Compact covers and function spaces

For a Tychonoff space X  , we denote by Cp(X)Cp(X) and Cc(X)Cc(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 Cp(X)Cp(X), we extend Okunevʼs results by showing that if there exists a surjection from Cp(X)Cp(X) onto Cp(Y)Cp(Y) (resp. from Lp(X)Lp(X) onto Lp(Y)Lp(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 Cc(X)Cc(X) onto Cc(Y)Cc(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 ℓpℓp-equivalence in the class of metric spaces (J. Baars, J. de Groot, J. Pelant). If X   is completely metrizable and ℓpℓp-equivalent to a first-countable Y, then Y is metrizable (V. Valov). Some additional results are presented.