Compact covers and function spaces

For a Tychonoff space X, we denote by C p ( X ) and C c ( X ) the space of continuous real-valued functions on X equipped with the topology of pointwise convergence and the compact-open topology respectively. Providing a characterization of the Lindelöf Σ-property of X in terms of C p ( X ) , we extend Okunevʼs results by showing that if there exists a surjection from C p ( X ) onto C p ( Y ) (resp. from L p ( X ) onto L p ( Y ) ) that takes bounded sequences to bounded sequences, then υY is a Lindelöf Σ-space (respectively K-analytic) if υX has this property. In the second part, applying Christensenʼs theorem, we extend Pelantʼs result by proving that if X is a separable completely metrizable space and Y is first countable, and there is a quotient linear map from C c ( X ) onto C c ( Y ) , then Y is a separable completely metrizable space. We study also a non-separable case, and consider a different approach to the result of J. Baars, J. de Groot, J. Pelant and V. Valov, which is based on the combination of two facts: Complete metrizability is preserved by ℓ p -equivalence in the class of metric spaces (J. Baars, J. de Groot, J. Pelant). If X is completely metrizable and ℓ p -equivalent to a first-countable Y, then Y is metrizable (V. Valov). Some additional results are presented.