Local Basis in a topological vector space.

Question

i'm studying local basis, but i'm stucked, because i already understood what a local basis means, if $\mathscr{B}$ is a local basis of $x \in X$, where $X$ is a topological vector space, then

1. $x \in B$, for all $B \in \mathscr{B}_x$.
2. If $U$ is an open set $U \in \tau$, such that $x \in U$, then $\exists B \in \mathscr{B}_x$ such that $x \in B \subseteq U$.

now i would like give another definition to local basis in terms of basics, my idea is: Let $\tau$ vectorial topology, such that is given by a basis $\mathscr{B}$, so the collection $\{x + B |B \in \mathscr{B}\}$ is a local basis for $x$, but i don't know if the basic $B \in \mathscr{B}$ must cotains $x$,i already now that $\{B| x \in B, \in \mathscr{B}\}$ is a local basis of $x$, i'm trying to relate this with the fact that $x + U$ is an open set, if $U \in \tau$, any help will be really appreciated, thank you so much.