Can we either add the empty set $\varnothing$arbitrarily to a topology, or is $\varnothing$ contained in any collection $B$ of subsets of a set $S$?

In my reading material (Appendix A of Introduction to Manifolds, Loring, pp. 322) the “only if” part for the claim

A collection $\mathcal{B}$ of subsets of a set $S$ is a basis for some topology $\mathcal{T}$ on $S$ if and only if i.) $S$ is the union of all the sets in $\mathcal{B}$, and ii.) given any two sets $B_1, B_2 \in \mathcal{B}$ and a point $p \in p \in B_1 \cap B_2$, there is a set $B \in \mathcal{B}$ such that $p \in B \subset B_1 \cap B_2$.

begins by defining $\mathcal{T}$ to be the set of all sets that are unions of the sets in $\mathcal{B}$. Then the author states that:

Then the empty set $\varnothing$ and the set $S$ are in $\mathcal{T}$ and $\mathcal{T}$ is clearly closed under arbitrary union.

By the property i.) $S$ is evidently in $\mathcal{T}$. But what about $\varnothing$? Is it tacitly assumed that $\varnothing$ belongs to all collections of elements, or can we just insert the empty set to $\mathcal{T}$ as we please?

The correct phrasing would be that $\mathcal T$ is the family of unions of subfamilies of $\mathcal B$. You can check that $\varnothing$ is the union of the empty family.