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

user122424
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    Who calls them that? – Hagen von Eitzen Dec 26 '18 at 17:57
  • In my experience one typically identifies $\omega^\omega$ with the irrational elements of $\mathbb R$; and then we call them "reals" because they are equinumerous, and in particular "isomorphic up to a countable set". – Mees de Vries Dec 26 '18 at 18:01
  • In books on forcing they always use reals for $\omega^\omega$. Am I wrong? – user122424 Dec 26 '18 at 18:01
  • @HagenvonEitzen Sadly, logicians do tend to refer to elements of Baire space, elements of Cantor space, and related objects as "reals." This is because all the relevant spaces are Borel-isomorphic, so in many (but not all) contexts they are essentially interchangeable. – Noah Schweber Dec 26 '18 at 18:01
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    @HagenvonEitzen, see e.g. Wikipedia, which states "This space is commonly used in descriptive set theory, to the extent that its elements are often called “reals.”" – Mees de Vries Dec 26 '18 at 18:02
  • Given a sequence $a_0,a_1,a_2,\ldots$ with $a_i\in\omega$, you can consider the continued fraction $a_0+\frac1{1+a_1+\frac1{1+a_2+\frac1\ldots}}$. This alrady takes you close (enough) to the goal ... – Hagen von Eitzen Dec 26 '18 at 18:03
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    @Noah I don't see anything sad about it... – Andrés E. Caicedo Dec 26 '18 at 18:07
  • @AndrésE.Caicedo I think it is at least somewhat sad - while for many people (including me) it's actually a clarifying abuse of terminology, I think it also makes the topic slightly more impenetrable for people approaching the topic from the outside. – Noah Schweber Dec 26 '18 at 18:16
  • @Noah I doubt that is really the case. It takes half a minute to clarify the usage, that is not really an obstacle. – Andrés E. Caicedo Dec 26 '18 at 18:17
  • @AndrésE.Caicedo I mean I'm ultimately basing this off of specific interactions with various people who I consider to be giving the subject a reasonable attack - my impression is that this is a conflation whose content we do tend to underestimate. And plenty of texts simply say something like "we refer to elements of $\omega^\omega$ as reals" - or simply refer to them this way without even mentioning it - without saying why this is fine. So I do think it's a reasonable point of confusion. But ultimately this is of course subjective, and I do think the conflation is ultimately quite useful. – Noah Schweber Dec 26 '18 at 18:19
  • @Noah I actually don't think texts should say more in general (introductory texts of course should, and my experience is that they typically do), since usually all that matters is that the space in question has the right size. In contexts where the difference is important, or the fact that the identification can be carried out via Borel maps, I've always seen it pointed out explicitly. – Andrés E. Caicedo Dec 26 '18 at 18:30
  • (@Noah I agree that it helps others if our language matches what they are used to, of couse. But I also think that the intuitions provided by our usage of language end up being more important.) – Andrés E. Caicedo Dec 26 '18 at 18:33
  • @MeesdeVries Please have a look at this. – user122424 Jan 03 '19 at 16:32

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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.

Noah Schweber
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