We prove that for each countably infinite, regular space X such that $C_p\left(X\right)$ is a $Z_{\sigma }$-space, the topology of $C_p\left(X\right)$ is determined by the class $F_0\left(C_p\left(X\right)\right)$ of spaces embeddable onto closed subsets of $C_p\left(X\right)$. We show that $C_p\left(X\right)$, whenever Borel, is of an exact multiplicative class; it is homeomorphic to the absorbing set ${\Omega }_{\alpha }$ for the multiplicative Borel class $M_{\alpha }$ if $F_0\left(C_p\left(X\right)\right)=M_{\alpha }$. For each ordinal α ≥ 2, we provide an example $X_{\alpha }$ such that $C_p\left(X_{\alpha }\right)$ is homeomorphic to ${\Omega }_{\alpha }$.

Arhangel’skiǐ proved that if $X$ and $Y$ are completely regular spaces such that $C_p\left(X\right)$ and $C_p\left(Y\right)$ are linearly homeomorphic, then $X$ is pseudocompact if and only if $Y$ is pseudocompact. In addition he proved the same result for compactness, $\sigma$-compactness and realcompactness. In this paper we prove that if $\phi :C_p\left(X\right)\to C_p\left(X\right)$ is a continuous linear surjection, then $Y$ is pseudocompact provided $X$ is and if $\phi$ is a continuous linear injection, then $X$ is pseudocompact provided $Y$ is. We also give examples that both statements do not hold...