Can one define a function that sends lets say a line in r2 to a volume in r3?

[Kidphysics]

Or perhaps there is a more general function that sends to the next hypervolume? Can it be bijective? Continuous?

 

[A. Bahat]

This may be of interest: http://en.wikipedia.org/wiki/Space-filling_curve

 

[Kidphysics]

Ah ^thanks, so by this logic we can define H(x,y)--> (h(x),h(y),h(yx),h(xy),h(xx),h(yy)...)?

thus we can go from R^2-->R^n?

 

[mathwonk]

it can either be bijective or be continuous but not both, I think.

 

[Bacle2]

it can either be bijective or be continuous but not both, I think.

Maybe you're referring to mapping a (closed) line segment? In that case, you would have a bijection between compact and Hausdorff, which is a homeomorphism?

To the OP: I'm curious: why are you considering a line embedded in ℝ², instead of considering ℝ itself?

 

[Bacle2]

Actually (to the OP), this should help answer your question:

http://en.wikipedia.org/wiki/Invariance_of_domain

 

[Bacle2]

Actually (to the OP), this should help answer your question:

http://en.wikipedia.org/wiki/Invariance_of_domain

 

[Bacle2]

Actually (to the OP), this should help answer your question:

http://en.wikipedia.org/wiki/Invariance_of_domain