Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: 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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The categories of this paper are: quant-ph
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The categories of this paper are: solv-int
9
The categories of this paper are: nlin
33
The categories of this paper are: plasm-ph
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The categories of this paper are: comp-gas
15
The categories of this paper are: nucl-ex
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The categories of this paper are: supr-con
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The categories of this paper are: hep-th
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The categories of this paper are: stat
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The categories of this paper are: q-fin
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The categories of this paper are: math-ph
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The categories of this paper are: eess
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The categories of this paper are: ao-sci
24
The categories of this paper are: dg-ga
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The categories of this paper are: hep-lat
1
The categories of this paper are: cs