Reference/Literature on Cantor Sets I recently came across Cantor sets in my analysis course while doing a hw problem and find them extremely fascinating. I am trying to find other notes/literature on cantor sets with no success. I only managed to find this so far: https://www.missouriwestern.edu/orgs/momaa/ChrisShaver-CantorSetPaper4.pdf.
Are there any books/lecture notes available that I can refer to know more/ read up on cantor sets?
 A: See the recent (published in 2013) book The Elements of Cantor Sets With Applications by Robert W. Vallin.
A: *

*There is a theorem that any compact metric space is surjected upon by the Cantor set: http://www.maa.org/sites/default/files/images/upload_library/22/Ford/Benyamini832-839.pdf (This is mind bending.)


There are many neat corollaries of this result included in the link.


*Hausdorff measures. (Already discussed in your link.) I think the end of Follands book covers this? I don't know too much about it.


Random fact about the Cantor set if you know some commutative algebra: $\operatorname{Spec} \mathbb{F}_2[x_0, \ldots x_n \ldots ] / (x_i(x_i - 1))_{i = 0}^{i = \infty}$ has an underlying topology which is homeomorphic to the Cantor set (both are homeomorphic to $\{0,1\}^{\mathbb{N}}$, where the $\{0,1\}$ has the discrete topology). 
Proof sketch: Let $R = \mathbb{F}_2[x_0, \ldots x_n \ldots ] / (x_i(x_i - 1))_{i = 0}^{i = \infty}$. All the prime ideals of $R$ are maximal, since a Boolean domain is a field, and in fact $\mathbb{F}_2$. The maximal ideals of $R$ are of the form $(x_i + a_i)_{i = 0}^{\infty}$, where
$a_i$ is a sequence in $\{0,1\}^{\mathbb{N}}$ - this is because each maximal ideal is a map to $\mathbb{F_2}$ and correspondingly is determined by where it sends the $x_i$. This gives the bijection. You can compare their topologies using the cylindrical sub-basis for the product topology, and the subbasis consisting of the $D(x_i)$ and the $D(1 + x_i)$ for the Zariski topology of $\operatorname{Spec} R$ (to verify that the latter is a subbasis, use that inside $R$, $V(f + g) = (V(f) \cap V(g)) \cup (V(f + 1) \cap V(g + 1))$, since every prime ideal contains either $f$ or $f+1$, and the standard fact $V(fg) = V(f) \cup V(g)$.).
