Embeddability properties of countable metric spaces

For subspaces X and Y of Q the notation X ⩽ h Y means that X is homeomorphic to a subspace of Y and X ∼ Y means X ⩽ h Y ⩽ h X . The resulting set P ( Q ) / ∼ of equivalence classes X ¯ = { Y ⊆ Q : Y ∼ X } is partially-ordered by the relation X ¯ ⩽ h Y ¯ if X ⩽ h Y . It is shown that ( P ( Q ) , ⩽ h ) is partially well-ordered in the sense that it lacks infinite anti-chains and infinite strictly descending chains. A characterization of ( P ( Q ) , ⩽ h ) in terms of scattered subspaces of Q with finite Cantor–Bendixson rank is given and several results relating Cantor–Bendixson rank to this embeddability ordering are established. These results are obtained by studying a local homeomorphism invariant (type) for countable scattered metric spaces.