In the Trèves' book, Topological Vector Spaces, Distributions and Kernels he defines:
A family $\mathscr{B}$ of subset of $E$ is a basis of a filter $\mathscr{F}$ on $E$ if the following two conditions are satisfied:
$BF_1)$ $\mathscr{B} \subset \mathscr{F}$, i.e., any subset which belongs to $\mathscr{B}$ must belong to $\mathscr{F}$.
$BF_2)$ Every subset of $E$ belonging to $\mathscr{F}$ contains some subset of $E$ which belongs to $\mathscr{B}$.
My question: Do I need to assume that $\mathscr{B}$ is non-empty? Or does that follow immediately from the definition?
By definition (at least all that I know of) of a filter $\mathscr{F}$ is non-empty (we can state that for sure that $E \in \mathscr{F}$) , so $BF_2$ forces that $\mathscr{B}$ is non-empty, to contain a subset of $E$.