Sunday, January 24, 2016

functions - Bijection from $[0,1]^3$ to $[0,1]$?





Is there any bijection from $[0,1]^3$ to $[0,1]$? How can I construct it?


Answer



Hint:



If there exists a surjection between $A$ to $B$ and a surjection between $B$ to $A$, then there exists a bijection between $A$ to $B$. In your case, space filling curves are surjections from $[0,1]$ to $[0,1]^3$. It should be easy to find a surjection going the other way.


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