Let $\mathcal{A}$ a collection of subsets of $\mathbb{X}$. Let $ \mathcal{T}$ the topology generated by colection $\mathcal{A}$ and $\mathcal{F}$ the $\sigma$-field generated by $ \mathcal{A}$.
Denote by , $\mathrm{Borel}(\mathcal{T})$ the $\sigma$-field of Borel sets of $\mathbb{X}$ whit respect the topology $\mathcal{T}$.
Question1: Is true that $\mathrm{Borel} (\mathcal{T}) =\mathcal{F}$?
Thank's.
Edit:
Question 2: Suppose now that $\mathbb{X}$ is countable and discrete with respect to some metric $d$ that generates the topology $\mathcal{T}$. Is true that $\mathrm{Borel} (\mathcal{T}) =\mathcal{F}$?
If answer is not, I have a more question.
Question 3 If the answer to question 2 is still there some condition (topological, metric or condition of measurability) on $\mathbb{X}$ or $\mathcal{A}$ it is enough that the answer is yes?