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

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Discussion Overview

The discussion revolves around the possibility of defining a function that maps a line in ℝ² to a volume in ℝ³, exploring concepts of continuity, bijectiveness, and potential generalizations to higher dimensions.

Discussion Character

  • Exploratory
  • Debate/contested
  • Mathematical reasoning

Main Points Raised

  • Some participants propose the idea of a general function that could map to a higher hypervolume, questioning whether such a function can be bijective or continuous.
  • One participant references space-filling curves as a potential related concept.
  • A participant suggests a specific mapping function H(x,y) that could extend from ℝ² to ℝⁿ, raising questions about the nature of such mappings.
  • There is a claim that a mapping can either be bijective or continuous, but not both, with a suggestion that this might depend on the nature of the line segment being mapped.
  • Another participant questions the choice of a line in ℝ² instead of ℝ itself, hinting at different perspectives on dimensionality in the mapping process.
  • Multiple participants provide links to external resources that may help clarify the discussion, specifically referencing the invariance of domain.

Areas of Agreement / Disagreement

Participants express differing views on the properties of the proposed mappings, particularly regarding bijectiveness and continuity. There is no consensus on the nature of the function or the best approach to the problem.

Contextual Notes

Some assumptions about the properties of the mappings and the definitions of continuity and bijectiveness remain unresolved. The discussion also reflects varying interpretations of dimensionality in the context of the mappings.

Kidphysics
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Or perhaps there is a more general function that sends to the next hypervolume? Can it be bijective? Continuous?
 
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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?
 
it can either be bijective or be continuous but not both, I think.
 
mathwonk said:
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?
 

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