Rational approximation to surfaces defined by polynomials in one variable

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 \({\gamma}\)-set.
• \({B_{\alpha}(X)}\) (= functions of Baire class \({\alpha}\) (\({1 < \alpha \leq \omega_1}\)) on a Tychonoff space X with the pointwise topology) is sequentially separable iff there exists a Baire isomorphism class \({\alpha}\) from a space X onto a \({\sigma}\)-set.
• \({B_{\alpha}(X)}\) is strongly sequentially separable iff \({iw(X)=\aleph_0}\) and X is a \({Z^{\alpha}}\)-cover \({\gamma}\)-set for \({0 < \alpha \leq \omega_1}\).
• There is a consistent example of a set of reals X such that C _p (X) is strongly sequentially separable but B₁(X) is not strongly sequentially separable.
• B(X) is sequentially separable but is not strongly sequentially separable for a \({\mathfrak{b}}\)-Sierpiński set X.