Maybe this is me being pedantic, but this thought has just occurred to me.
The Borel $\sigma$-algebra on X is the smallest $\sigma$-algebra containing all open sets. When we talk about Borel $\sigma$-algebra on $\mathbb{R}^n$, we always assume open sets are the topology generated by the Euclidean norm. Why is this the case? I assume this is just convention.
are there examples when we say 'Borel $\sigma$-algebra' on a sample space rather than a subset of $\mathbb{R}^n$, without referring to the topology? I would assume if we take a normed space, we assume the Borel $\sigma$-algebra to be the one wrt the topology generated by the norm.