Are these definitions of Borel $\sigma$-field and Borel sets correct$?$
I encountered a bit different definition of Borel $\sigma$-field and Borel sets. Possibly the reason is that I'm not accustomed to these definitions w.r.t. topological spaces. In any case, I'd like to verify these definitions $($marked in the red rectangle$)$. Please let me know whether they are correct or not. Thank you.![enter image description here](