Is there, e.g. in $\mathbb R^4$ a simple curve that does not contain the origin and intersects every subspace of dimension 2?


Sorry if the question is too easy, but I just cannot figure it out.
In three dimensions such a curve exists, but I cannot imagine four dimensions. Is it possible to somehow lift-up the Peano-curve? What about higher dimensions and higher dimensional subspaces?



Can a simple curve intersect every subspace of dim 2 and avoid the origin?
Consider closed space filling curve $\theta:\mathbb S^1\to\mathbb S^3$.
You can choose $\theta$ so that $\theta^{-1}(x)$ is finite for any $x\in\mathbb S^3$
and $|\theta^{-1}(x)|=1$ for all but countable set in $\mathbb S^3$.


Then it is easy to find a function $\rho:\mathbb{S}^1\to\mathbb R\_+$
so that the curve $\gamma(x)=(\theta(x),\rho(x)$ in polar coordinates 
is the curve you are looking for.


