I'm writing my bachelor thesis on various topics from set theory and descriptive set theory (mainly topological games), and I just read a paper in which the Baire Space $\mathbb{N}^\mathbb{N}$ is dubbed $\mathbb{R}$. Here is how it justifies it:
Since any two uncountable polish spaces are borel ishomorphic and since many set-theoretic questions are invariant under borel isomorphism it is customary to dub any uncountable polish space $\mathbb{R}$. Also in set theory it is often desirable to deal with a version of $\mathbb{R}$ which is transparent from a combinatorial point of view, so we will work with the Baire space $\mathbb{N}^\mathbb{N}$.
I don't really understand what they mean by "transparent from a combinatorial point of view". I have an intuition on why it is desirable to work with the Baire space. In fact I'd say that it is because it is pretty close to $\mathbb{R}$ (it is homeomorphic to the irrational numbers with the relative topology) and, in a certain sense, it allows to contruct its elements in a much simpler way than in $\mathbb{R}$. This emerges for example in the elegant characterization of continuous functions in the Baire space. So my questions are:
- What do they mean by "transparent from a combinatorial point of view"?
- Is my intuition on the construction of the baire space elements formalizable in some way? It is in some way related with the total disconnection of the space?
- If the answer to the previous question is affermative then the reason why the Baire space is desirable can rely also on this consideration (on the element construction)?
Thanks!
EDIT: this is the link to the article I was quoting: article I want to point out that the article doesn't say anything more than the quote I posted in this regard.
