Reflections on dyadic compacta

Let M be an elementary submodel of the universe of sets, and 〈X,T〉 a topological space in M. Let XM be X∩M with topology generated by {U∩M: U∈T∩M}. Let D be the two-point discrete space. Suppose the least cardinal κ of a basis for XM is a member of M, and XM is an uncountable continuous image of Dκ. Then X=XM if either 0# does not exist or κ is less than the first inaccessible cardinal. A corollary is that if GM is a compact group and the least cardinal of a basis for GM is in M, then G=GM.