Čech-complete spaces and the upper topology

Let X be a topological space and let K(X) be the set of all compact subsets of X. The purpose of this note is to prove the following: if X is regular and q-space, then X is Lindelöf and Čech-complete if and only if there exists a continuous map f from a Lindelöf and Čech-complete space Y to the space K(X) endowed with the upper topology, such that f(Y) is cofinal in (K(X), ⊂). This result extends the following result of Saint Raymond and Christensen: if X is separable metrizable, then X is a Polish space if and only if the space K(X) endowed with the Vietoris topology is the continuous image of a Polish space.