I'm currently going through Munkres' Topology, and his discussion of bases for topologies has me a little confused. He defines a basis $\mathfrak{B}$ for a topology $\mathscr{T}$ on a set $X$ as a collection of subsets that satisfy the given requirements. These elements of $\mathfrak{B}$ are not defined as open, but in the following lemma which claims that $\mathscr{T}$ equals the collection of all unions of elements from $\mathfrak{B}$, he claims in the proof of this lemma that the elements of $\mathfrak{B}$ are in $\mathscr{T}$, and therefore all of the basis elements are open.
Why is this true? I know the basis elements are open in the topology generated by $\mathfrak{B}$, but the lemma refers to an arbitrary topology on some set X. Why should I assume that these are open sets? I appreciate any help. Thanks.