Projective σ-compactness, ω1-caliber, and Cp-spaces

A space is called projectively σ-compact, if every separable metrizable continuous image of this space is σ-compact. In particular, we establish when Cp(X) is projectively σ-compact. In the last section, a theorem on cardinality of Lindelöf spaces is proved and then applied to obtain a result in Cp-theory.