$\omega^\omega$ correspondence with $\mathbb R$

Question

How does the natural continuous bijection between $\omega^\omega$ and $\mathbb R$ look like? I.e. why elements of $\omega^\omega$ are called reals?

Answer 1

Contra the fifth word of your question, there is no natural continuous bijection between $\omega^\omega$ and $\mathbb{R}$. Indeed, there isn't even a continuous injection from $\mathbb{R}$ to $\omega^\omega$ since the former is connected but the latter is totally disconnected.

However, there are still reasonably-low-complexity bijections between the two - namely, with respect to the natural topologies they are Borel isomorphic (that is, there is a Borel bijection between the two). We can see this by finding Borel injections in each direction, and then checking that the Cantor-Bernstein construction turns a pair of Borel injections into a Borel bijection.

Cooking up specific Borel injections each way is a good exercise. But here are a couple hints in each direction:

• The map $bin$ sending a real in $(0,1)$ to its canonical (= not-eventually-all-$1$s) binary representation is an injection into $\omega^\omega$ (in fact, into $2^\omega$). $(0,1)$ and $\mathbb{R}$ are clearly homeomorphic; can you show that $bin$ is Borel? (Note that we know per the above that $bin$ can't be continuous - the discontinuity crops up at the dyadic reals, do you see why?)

• In the other direction, the most commonly used map is via continued fraction expansions. However, it may be more intuitive to go a bit more combinatorial. First, we can go from $\omega^\omega$ to $2^\omega$ by "counting $0$s:" given $f\in\omega^\omega$, we write a binary sequence such that the number of $0$s between the $n$th and $(n+1)$th $1$s is $f(n)$. As an example, $f=(1,4,3,0,2,...)$ goes to $$(1,0,1,0,0,0,0,1,0,0,0,1,1,0,0,1,...)$$ This is obviously an injection, and is easily checked to be Borel; now just compose with the usual continuous map from $2^\omega$ to $\mathbb{R}$.

Thinking along these lines, it's also easy to check that in fact $\omega^\omega$ is homeomorphic to the set of irrationals (again, each with the usual topology).

Introductory texts on descriptive set theory - e.g. Kechris and Moschovakis - go into this in more detail. In particular, this is a specific example of the more general fact that any two uncountable Polish spaces are Borel isomorphic.

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All of this says that the difference between the reals and the reals (hehe) is fairly minor - we can conflate the two either via bijection of fairly low complexity, or by ignoring a small (= countable) set. This means that in many situations (e.g. forcing and descriptive set theory) it essentially doesn't matter which we use.

There are situations, of course, where the difference is meaningful - e.g. in computable structure theory (here/here) - but they are in practice rare enough that the abuse of terminology doesn't lead to trouble in practice.