forms of spheres

[houserichichi]

...spheres only have two forms, solid and hollow...

Should it not have three forms? Solid (with boundary), solid (without boundary), and hollow (empty with boundary). There's always empty (empty without boundary).

Split from "Wiki discussion" by Rob.

[Keiji]

A "Solid without boundary" sphere of radius r is the same as a "Solid with boundary" sphere of radius:

r+1/∞ > r

as far as I am concerned. 1/∞ gives three results: zero, +x and -x, where x is an infinitely small positive number. I'm sure many of you will disagree with this, but I think it is very helpful to consider this true.

[Nick]

A "Solid without boundary" sphere of radius r is the same as a "Solid with boundary" sphere of radius:

Yeah, I agree with that.

r+1/∞ > r

as far as I am concerned. 1/∞ gives three results: zero, +x and -x, where x is an infinitely small positive number. I'm sure many of you will disagree with this, but I think it is very helpful to consider this true.

I thought you couldn't use infinity in mathematics... :?

[houserichichi]

Solid with boundary at radius r is {x | x <= r}

Solid without boundary at radius r is {x | x < r}

Hollow with boundary at radius r is {x | x = r}

Hollow without boundary at radius r is {}

At the very least, the first two are not one in the same. The third is obvious. The fourth may be omitted at will, in my books.

[PWrong]

Should it not have three forms? Solid (with boundary), solid (without boundary), and hollow (empty with boundary). There's always empty (empty without boundary).

I agree that an open ball is different from a closed ball, but I don't think it's worth classifying them separately. They're both balls, and they're both 3D objects, whereas the hollow sphere is only 2D.

I might mention the definition of a boundary.
The boundary of A is the set of points x in R^n such that every ball with center x contains points in A and point outside A.
A closed set contains all of it's boundary points. An open set doesn't.
This site has an explanation and a picture.

[jinydu]

My understanding is that the distinction is important in mathematics because it is easier to do calculus on open sets than closed sets. For example, when minimizing a differentiable function on the interval [0, 1], you have to check all the points where the derivative of the function is 0, and both of the endpoints. On the other hand, if the interval is (0, 1) instead, you don't have to check the endpoints.

The situation gets even more pronounced in multivariable calculus; since boundaries then contain infinitely many points.

[bo198214]

Maybe I should make some standardizing comments.

First, in mathematics the k-dimensional sphere always refers to the k-dimensional object {x: |x| = r}. { x : |x| <= r } is called the closed ball and { x : |x| < r } the open ball. So the k-dimensional sphere, or short S_k, is the boundary of the k+1 dimensional closed ball. Mostly used in topology the radius r does not matter and the sphere is up to isomorphism. It makes sense to use these conventions also throughout this forum.

Next the term closed is ambiguous. If we talk of a closed surface/manifold, it means that it has no boundary. If we talk of a closed (general) set, it means that the complement is open, which means in turn that each point has an environment that also belongs to the set (where environment is a base topological term).
To make it more difficult the term boundary is differently (i.e. more specific) defined for manifolds than for general sets (which was PWrong's definition).

[pat]

Hollow without boundary at radius r is {}

I thought that was the hollow simplex without boundary.

[moonlord]

Guess what, I was damn sure that was the hollow hypercube without boundary .

[bo198214]

Oh people! It was the gardener! You can verify it in the papers of A. C. Doyle

The hollow gardener without boundary ... *head scratch*

[houserichichi]

Isomorphisms abound!

[bo198214]

Help! The hollow isomorphism are attacking without boundary!*Screamingly runs around*