Gerlits-Nagy theorem: $C_p(X)$ is Frechet-Urysohn exactly for gamma-spaces

Call an open cover $\mathcal{U}$ of a space $X$ an omega-cover if $X\mathrm{\notin }\mathcal{U}$ and every finite subset of $X$ is contained in some member of $\mathcal{U}$, and a gamma-cover if it is infinite and every point of $X$ lies in all but finitely many of its members. A Tychonoff space $X$ has the gamma-property if every open omega-cover of $X$ contains a gamma-subcover.

Gerlits-Nagy theorem. For a Tychonoff space $X$ the following are equivalent: $C_p(X)$ is Frechet-Urysohn; $C_p(X)$ is sequential; $C_p(X)$ is a $k$-space; and $X$ has the gamma-property.

Not proved in this library. Recorded with a citation.

What would prove it. A translation between covering properties of $X$ and convergence properties at the zero function of $C_p(X)$: a sequence of functions tending to zero pointwise corresponds to a sequence of sets covering finite subsets, and the gamma-property is exactly what allows a sequence to be extracted from a cluster point.

Why it matters here. It is the model case of a selection principle: three convergence properties of a function space, which are distinct for general topological spaces, coincide for $C_p(X)$ and are equivalent to a single combinatorial covering property of $X$. That equivalence is what makes the whole family of selection principles worth studying, and it is one more instance of the dictionary that topological information about $X$ is recoverable from a function space over $X$.