Infinity: Limitless or Limited? Exploring the Concept of Infinity in Mathematics

[gravenewworld]

If I have a sphere and I remove a point from it, it still has volume right? But topologically speaking a sphere with a point removed is the same as a plane, so where does the volume go when I stretch the sphere into a plane?

 

[MeJennifer]

If I have a sphere and I remove a point from it, it still has volume right? But topologically speaking a sphere with a point removed is the same as a plane, so where does the volume go when I stretch the sphere into a plane?

Do you mean a sphere or a ball?
If you mean a ball then the volume of the ball and the volume of a ball with one point removed (whatever you mean by removing a point) is the same.
Remember that a point does not occupy any space!

If you are really talking about a sphere then well we just learned that a sphere has no volume just like a circle has no area.

And by the way removing a point from a sphere does not reduce the area of the object.

 

[river_rat]

If I have a sphere and I remove a point from it, it still has volume right? But topologically speaking a sphere with a point removed is the same as a plane, so where does the volume go when I stretch the sphere into a plane?

Ok firstly a sphere has zero volume (a sphere is a closed subset of R^3 and every closed subset is measurable under the normal legesgue measure). However volume is not a topological invariant (think length of (0, 1) which is homeomorphic to the entire real line) but is rather invariant under isometries (ie. rigid motions) which continuous functions need not be. Remember that not even angles need to be preserved under continuous mappings, so volumes being preserved is very special indeed.

 

[gravenewworld]

Ok firstly a sphere has zero volume (a sphere is a closed subset of R^3 and every closed subset is measurable under the normal legesgue measure). However volume is not a topological invariant (think length of (0, 1) which is homeomorphic to the entire real line) but is rather invariant under isometries (ie. rigid motions) which continuous functions need not be. Remember that not even angles need to be preserved under continuous mappings, so volumes being preserved is very special indeed.

lol sounds like i need to stick to logic and algebra. analysis and topology were always my weakness.