Application of selection principles in the study of the properties of function spaces

For a Tychonoff space X , we denote by C p ( X ) the space of all real-valued continuous functions on X with the topology of pointwise convergence. In this paper we prove that: If every finite power of X is Lindelöf then C p ( X ) is strongly sequentially separable iff X is γ -set. B α ( X ) (= functions of Baire class α ( 1 < α ≤ ω 1 ) on a Tychonoff space X with the pointwise topology) is sequentially separable iff there exists a Baire isomorphism class α from a space X onto a σ -set. B α ( X ) is strongly sequentially separable iff i w ( X ) = ℵ 0 and X is a Z α -cover γ -set for 0 < α ≤ ω 1 . There is a consistent example of a set of reals X such that C p ( X ) is strongly sequentially separable but B 1 ( X ) is not strongly sequentially separable. B ( X ) is sequentially separable but is not strongly sequentially separable for a b -Sierpiński set X .