Alternate presentation of the Borel $\sigma$-Algebra over the unit interval

Question

Just after a few month of not touching the subject, I'm already rusty on the key ideas: My off the top definition for the Borel $\sigma$-Algebra over the unit interval is $$ \mathcal{B}_{[0,1]} = \sigma([a,b[ \: | \: a<b \in [0,1])$$.

Can somebody share a presentation along the lines of $\mathcal{B}(\mathbb{R})\cap \mathcal{A} = \mathcal{B}_{[0,1]}$?

I started with $\mathcal{B}(\mathbb{R})\cap [0,1] = \mathcal{B}_{[0,1]}$ which made no sense, considered both $\mathcal{B}(\mathbb{R})\cap \sigma([0,1])$ and $\mathcal{B}(\mathbb{R})\cap \mathfrak{P}([0,1])$, which are both wrong?

Answer 1

$\mathcal B([0,1]) = \{A \cap [0,1]: A \in \mathcal B(\mathbb R)\} = \{B \in \mathcal B(\mathbb R): B \subseteq [0,1]\}$. If $\mathfrak P([0,1])$ means the power set of $[0,1]$, that is $\mathcal B(\mathbb R) \cap \mathfrak P([0,1])$.