Network Properties of Function Spaces Endowed with the Fell Hypograph Topology

For a Tychonoff space X, we study the network properties of the space C↓F(X) of continuous functions from X to R, endowed with the Fell hypograph topology. We prove that the function space C↓F(X) is cosmic iff C↓F(X) is an ℵ0-space iff C↓F(X) is a P0-space iff X is a weakly locally compact ℵ0-space.