-2

The concept of space-filling curves is well-known: There exist continuous maps from $[0,1]$ that fill a box in $n$ dimensions. Does there exist a more general concept of filling $n$-dimensional space by a continuous injective map from the $m$-box into the $n$-box, where $m<n$?

Jas Ter
  • 1,575
  • What is the math question here? Are you asking for some interesting examples? – Moishe Kohan Jun 03 '24 at 13:12
  • I am basically looking for terminology or existing results. I have found very little. Of course, we can give specific examples: It is trivial to fill 3-space with a plane-filling curve by extrusion, which gives a space filling surface, and I can fill 4-space with a plane-filling curve by using the curve twice, for each 2d coordinate separately. And we can similarly fill 5-space by 2-space (by filling 2 and 3 space). But the general question interests me. – Jas Ter Jun 03 '24 at 13:19
  • 1
    If you replace the nonsensical "injective" by "surjective" then these are "dimension-raising maps". See my answer here for an interesting example. – Moishe Kohan Jun 03 '24 at 13:33

2 Answers2

1

If $f:I\to I^n$ is your continuous map from the interval onto an $n$-dimensional box, then $f\circ \pi_1 : I^m\to I^n$ is a continuous map from an $m$-box onto an $n$-box (here $\pi_1:I^m\to I$ is projection onto the first coordinate).

MPW
  • 44,860
  • 2
  • 36
  • 83
1

As far as I can tell, you are asking whether there exists a continuous, surjective, injective function $f : [0,1]^m \to [0,1]^n$ with $m<n$.

No such function exists. A theorem of point set topology (found in Munkres Topology for example) says that any continuous, surjective, injective function from a compact space to a Hausdorff space is a homeomorphism. But it is a theorem of algebraic topology (found in Hatcher's Algebraic Topology for example) that $[0,1]^m$ and $[0,1]^n$ are not homeomorphic.

Lee Mosher
  • 135,265