Let $T = (X, \mathcal{T})$ be a D1106: Topological space.
Let $\mathsf{SA} : = \mathsf{SA}(X)$ be the D484: Set of sigma-algebras on $X$.
The Borel sigma-algebra on $X$ with respect to $T$ is the D11: Set \begin{equation} \sigma \langle \mathcal{T} \rangle : = \bigcap \{ \mathcal{F} : \mathcal{T} \subseteq \mathcal{F} \in \mathsf{SA} \} \end{equation}

Let $T = (X, \mathcal{T})$ be a D1106: Topological space.
The Borel sigma-algebra on $X$ with respect to $T$ is the D11: Set \begin{equation} \sigma \langle \mathcal{T} \rangle : = \bigcap \{ \mathcal{F} :\mathcal{T} \subseteq \mathcal{F} \text{ and } \mathcal{F} \text{ is a sigma-algebra on } X \} \end{equation}