Metrizable-like locally convex topologies on C(X)

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 τk if and only if X is hemicompact. Less demanding but still applicable problem asks whether τk has an NN-decreasing base at zero (Uα)α∈NN, 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 τp and the bounded-open topology τb such that (C(X),T) is either metrizable or is an (LM)-space or even has a G-base.