Projective versions of selection principles

All spaces are assumed to be Tychonoff. A space X is called projectively P (where P is a topological property) if every continuous second countable image of X is P . Characterizations of projectively Menger spaces X in terms of continuous mappings f : X → R ω , of Menger base property with respect to separable pseudometrics and a selection principle restricted to countable covers by cozero sets are given. If all finite powers of X are projectively Menger, then all countable subspaces of C p ( X ) have countable fan tightness. The class of projectively Menger spaces contains all Menger spaces as well as all σ-pseudocompact spaces, and all spaces of cardinality less than d . Projective versions of Hurewicz, Rothberger and other selection principles satisfy properties similar to the properties of projectively Menger spaces, as well as some specific properties. Thus, X is projectively Hurewicz iff C p ( X ) has the Monotonic Sequence Selection Property in the sense of Scheepers; βX is Rothberger iff X is pseudocompact and projectively Rothberger. Embeddability of the countable fan space V ω into C p ( X ) or C p ( X , 2 ) is characterized in terms of projective properties of X.