Is there a C^infty map that is one to one from R^n to R

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In summary, it is not possible to have a C^infty map that is one to one from R^n to R, as shown by the invariance of domain theorem and Sard's theorem. Additionally, it is not even possible to have a continuous injection from R^n to R when n>1, as proven by various arguments such as the fact that a continuous bijection between compact and Hausdorff spaces is a homeomorphism.
  • #1
mathsq
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Is there a C^infty map that is one to one from R^n to R?

Thanks.
 
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  • #2
Some comments; maybe someone else can extend:

I think no, but I have no proof. By invariance of domain, the map cannot be onto, since ℝn is open in itself, and, if it where onto ℝ, you could compose it with the inclusion (an inclusion, actually) into ℝn (any n>1 will do ) , concluding that there is a copy of
ℝ that is open in ℝn, i.e., assume f(ℝn)=ℝ, and then let i be the standard inclusion into ℝn, i.e, i(x)=(x,0,0,...,0). Then i°f is a map whose image is a copy of ℝ, which is open in ℝn.

Or, you can use Sard's theorem (we had a similar problem here a while back): Since every point in ℝn is a critical point of the differentiable map f, the image of ℝn under this map must have measure zero in ℝ.
 
  • #3
There isn't even a continuous injection from [itex]\mathbb{R}^n \rightarrow \mathbb{R}[/itex] when n>1. For suppose there were. Let f be the restriction of this map to, say, the closed unit ball. If A is a closed subset of the closed unit ball, then A is compact, so f(A) is compact and therefore closed. Hence f is a closed map and therefore a homeomorphism onto its image. This implies the unit ball in [itex]\mathbb{R}^n[/itex] is homeomorphic to a subset of [itex]\mathbb{R}[/itex]. But this is impossible, since any subset of [itex]\mathbb{R}[/itex] that has at least two points is either disconnected or can be made disconnected by removing a single point, whereas the closed unit ball in [itex]\mathbb{R}^n[/itex] is connected and remains so if any single point is removed.
 
  • #4
Right. A small variant of Citan Uzuki's proof: use that a continuous bijection between compact and Hausdorff is a homeo., so that the image of any compact subset of Rn, say Snl restricts to a homeo between S^n and the Hausdorff subspace f(Sn) , but then you can use Citan Uzuki's argument, or Borsuk-Ulam that does not allow you to embedd S^n in nothing smaller than Rn+1.
 
  • #5
Thank you! That makes sense.
 
  • #6
there are easier variants as well. any continuous map takes connected sets to connected sets. But the complement of a point p of R^2 is connected whereas its image in R is not by any injective map, since it is separated by f(p), if p is chosen semi carefully.
 

Related to Is there a C^infty map that is one to one from R^n to R

1. Can a C∞ map be one-to-one from Rn to R?

Yes, it is possible for a C∞ (smooth) map to be one-to-one from Rn (n-dimensional Euclidean space) to R (real numbers). This means that the map is injective, where each element in the domain (Rn) is mapped to a unique element in the range (R).

2. What does it mean for a map to be C∞?

A C∞ map is a function that has derivatives of all orders (infinitely differentiable) at every point in its domain. This means that the function is smooth and has a well-defined tangent line at every point.

3. How is a C∞ map different from a differentiable map?

A differentiable map is a function that has a derivative at every point in its domain, but the derivative may not be continuous. A C∞ map, on the other hand, has derivatives of all orders at every point, meaning that it is not only differentiable but also infinitely differentiable.

4. What is the significance of a C∞ one-to-one map in mathematics?

A C∞ one-to-one map has important implications in the study of topology and differential geometry. It can also be used in various mathematical models and applications, such as in physics and engineering, where smooth and injective mappings are required.

5. Is there a simple example of a C∞ one-to-one map from Rn to R?

One example of a C∞ one-to-one map from Rn to R is the exponential function f(x) = ex. This function is smooth and injective, mapping each real number to a unique positive real number. Another example is the hyperbolic tangent function, tanh(x), which maps the real line to the interval (-1,1) in a smooth and one-to-one manner.

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