Separability and submetrizability in locally convex spaces

We introduce the property of countable separation for a locally convex Hausdorff space X and relate it to the existence of a metrizable coarser topology. Building on this, we demonstrate how the separability of X is equivalent to the existence of a locally convex topology on the dual $$X'$$ X ′ that is metrizable and coarser than the weak topology σ (X', X) σ ( X ′ , X ) . This result generalizes known conditions for separability and provides a precise duality between separability and metrizability. We also show how to derive new and known conditions for the separability of X from this characterization.