Corollary

If p is a continuous semi-norm on E and q is a semi-norm such that $q \leq p$, then q is continuous in E.

When p is an ultra-semi-norm on E, then the sets W(p, $\alpha$) and V(p, $\alpha$) are both open and closed. For, we have seen that W(p, $\alpha$) is open; on the other hand if z is a cluster point of W(p, $\alpha$), then there exists $y \in W(p, \alpha)$ such that $p(y - z) < \alpha$, and from (SN$_{II}'$) we have $p(z) < \alpha$, thus W(p, $\alpha$) is closed. Also, V(p, $\alpha$) is closed since p is continuous; further if $p(x) \leq \alpha$ and $p(y) \leq \alpha$, then $p(x + y) \leq \alpha$ by (SN$_{II}'$), and this shows that V(p, $\alpha$) is open.