Definition:Locally Convex Space/Standard Topology

Definition
Let $\struct {X, \mathcal P}$ be a locally convex space. Define the collection of sets $\tau \subseteq \map {\mathcal P} X$ by:


 * $U \in \tau$ $U \subseteq X$ and for each $x \in U$, there exists finitely many $p_1, p_2, \ldots, p_n \in \mathcal P$ and $\epsilon > 0$ such that:


 * $\set {y \in X : \map {p_k} {y - x} < \epsilon, \text { for all } k \in \set {1, 2, \ldots, n} } \subseteq U$

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

Also see

 * Locally Convex Space Induces Topology verifies that this collection $\tau$ is indeed a topology on $X$