Separating family of seminorms (${\bigcap }_{p\in \mathcal{P}}\ker p=\{0\}$)

A family of seminorms separates points when every nonzero vector $z$ has $p(z)>0$ for some member $p$. Equivalently, its kernels of seminorms have intersection $\{0\}$, or any two distinct vectors $x,y$ have $p(x-y)>0$ for some $p$. Individual seminorms can have large kernels of seminorms; the whole family must have no common invisible direction.

The associated seminorm topology is Hausdorff if and only if the family separates points. Indeed, if $p(x-y)=a>0$, the neighbourhoods $\{u:p(u-x)<a/3\}$ and $\{v:p(v-y)<a/3\}$ are disjoint by the triangle inequality. Conversely, if every $p$ vanishes on a nonzero $z$, then zero and $z$ belong to exactly the same open sets and cannot be separated. This criterion keeps separation distinct from completeness: detecting all directions does not guarantee that all Cauchy sequences have limits.