Embeddability properties of countable metric spaces

Embeddability properties of countable metric spaces Article

Gillam, WD. (2005). Embeddability properties of countable metric spaces . TOPOLOGY AND ITS APPLICATIONS, 148(1-3), 63-82. 10.1016/j.topol.2004.08.001

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 X̄ = {Y ⊆ ℚ: Y ∼ X} is partially-ordered by the relation X̄ ≤h. Ȳ 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. © 2004 Elsevier B.V. All rights reserved.