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

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.
