The projective Menger property and an embedding of Sω into function spaces

For a Tychonoff space X, we denote by Cp(X) the space of all real-valued continuous functions on X with the topology of pointwise convergence. In this paper, we show that (1) Cp(X,I) is projectively Menger if and only if X is b-discrete (i.e., every countable subset of X is closed and C⁎-embedded in X), (2) there is a Menger space L such that the sequential fan Sω can be embedded into Cp(L). The first (1) enables us to give a direct proof of Arhangel'skii's theorem [2, Theorem 6]: If Cp(X) is Menger, then X is finite. The second (2) is an affirmative answer to Arhangel'skii's problem [5, Problem II.2.7] under CH (the continuum hypothesis).