Solution
= Solution
By the covering condition, $X=\bigcup_{B\in\mathcal B}B$ belongs to $\mathcal T$. The empty set belongs to $\mathcal T$ as the empty union. An arbitrary union of sets that are themselves unions of members of $\mathcal B$ is still a union of members of $\mathcal B$. Thus $\mathcal T$ contains $X,\varnothing$ and is closed under arbitrary unions, as required for a <topology>.