Topological classification of function spaces with the Fell topology I

For an infinite Tychonoff space X, let ↓CF(X) denote the collection of the hypographs of all continuous maps from X to [0,1] with the Fell topology. We shall prove that, if ↓CF(X) is metrizable, then there exists a copy MQ of the Hilbert cube Q=[−1,1]ω such that ↓CF(X) is homotopy dense in MQ, that is, there exists a homotopy H:MQ×[0,1]→MQ such that H(y,0)=y and H(y,t)∈↓CF(X) for every y∈MQ and t∈(0,1]. We also give methods to construct non-metrizable spaces X such that ↓CF(X) are metrizable.