by adam warlock
I was talking about the surface of a sphere and not about the sphere. But even if you are talking about a sphere it is closed.

A sphere is given by the condition x^2+y^2+z^2 <= R^2 (let me you know if you disagree). The set of the limit points of a sphere (which is the sphere itself) is contained in the sphere (trivially). Thus, by definition, a sphere is closed.

The same is true for the surface of the sphere.

So what are 'limit points' if not 'boundaries'?

I am fairly sure that if the surface of the sphere if taken from a 2D perspective it is both open and closed.

Originally posted by Palynka
Can you define this mathematic property precisely or point me to a precise definition of it?

The way Einstein is talking about it there seems to me about boundaries not about boundedness.

Edit- It's also hardly my claim that a sphere is bounded.

Einstein certainly isn't talking about boundaries in the topological sense

In a short notice this is all I found. If I can find a better source than Merriam-Webester I'll post it, though.

http://books.google.pt/[WORD TOO LONG]

But I'm actually starting to think that boundless isn't the best word and I'm starting to think I'll maybe use "without boundary" from now on.
But I don't like to say without boundary because a spherical surface does have a boundary. 😕 😕 😕

Originally posted by twhitehead
So what are 'limit points' if not 'boundaries'?

I am fairly sure that if the surface of the sphere if taken from a 2D perspective it is both open and closed.

I'm using standard topology definitions, thus limit points aren't boundaries. Limit points of a set X in a space S are points x such as all neighborhoods of x contain points in X other than x itself.

A boundary of a set X is a set A such as for all of its elements every neighborhood contains points in X and points not in X.

These two concepts are related since every closed set contains its boundary but they are in no way equivalent.

What are your definitions of open and closed?

Originally posted by adam warlock
I'm using standard topology definitions, thus limit points aren't boundaries. Limit points of a set X in a space S are points x such as all neighborhoods of x contain points in X other than x itself.

A boundary of a set X is a set A such as for all of its elements every neighborhood contains points in X and points not in X.

These two concepts are related since every closed set contains its boundary but they are in no way equivalent.

Thanks for the definitions. I see I was mistaken. As I said, its been years .....
But I now realize the all important "in a space S".
I would say that if the space is the surface of the sphere then it is both open and closed by definition. It also has no boundaries - by definition. (there cannot be points outside the space).
But this would apply just as well to a flat square where the square is the space.

If the surface of the sphere is not the space, but the space is three dimensional space, then clearly every point is a boundary point.

What are your definitions of open and closed?
I think I'd go with wikipedia:
http://en.wikipedia.org/wiki/Open_set
http://en.wikipedia.org/wiki/Closed_set

I think the word you are looking for is continuous but I can't immediately find a definition for that.

Originally posted by twhitehead
Thanks for the definitions. I see I was mistaken. As I said, its been years .....
But I now realize the all important "in a space S".
I would say that if the space is the surface of the sphere then it is both open and closed by definition. It also has no boundaries - by definition. (there cannot be points outside the space).
But this would apply just a ...[text shortened]... you are looking for is continuous but I can't immediately find a definition for that.

"I would say that if the space is the surface of the sphere then it is both open and closed by definition. It also has no boundaries - by definition."
It isn't open because the set of the interior points to the surface of a sphere doesn't coincide with the surface of the sphere. Just look at your won link that defines an open set.
Also: how would you embed the surface of a sphere in the surface of the sphere itself?

"If the surface of the sphere is not the space, but the space is three dimensional space, then clearly every point is a boundary point."
Right on!

Originally posted by adam warlock
Einstein certainly isn't talking about boundaries in the topological sense

In a short notice this is all I found. If I can find a better source than Merriam-Webester I'll post it, though.

http://books.google.pt/[WORD TOO LONG] like to say without boundary because a spherical surface does have a boundary. 😕 😕 😕

Edited out after I read your last paragraph.

A sphere has a boundary in R^3. That's clear, right? BUT what you're thinking of is taking the sphere as a topological space and checking if it has a boundary within that same space. And there it doesn't because there is simply no "outside" (I think 😕).

Originally posted by Palynka
Edited out after I read your last paragraph.

A sphere has a boundary in R^3. That's clear, right? BUT what you're thinking of is taking the sphere as a topological space and checking if it has a boundary within that same space. And there it doesn't because there is simply no "outside" (I think 😕).

"A sphere has a boundary in R^3. That's clear, right?"
Right!

"BUT what you're thinking of is taking the sphere as a topological space and checking if it has a boundary within that same space."
No, no, no! I'm always thinking that the surface is embedded in a higher space. And in that sense we can say that the surface of a sphere is boundless (has no limits) and is finite. Just like Einstein says in that link. I've seen this language plenty of times so far, but to re-iterate I'm starting to think it is confusing.

But I refuse to use boundaryless, since the surface of a sphere has a boundary! 😵

"And there it doesn't because there is simply no "outside""
I think I can say that in that case most of the topological notions we are using are meaningless, since they presuppose that the set X is embedded in a topological space. But I'm not really sure on this one... 😕