Transformations between spaces

[Curl]

Is it possible, in general, to have a one-to-one transformation from Rⁿ to R^m for n>m?

I'm thinking in the context of geometry, where you want to map a bounded region from a higher space to a lower space.

 

[chiro]

Is it possible, in general, to have a one-to-one transformation from Rⁿ to R^m for n>m?

I'm thinking in the context of geometry, where you want to map a bounded region from a higher space to a lower space.

This is an interesting question.

Intuitively since this is bounded, you would think that it is possible, but given the nature of real numbers, I'm not so sure it is.

It is definitely not possible if m = 0. You can not map an interval onto a point: it is impossible to go from dimension > 0 to 0 and back again.

My guess is if you wanted to prove this, you should use an inductive argument that shows that you can't go from m to m-1 in terms of one-to-one properties, and then if that is correct, you're finished.

 

[Deveno]

i think you can do this:

imagine you have the peano space-filling curve (well, actually the hilbert one works better)

(http://en.wikipedia.org/wiki/Space-filling_curve)

now this map may not be injective, but it IS surjective, so we can define a right-inverse from R²→R that is injective. using some form of the Cantor–Bernstein–Schroeder theorem, we can obtain a bijection between the two.

i don't think you can do this with differentiable curves, though.

 

[disregardthat]

Deveno, this right-inverse won't necessarily be continuous though. I assume this question is about continuous functions.

 

[Deveno]

well, if we have to have a homeomorphism, we can't do it. to see this, just remove one point from R², the injection we are seeking will map to two disconnected sets, but R²-{x₀,y₀} is connected.

 

[disregardthat]

well, if we have to have a homeomorphism, we can't do it. to see this, just remove one point from R², the injection we are seeking will map to two disconnected sets, but R²-{x₀,y₀} is connected.

An injection isn't necessarily an homeomorphism. Also, such an injective function need not necessarily be surjective. But, as it turns out, injections from R^n to R^n have open images and are homeomorphisms onto their image by the invarience of domain theorem.

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

As R^m can be seen as a subspace of R^n, such an injective function f would be homeomorphic to an open subset of R^m. This set would then be open and connected in R^m (since R^n is connected), thus generating an homeomorphism from R^n to R^m. But this is impossible due to different fundamental groups.

 

[Deveno]

An injection isn't necessarily an homeomorphism. Also, such an injective function need not necessarily be surjective. But, as it turns out, injections from R^n to R^n have open images and are homeomorphisms onto their image by the invarience of domain theorem.

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

As R^m can be seen as a subspace of R^n, such an injective function f would be homeomorphic to an open subset of R^m. This set would then be open and connected in R^m (since R^n is connected), thus generating an homeomorphism from R^n to R^m. But this is impossible due to different fundamental groups.

yes, i know that a continuous injection need not be a homeomorphism (even a continuous bijection need not be, the inverse also has to be continuous). but, as you point out, we can easily translate our proposed injection into a homeomorphism, so the connectedness should run both ways (i seem to recall, dimly, this (the open mapping property of continuous injection in Rⁿ) involving a fixed point theorem, somewhere. i dunno, I've slept since then).

i don't understand your remark about fundamental groups, because π₁(Rⁿ) is trivial.

 

[disregardthat]

i don't understand your remark about fundamental groups, because π₁(Rⁿ) is trivial.

You're right, faulty train of thought there. I was thinking about fundamental groups as one remove a certain subset from R^m. It works for m = 1 by removing a point, m = 2 by removing a line I guess, but I don't see how to formalize it for general m. It can actually be seen easily by the invariance of domain theorem however. R^m can be seen as a closed subspace of R^n, so a homeomorphism would imply an injective function from R^n to R^n with image R^m which must be open by the theorem.

 

[Jamma]

There is no homeomorphism between R^n and R^m for n not equal to m.

However, they obviously have the same cardinality (ignoring the n=0 case) so there will be bijections between then.

 

[Jamma]

Oh, and to see the above, just remove a point for R^n and R^m. If they were homeomorphic before, they will be after removing a point too (just precompose the homeomorphism with a homeomorphism which takes the removed point of R^n to the preimage of the removed point of R^m and then restrict).

But each of R^n and R^m with a point removed are homotopy equivalent to S^{n-1} and S^{m-1} respectively, which aren't homeomorphic for n=/=m (just look at homology groups).

 

[lavinia]

Oh, and to see the above, just remove a point for R^n and R^m. If they were homeomorphic before, they will be after removing a point too (just precompose the homeomorphism with a homeomorphism which takes the removed point of R^n to the preimage of the removed point of R^m and then restrict).

But each of R^n and R^m with a point removed are homotopy equivalent to S^{n-1} and S^{m-1} respectively, which aren't homeomorphic for n=/=m (just look at homology groups).

You can make this argument even stronger. R^{m} is a subspace of R^{n} so if there were a continuous map that was 1-1 (or even only 1-1 on R^{m}) from R^{n} onto R^{m} then S^{m-1} would be a retract of S^{n-1}. But your homology argument shows that this is impossible.

 

[disregardthat]

Regarding the terminology one-to-one and 1-1, does this mean bijective or simply injective?

 

[Jamma]

1-1 = one- to- one = injective
onto = surjective

 

[Jamma]

You can make this argument even stronger. R^{m} is a subspace of R^{n} so if there were a continuous map that was 1-1 (or even only 1-1 on R^{m}) from R^{n} onto R^{m} then S^{m-1} would be a retract of S^{n-1}. But your homology argument shows that this is impossible.

Sorry Lavinia, could you explain this a little more? For a retract S^{n-1} to S^{m-1}, wouldn't we need the points of S^{m-1} contained in S^{n-1} to be invariant under the induced map? How do we know that the subspace R^m is left invariant under such a homeomorphism?

 

[lavinia]

Regarding the terminology one-to-one and 1-1, does this mean bijective or simply injective?

1-1 means injective to me but I also said onto - I hope

 

[lavinia]

Sorry Lavinia, could you explain this a little more? For a retract S^{n-1} to S^{m-1}, wouldn't we need the points of S^{m-1} contained in S^{n-1} to be invariant under the induced map? How do we know that the subspace R^m is left invariant under such a homeomorphism?

My idea was to remove the origin so that the inclusion of R^{m} into R^{n} followed by the 1-1 map from R^{m} onto R^{m} induces an isomorphism on the top homology of R^{m} minus a point - so I was a little hasty because this is not exactly a retract - sorry. But the idea is the same.

I guess once could say that this map when restricted to the m-1 sphere is a continuous bijection and therefore since the sphere is compact, is a homeomorphism onto its image. This shows that the map must induce an isomorphism on homology from the m-1 sphere onto its its image which is itself homeomorphic to an m-1 sphere. But this map is factored through n space minus a point and its m-1st homology is zero.

Do you think this is wrong? I suddenly got worried. But I think its OK.

 

[Curl]

Alright, can anyone show an example how this can be done?
Can anyone show how to map the unit disk to an interval, or the unit square to an interval on R?

Or how about a cube or sphere mapped to R^2?

 

[Deveno]

i posted a link in an earlier post. the function is extremely non-differentiable, and it's inverse isn't continuous (imagine taking the points of the unit square, and mapping them onto an infinite number of "cantor-like" sets on the unit interval). it's actually the inverse you're looking for. basically, the idea is this: you take the unit interval, and bend it so many times that it fills space (the unit square, or other region of the plane).

this gives us a surjection. using the axiom of choice, pick a point in the preimage of any point in the plane region, this gives a bijection between a subset of a real interval and a region of the plane. some possible things to look at:

Hilbert curve
Peano curve
Moore curve
Dragon curve
Terdragon curve
Gosper curve
Lesbegue curve (this one is "almost differentiable", which is kinda scary)

one thing i did not know until recently, is that these types of things are actually used to convert higher-dimensional information such as maps or IP addresses, into linear sequences.

 

[lavinia]

Alright, can anyone show an example how this can be done?
Can anyone show how to map the unit disk to an interval, or the unit square to an interval on R?

Or how about a cube or sphere mapped to R^2?

As my previous post showed, there is no continuous bijection from R^m to R^m with m <n. The same type of arguments apply for the unit disk or a square or a cube or whatever.

There are continuous space filling maps as was previously posted but they are not 1-1.

There are bijection between any n dimensional region and any m dimensional region, m>0, but these can not be continuous.

 

[Jamma]

Basically, in summary,

There are no homeomorphisms between different dimensional spaces.
For n<m there are continuous maps R^n onto R^m but they will always fail to be one to one.
In general, of course there are non-continuous bijections between them too.

But the original question was can there exist a transformation R^n into R^m which is one to one for n>m. I'll assume we mean continuous here. We don't seem to have answered this question- space filling is mapping lower dimensional things to cover higher ones.

The answer, I think, is clearly that you can't have a continuous map from R^n into R^m for n>m which is one to one. Is there an easy way to see this?

 

[Deveno]

Basically, in summary,

There are no homeomorphisms between different dimensional spaces.
For n<m there are continuous maps R^n onto R^m but they will always fail to be one to one.
In general, of course there are non-continuous bijections between them too.

But the original question was can there exist a transformation R^n into R^m which is one to one for n>m. I'll assume we mean continuous here. We don't seem to have answered this question- space filling is mapping lower dimensional things to cover higher ones.

The answer, I think, is clearly that you can't have a continuous map from R^n into R^m for n>m which is one to one. Is there an easy way to see this?

suppose f:R²→R is a continuous injection. then f(R²) is connected (since R² is), so we may assume it contains an interval (a,b).

choose any x in (a,b), and consider the set A = R² - f^-1(x).

this is still connected, but f(A) is not, contradicting the fact is f is continuous. therefore, there can be no such continuous injection.

to generalize to m dimensions, remove a set of m-1 dimensions from f(Rⁿ) (a line from a region of R², a plane from a region of R³...in general, a hyperplane of R^m), such that f(Rⁿ) is disconnected after removing the hyperplane. Rⁿ is still connected, even after removing the pre-image of the hyperplane.

 

[lavinia]

suppose f:R²→R is a continuous injection. then f(R²) is connected (since R² is), so we may assume it contains an interval (a,b).

choose any x in (a,b), and consider the set A = R² - f^-1(x).

this is still connected, but f(A) is not, contradicting the fact is f is continuous. therefore, there can be no such continuous injection.

to generalize to m dimensions, remove a set of m-1 dimensions from f(Rⁿ) (a line from a region of R², a plane from a region of R³...in general, a hyperplane of R^m), such that f(Rⁿ) is disconnected after removing the hyperplane. Rⁿ is still connected, even after removing the pre-image of the hyperplane.

good argument. The homology argument also works. But yours reveals the idea simply.

 

[Jamma]

I don't think the argument works- I agree with you for mapping into R but not your general argument. You need to ensure that removing a set of dimension m-1 still leaves the original space minus the preimage of this set connected.

Indeed, we have already shown that there exist odd space filling maps from lower dimensional spaces to higher ones- how do we know that removing a circle in the plane doesn't remove a sphere from the preimage in 3-space, for example? (the preimage isn't even necessarily continuous).

Also, how do we know we can remove these sorts of sets- I don't see an obvious generalisation here. What if the higher dimensional space mapped to the plane which leaves a fractal like image, for which we can't find circles or hyperplanes to remove?

Your argument seems like the right one, but with perhaps some annoying tweaks missing.

 

[lavinia]

I don't think the argument works- I agree with you for mapping into R but not your general argument. You need to ensure that removing a set of dimension m-1 still leaves the original space minus the preimage of this set connected.

Indeed, we have already shown that there exist odd space filling maps from lower dimensional spaces to higher ones- how do we know that removing a circle in the plane doesn't remove a sphere from the preimage in 3-space, for example? (the preimage isn't even necessarily continuous).

Also, how do we know we can remove these sorts of sets- I don't see an obvious generalisation here. What if the higher dimensional space mapped to the plane which leaves a fractal like image, for which we can't find circles or hyperplanes to remove?

Your argument seems like the right one, but with perhaps some annoying tweaks missing.

So you are saying that the map on the removed hyperplane could be space filling. Hmmm That seems like a good point. We are back to the homology argument.