Definition:Locally Convex Space/Standard Topology

Definition
Let $\struct {X, \mathcal P}$ be a locally convex space. For each $p \in \PP$, $\epsilon > 0$ and $y \in X$, define:


 * $\map {B_p} {\epsilon, x} = \set {y \in X : \map p {y - x} < \epsilon}$

and:


 * $\SS = \set {\map {B_p} {\epsilon, x} : p \in \PP, \, \epsilon > 0, \, x \in X}$

Let $\tau$ be the topology generated by $\SS$.

We call $\tau$ the standard topology on $\struct {X, \mathcal P}$.