Definition:Topology Induced by Pseudometric

Definition
Let $\struct {X, d}$ be a pseudometric space.

Let $\tau_d$ be the set of all $X \subseteq S$ which are open in the sense that:
 * $\forall y \in X: \exists \epsilon > 0: \map {B_\epsilon} y \subseteq X$

where $\map {B_\epsilon} y$ is the open $\epsilon$-ball of $y$.

We call $\tau_d$ the topology on $X$ induced by $d$.

Also see

 * Pseudometric induces Topology shows that $\tau_d$ is indeed a topology on $X$
 * Definition:Topology Induced by Metric