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I'm familiar with the fact that, if I'm not mistaken, there is a one-to-one correspondence between the unit interval $[0, 1]$ and the unit sphere $S^2$ though I'm not sure explicitly how to find it. My professor told me that in fact there is a one-to-one correspondence between the standard middle third Cantor set $C$ and the unit sphere $S^2$. Is that correct, and if so, how does one construct such a correspondence? This seems to be an interesting idea to investigate further, but I know of no articles that make any mention of this.

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Would the one-to-one correspondence between the unit interval and the unit sphere be some sort of stereographic projection? –  Libertron Jan 31 '12 at 23:39
    
You’re not going to get a particularly nice construction, but there’s a bijection simply because both sets have cardinality $2^\omega=\mathfrak{c}$. –  Brian M. Scott Jan 31 '12 at 23:47
    
Is there some additional structure that this mapping is supposed to preserve? All sets of the same cardinality have a 1-1 correspondence, though the exact details are usually not interesting (the map may not even have an explicit form, but simply is known to exist via indirect means). –  Nick Alger Jan 31 '12 at 23:50
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@BrianM.Scott: Even if it's not "nice," what exactly then would the construction of this map be? I agree with your cardinality argument, by the way. –  Libertron Feb 1 '12 at 0:23
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Since the Lebesgue measure of the $C$ is $0$ and the measure of $S^2$ is $4\pi$, Lebesgue measure cannot be preserved. What measure does the mapping preserve? –  robjohn Feb 1 '12 at 1:09

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It's perhaps a surprising fact that the standard Cantor set has a continuous surjection onto any compact metric space, and in particular the sphere $S^2$. I'm not sure if the map that ends up getting constructed is an injection, but this at least gets you half of the way, continuously! This is discussed in some books on general topology, for instance in Hocking and Young's book Topology . There's an article by Yoav Benyamini that discusses some further consequences of this fact that you can find here.

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It can't be a continuous bijection, since it would then (using compact-and-Hausdorff-ness) be a homeomorphism, and we would have a connected space homeomorphic to a disconnected one. –  user16299 Feb 1 '12 at 1:08

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