Embeddability properties of countable metric spaces

For subspaces X and Y of Q the notation X⩽hY means that X is homeomorphic to a subspace of Y and X∼Y means X⩽hY⩽hX. The resulting set P(Q)/∼ of equivalence classes X¯={Y⊆Q:Y∼X} is partially-ordered by the relation X¯⩽hY¯ if X⩽hY. 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.