Metrizable-like locally convex topologies on C(X)

Abstract The classic Arens theorem states that the space C ( X ) of real-valued continuous functions on a Tychonoff space X is metrizable in the compact-open topology τ k if and only if X is hemicompact. Less demanding but still applicable problem asks whether τ k has an N N -decreasing base at zero ( U α ) α ∈ N N , called in the literature a G -base. We characterize those spaces X for which C ( X ) admits a locally convex topology T between the pointwise topology τ p and the bounded-open topology τ b such that ( C ( X ) , T ) is either metrizable or is an ( L M ) -space or even has a G -base.