Countable ordinal spaces and compact countable subsets of a metric space

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 $\mathscr{K}_E$, up to homeomorphisms, of compact countable subsets of $E$ is less than or equal to $\aleph_1$, i.e. $|\mathscr{K}_E| \le \aleph_1$. 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 $|\mathscr{K}_{E_{\kappa}}|= \kappa$.

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