Let $\mathfrak B_{[a,b]}$ be the Borel $\sigma$-algebra of some non degenerate interval $[a,b]$ in $\mathbb R$. Let $\mathfrak B_{\mathbb R}$ be the Borel $\sigma$-algebra of $\mathbb R$. We want to show that $\mathfrak B_{[a,b]} \subseteq \mathfrak B_{\mathbb R}$. I thought of showing that $\mathcal T_{[a,b]}$ (where $\mathcal T_{[a,b]}$ is the set of open sets in the metric space $([a,b],d)$) is contained in $B_{\mathbb R}$ which would thus imply that $\mathfrak B_{[a,b]} \subseteq \mathfrak B_{\mathbb R}$ by minimality. But a peer told me that the two sets are different and you can't compare them? I don't see anything wrong with this though. Indeed, every set in $\mathcal T_{[a,b]}$ is of the form $[a,b] \cap U$ for some open set $U$ in $\mathbb R$.
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