Show the following 3 spaces are homeomorphic.

Question

I want to show the following 3 spaces are homeomorphic to one another.

1. $\{0,1\}^{\omega}$, i.e., $\{0,1\}^{\mathbb{N}}$, the set of infinite sequences of 0's and 1's, in the dictionary order topology.

2. $\{0,1\}^{\omega}$, the same set, in the product (cylinder) topology.

3. The Cantor Set $C$.

I've been stuck in this question for quite a while. Sincerely appreciate any help! Thanks a lot.

Answer 1

This answer of mine sets out the basics for the Cantor set definition and outlines the homeomorphism with $\{0,1\}^\omega$ as well.

It remains to notice that the lexicographic order is just the usual order that $C$ inherits from $[0,1]$, under this homeomorphism. So we get an order isomorphism that way.

There is also Brouwer's theorem that all spaces $X$ that are compact, Hausdorff, have a countable base of clopen sets and do not have isolated points are homeomorphic to the Cantor set. But this is not commonly taught (only as a matter of course in descriptive set theory, where it's used extensively).