1. ## Cantor set questions

Defining a "Cantor space" as a non-empty, complete, totally disconnected perfect set $C\subseteq[0,1]$ with d(x,y)=|x-y| in R (if I'm right this would only be THE Cantor set (i.e. infintiely removing all mittle thirds fron [0,1]) up to homomorphism, either way we don't "officially" know this yet) prove or disprove:

(i) finite unions of Cantor sets are also Cantor sets.
(ii)countable unions of Cantor sets are also Cantor sets.
(iii)the intersection of two Cantor sets is either empty or also a Cantor set.
(iv)every Cantor set is a Lebesque null set.
(v)the complement of a Cantor set is never a Lebesque null set.
(vi) every infinite, closed, totally disconnected set $A\subseteq[0,1]$ is uncountable.
(vii) every Cantor set is uncountable.
(viii)every non-empty, complete, perfect set $A\subseteq[0,1]$ is uncountable.

I'm pretty sure (i) is true and (ii) is false (due to the fact that Q is countable). If I'm right, (v) follows directly from (iv), which I'm pretty sure is true. I think there are simple counterexamples for (vi) and (viii) but I'm at a loss for ideas of what they would be, and (vii) would be trivial if I could use that statement up there in parenthesis.