TITLE: $\{0,1\}^\mathbb{N}$ is homeomorphic to which subset of $\mathbb{R}$.
QUESTION [6 upvotes]: In an interview the interviewer asked me the following but I failed to give the answer.   
$\{0,1\}^\mathbb{N}$ with product topology is homeomorphic to which subset of $\mathbb{R}$?
Can anyone give me the answer and explain me please? thanks for your kind help.

REPLY [3 votes]: This idea was dealt with by Roger Penrose in page 370 of his book "The Road to Reality: A Complete Guide to the Laws of the Universe".
The set is called the Cantor Set, and according to Wikipedia: As a topological space, the Cantor set is naturally homeomorphic to the product of countably many copies of the space $\{0, 1\}$, where each copy carries the discrete topology. This is the space of all sequences in two digits.