Countable ordinal spaces and compact countable subsets of a metric space

Abstract

We show in detail that every compact countable subset of a metric space is homeomorphic to a countable ordinal number, which extends a result given by Mazurkiewicz and Sierpinski for finite-dimensional Euclidean spaces. In order to achieve this goal, we use Transfinite Induction to construct a specific homeomorphism. In addition, we prove that for all metric space $(E,d)$, the cardinality of the set of all the equivalence classes ${\mathcal{K}}_E$, up to homeomorphisms, of compact countable subsets of $E$ is less than or equal to ${\aleph }_1$, i.e. $|{\mathcal{K}}_E|\le {\aleph }_1$. We also show that for all cardinal number $\kappa$ smaller than or equal to ${\aleph }_1$, there exists a metric space $(E_{\kappa },d_{\kappa })$ such that $|{\mathcal{K}}_{E_{\kappa }}|=\kappa$.

15 pages