A sequential property of $\mathsf {C}_p(X)$ and a covering property of Hurewicz

Abstract:

$\mathsf {C}(X)$ has the monotonic sequence selection property if there is for each $f$, and for every sequence $(\sigma _n:n<\omega )$ where for each $n$ $\sigma _n$ is a sequence converging pointwise monotonically to $f$, a sequence $(f_n:n<\omega )$ such that for each $n$ $f_n$ is a term of $\sigma _n$, and $(f_n:n<\omega )$ converges pointwise to $f$. We prove a theorem which implies for metric spaces $X$ that $\mathsf {C}(X)$ has the monotonic sequence selection property if, and only if, $X$ has a covering property of Hurewicz.