About closed subsets of spaces of first category

The space h ( X , k ) is the smallest h-homogeneous space of first category and of weight k that contains a space X as a closed subset. We prove that if Y is a metric space of first category such that every nonempty open subset of Y contains a closed copy of X and has weight ⩾k, then Y contains a closed copy of h ( X , k ) . This allows us to give an internal characterization of h ( X , k ) . We also establish some relations between homogeneous and h-homogeneous spaces.