Suppose $\mathscr{P}$ is a separating family of seminorms on a vector space $X$. Associate to each $p\in\mathscr{P}$ and each positive integer $n$ the set
$$V(p,n)=\left\{x;p(x)<\frac{1}{n}\right\}.$$
Let $\mathscr{B}\subset\mathcal{P}(X)$ be the collection of all finite intersections of the sets $V(p,n)$.
Let $\tau$ be the family of all unions of translates of members of $\mathscr{B}$, that is, $V\in \tau$ if only if there are a family of indexes $J$, a sequence $(x_j)_{j\in J}$ and a collection $(B_j)_{j\in J}$ on $\mathscr{B}$ such that
$$V=\bigcup_{j\in J}(x_j+B_j).$$
I assume that $\tau$ is a topology on $X$.
Let $x,y\in X$. Let $V$ a neighborhood of $x+y$. By definition there are a family of indexes $J$, a sequence $(x_j)_{j\in J}$ and a collection $(B_j)_{j\in J}$ on $\mathscr{B}$ such that
$$x+y\in V=\bigcup_{j\in J}(x_j+B_j).$$
Then, there is $i\in J$ such that $$x+y\in x_i+B_i.$$
I need to found a neighborhood $V_1$ of $x$ and a neighborhood $V_2$ of $y$ such that $$V_1+V_2\subset x_i+B_i.$$ This is possible?
On the other hand, there is $B\in \mathscr{B}$ such that $(x+y)+B\subset V$?