The space $C_p(X)$ is cofinally $\sigma$-compact if and only if it is $\sigma$-pseudocompact

Source abstract

We prove that the function space $C_p(X)$ of a Tychonoff space $X$ is cofinally $\sigma$-compact if and only if it is $\sigma$-pseudocompact. It is also established that cofinal $\sigma$-compactness of $C_p(X, [0,1])$ is equivalent to its pseudocompactness. If $X$ is a locally Lindelöf domain representable GO space of countable extent, then it is cofinally Polish. In particular, any locally Lindelöof Cech-complete GO space of countable extent is cofinally Polish. Our results provide answers to two published open questions.