Geometric Issues with a line, a plane and a sphere...

[dom_quixote]

I - A point divides a line into two parts;
II - A line divides a plane into two parts;
III - Does a smaller sphere divide a larger sphere into two parts, like layers of an onion?

Note that the first two statements, the question of infinity must be considered.

For the third statement, is the division of three-dimensional space into two parts correct?

IV - Could the straight line be an arc formed by a circle of infinite radius?

 

[PeroK]

I - A point divides a line into two parts;
II - A line divides a plane into two parts;

Okay

III - Does a smaller sphere divide a larger sphere into two parts, like layers of an onion?

Why not?

For the third statement, is the division of three-dimensional space into two parts correct?

If I understand the idea, you have three parts: the volume inside the smaller sphere; the volume (annulus) between the spheres; the volume outside the larger sphere.

IV - Could the straight line be an arc formed by a circle of infinite radius?

Infinite radius makes no sense mathematically. A radius is, by definition, a finite number.

 

[Lnewqban]

A line divides a plane into two parts, as long as that line is contained within that plane.

Please, see:
https://en.m.wikipedia.org/wiki/Coplanarity

 

[Office_Shredder]

Infinite radius makes no sense mathematically. A radius is, by definition, a finite number.

Thinking of a line as an infinite radius circle is actually kind of useful for some applications, but I think you need a much more technical understanding of certain topics than the OP question reflects.

 

[PeroK]

Thinking of a line as an infinite radius circle is actually kind of useful for some applications, but I think you need a much more technical understanding of certain topics than the OP question reflects.

If we take the x-axis as a sphere of infinite radius, then the line $y = 1$ must be another sphere of infinite radius. In what way are these two spheres different? Different centre? And/or different infinite radius?

In what way are they circles?

 

[Office_Shredder]

If we take the x-axis as a sphere of infinite radius, then the line $y = 1$ must be another sphere of infinite radius. In what way are these two spheres different? Different centre? And/or different infinite radius?

In what way are they circles?

For example, stereographic projection sends circles to circles, except ones that pass through the pole become lines. Relatedly, mobius transformations send circles and lines to circles and lines.

 

[PeroK]

For example, stereographic projection sends circles to circles, except ones that pass through the pole become lines. Relatedly, mobius transformations send circles and lines to circles and lines.

You can map a circle to the half open interval $[0,1)$, but that doesn't make the interval itself a circle.

 

[FactChecker]

I - A point divides a line into two parts;
II - A line divides a plane into two parts;
III - Does a smaller sphere divide a larger sphere into two parts, like layers of an onion?

Note that the first two statements, the question of infinity must be considered.

I'm not sure that I agree. If I say that I have a path that connects the two parts without crossing the dividing line, do I have to consider infinity to determine if that is correct?

For the third statement, is the division of three-dimensional space into two parts correct?

Are the spheres concentric? Do you want to restrict their relative position?

IV - Could the straight line be an arc formed by a circle of infinite radius?

You might be able to develop this idea. In the right situation, with the right development, it might make good sense.

 

[Office_Shredder]

You can map a circle to the half open interval $[0,1)$, but that doesn't make the interval itself a circle.

https://en.m.wikipedia.org/wiki/Generalised_circle

If you can find me an equivalent version for circles and half intervals, I will concede the point!

 

[PeroK]

https://en.m.wikipedia.org/wiki/Generalised_circle

If you can find me an equivalent version for circles and half intervals, I will concede the point!

You don't have to. I'll concede the point!

 

[dom_quixote]

Yes, I imagined the problem with two concentric spheres. I think it is possible to equalize the volume of the two spheres, but one of them will be solid and the other will be hollow.

Are the spheres concentric? Do you want to restrict their relative position?