Invertible Borel set preserved mapping between $\times_{n \in \mathbb{N}} \mathbb{C}$ and $[0, 1]$ The following question is from Linear Operators Part II editted by Dunford & Schwartz, Exercise 15 in Chapter X.
Define $P$ to be the topological space $\times_{n \in \mathbb{N}}\,\mathbb{C}$ (where $\mathbb{C}$ is equipped with the usual topology). Show that there is a bijective mapping $h: P \rightarrow [0, 1]$ such that $h(A)$ is a Borel set iff $A$ is a Borel set.
Edit: Thanks for Eric's comment, I was reminded that this mapping does not have to be a homeomorphism because it only require the image of a Borel set to be Borel. Hence such a mapping could still exist.
 A: Both $P$ and $[0,1]$ are Polish spaces, that is separable topological spaces admitting a complete metric. Then these spaces are Borel isomorphic by Theorem 15.6 from [Kech] (see below).
But first we recall the following definitions. A measurable space $(X,\mathcal S)$ is a set $X$ with a $\sigma$-algebra $\mathcal S$ on it. Measurable spaces $(X,\mathcal S)$ and $(Y,\mathcal T)$ are isomorphic, if there exists a bijection $f$ from $X$ to $Y$ such that $f(A)\in\mathcal T$ and $f^{-1}(B)\in\mathcal S$ for each $A\in\mathcal S$ and $B\in\mathcal T$. A measurable space $(X,\mathcal S)$ is a standard Borel space, if it is isomorphic to a measurable space $(Y,\mathcal B(Y))$, where $Y$ is a Polish space and $\mathcal B(Y)$ is the $\sigma$-algebra of Borel subsets of $X$.
Now we cite [Kech]. Here $\mathbb I=[0,1]$, $\mathcal C$ is the Cantor set, and $\mathcal N=\Bbb N^{\Bbb N}$ endowed with the product topology.
(Remark that for the particular case from the question it suffices to find a Borel injection from $P$ to $[0,1]$ and then apply Theorem 15.7).

Referenced claims and their proofs:
6.4–5, 7.8,
7.9.part1, 7.9.part2, 14.12, and 15.1–2.
References
[Kech] A. Kechris, Classical Descriptive Set Theory, – Springer, 1995.
