A non-separable Christensenʹs theorem and set tri-quotient maps

For every space X let K ( X ) be the set of all compact subsets of X. Christensen [J.P.R. Christensen, Necessary and sufficient conditions for measurability of certain sets of closed subsets, Math. Ann. 200 (1973) 189–193] proved that if X , Y are separable metrizable spaces and F : K ( X ) → K ( Y ) is a monotone map such that any L ∈ K ( Y ) is covered by F ( K ) for some K ∈ K ( X ) , then Y is complete provided X is complete. It is well known [J. Baars, J. de Groot, J. Pelant, Function spaces of completely metrizable space, Trans. Amer. Math. Soc. 340 (1993) 871–879] that this result is not true for non-separable spaces. In this paper we discuss some additional properties of F which guarantee the validity of Christensenʹs result for more general spaces.