Borel Sigma-Algebra on Euclidean Space

Let $n\ge1$ be a natural number. The Borel $\sigma$-algebra on Euclidean space $\mathbb{R}^n$, written $\mathcal{B}(\mathbb{R}^n)$, is the $\sigma$-algebra generated by the family of all Euclidean open subsets of $\mathbb{R}^n$; its members are the Borel subsets of $\mathbb{R}^n$. For $n=1$, under the identification of the real line with $\mathbb{R}^1$, this defining clause is the one used for the Borel $\sigma$-algebra on the real line, and the two notations $\mathcal{B}(\mathbb{R}^1)$ and $\mathcal{B}(\mathbb{R})$ denote the same $\sigma$-algebra.