Assuming the Axiom of Choice, the Borel sigma-algebra on R^n has cardinality continuum for n at least one

By [L1] and repeated use of the product identity in [L8], endpoint tuples show that the rational open boxes form an at most countable family. It is infinite because $q\mapsto (q,q+1{)}^n$ injects $\mathbb{Q}$ into that family when $n\ge 1$. Thus it is countably infinite, and [L2] and [L3] give $\mathrm{\mid }\mathcal{B}({\mathbb{R}}^n)\mathrm{\mid }\le {\mathrm{\aleph }}_0^{{\mathrm{\aleph }}_0}$.

Characteristic functions inject $\mathcal{P}(\mathbb{N})$ into ${\mathbb{N}}^{\mathbb{N}}$. Conversely, the graph map injects ${\mathbb{N}}^{\mathbb{N}}$ into $\mathcal{P}(\mathbb{N}\times \mathbb{N})$, which is equinumerous with $\mathcal{P}(\mathbb{N})$ by [L4] and [L8]. Hence [L5] and [L6] give ${\mathrm{\aleph }}_0^{{\mathrm{\aleph }}_0}=\mathfrak{c}$.

For $S\subseteq \mathbb{N}$, let $E_k(S)$ be the singleton $\{(k,0,\dots ,0)\}$ when $k\in S$ and the empty set otherwise. The point is defined because $n\ge 1$, each $E_k(S)$ is closed and hence Borel by [L2], and $\mathrm{\Psi }(S):={\bigcup }_{k\in \mathbb{N}}{E}_k(S)$ is Borel. Distinct subsets give distinct unions, so $\mathrm{\Psi }$ injects $\mathcal{P}(\mathbb{N})$ into $\mathcal{B}({\mathbb{R}}^n)$.

Steps 1.1 and 1.2 give an injection $\mathcal{B}({\mathbb{R}}^n)\to \mathcal{P}(\mathbb{N})$, while step 1.3 gives the reverse injection. Applying [L5] proves equality with $\mathfrak{c}$.