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19 votes
2 answers
1k views

Are there space filling curves for the Hilbert cube?

There is a surjective continuous map $[0;1]\rightarrow [0;1]^2$ ("space filling curve"). Using such a map one can easily get space filling curves for all finite dimensional cubes. So my question is: ...
HenrikRüping's user avatar
19 votes
1 answer
772 views

convexity of images of space-filling curves

Suppose $f:[0,1]\to[0,1]^2$ is continuous and for each $t\in[0,1]$, the area of $\lbrace f(s) : 0\le s\le t \rbrace$ is $t$. For what sets of values of $t\in[0,1]$ can $\lbrace f(s) : 0\le s\le t \...
Michael Hardy's user avatar
8 votes
1 answer
627 views

Space filling curve whose all level sets are finite (countable)

Is there a continuous surjective function $f:[0,1] \to [0,1]^2$ such that every level set $f^{-1}(y)$ is a finite set? If the answer is no, what about if we replace the finiteness of level sets by "...
Ali Taghavi's user avatar
7 votes
2 answers
1k views

Unusual space-filling curve

Around 1998, I encountered a (forgotten) reference to a particularly strange space-filling curve. Consider a foliation as a collection of continuous nonintersecting curves that start at $(0,0)$ and ...
Steven Heston's user avatar
6 votes
3 answers
1k views

Topological dimension of the image of continuous surjective functions

Consider two topological spaces $(X,\tau)$ and $(Y,\omega)$ and a continuous surjective function $f\colon X\to Y$. Let $\mathrm{dim}(X)$ and $\mathrm{dim}(Y)$ denote the Lebesgue covering dimension ...
JLM's user avatar
  • 71
3 votes
1 answer
669 views

Do Peano curves provide a counterargument to Grothendieck's critique?

This question arose in the context of an earlier question on Grothendieck's critique of the traditional foundations of topology. Can the paper Group Invariant Peano Curves by Cannon and Thurston be ...
Mikhail Katz's user avatar
  • 16.6k