According to Willard,
If $(X,\tau)$ is a topological space, a base for $\tau$ is a collection $\mathscr{B} \subset \tau$ such that $\tau=\{ \bigcup_{B \in \mathscr C} : \mathscr C \subset \mathscr B\}$. Evidently, $\mathscr B$ is a base for $X$ iff whenever $G$ is an open set in $X$ and $p \in G$ there is some $B \in \mathscr B$ such that $p \in B \subset G$.
Question 1: Is it safe to assume that in the sentence beginning with "Evidently" it is assumed that $\mathscr B \subset \tau$, for otherwise the iff statement is not true.
Question 2: I've been told that not all basic sets are open, but it seems by the above definition that they are defined to be open.
Comment on Question 2: There is also a definition of being a base for "a" topology. Is this what was meant by not all basic sets are open? Is this just a semantic issue, i.e. the basic sets are open in the topology that the base is a base for but not open in general? Or can there be a base for a topology where the basic sets are not open in that same topology?