real analysis - Baby Rudin Problem 2.17: Homeomorphism to a Cantor Set?

Problem 2.17 asks us to consider some properties of numbers in $[0,1]$ in base $10$ with decimal expansions consisting only of $4$ and $7$. Let's call the set of all these numbers $E$.

The question asks us something different, but is this set homeomorphic to the Cantor set? In particular, if we write each element in the Cantor set in base $3$, and send $0$s to $4$s and $2$s to $7$s, will we get a homeomorphism with $E$?

Also, what about the set of base-$10$ numbers with decimal expansions consisting only of $3$ or more separate digits, for example? Will these be homeomorphic to the Cantor set, or to something else?

Answer

Every metric space that is compact, totally disconnected (the only non-empty connected subsets are singletons) and has no isolated points is homeomorphic to the Cantor set. This is a classical result due to Brouwer.

Your set fulfills these criteria: it's a closed subset of $[0,1]$ (so compact metric), contains no non-trivial intervals (which implies totally disconnected) and every neighbourhood of a point in it contains (many) other points from the set (so no isolated points).

The homeomorphism you suggested also works.

Other applications: the Cantor set $C$ is homeomorphic to $\{0,1\}^\mathbb{N}$, homeomorphic to any of its finite or countable powers, etc. etc.

Also, the final question is answered as well: the same arguments apply and we also get a Cantor set, whether we restrict to 2 or 3 digits (up to 9 I think will work). It's pretty ubiquitous.