A question on the sets $V(p,\epsilon)$ in the book of Rudin I am reading the book of Rudin's functional analysis. Let us start with a vector space $X$ over the reals and we let $P$ be a separating family of seminorms on $X$. For each $p\in P$ and $\epsilon >0$, write 
$$V(p,\epsilon)=\{x\in X: p(x)<\epsilon\}.$$
Rudin claimed that the family $P$ induces a locally convex topology $\tau$, hence making $X$ a locally convex TVS, with the property that every $p$ in $P$ is $\tau$-continuous.
Question. Do the sets $V(p,\epsilon)$ belong to $\tau$?

The sets $V(p,\varepsilon)$ belong to $\tau$ by definition. $\tau$ consists of arbitrary unions of finite intersections of elements of the form $V(p,\varepsilon,x)$ for some $x\in X, p\in P$ and $\varepsilon>0$, where $V(p,\varepsilon,x)=\{u\in X,p(x-u)<\varepsilon\}$.
