  Premetrics and premetrisable spaces have been long studied and their
topological interrelationships are well-understood. Consider the category ${\bf
Pre}$ of premetric spaces and $\epsilon$-$\delta$ continuous functions as
morphisms. The absence of the triangle inequality implies that the faithful
functor ${\bf Pre} \to {\bf Top}$ - where a premetric space is sent to the
topological space it generates - is not full. Moreover, the sequential nature
of topological spaces generated from objects in ${\bf Pre}$ indicates that this
functor is not surjective on objects either. Developed from work by Flagg and
Weiss, we illustrate an extension ${\bf Pre}\hookrightarrow {\bf P} $ together
with a faithful and surjective on objects left adjoint functor ${\bf P} \to
{\bf Top}$ as an extension of ${\bf Pre} \to {\bf Top}$. We show this
represents an optimal scenario given that ${\bf Pre} \to {\bf Top}$ preserves
coproducts only. The objects in ${\bf P}$ are metric-like objects valued on
value distributive lattices whose limits and colimits we show to be generated
by free locales on discrete sets.
