


Paper Title:
Countable Ordinal Spaces and Compact Countable Subsets of a Metric Space
Author(s):
B. AlvarezSamaniego, A. Merino
Nucleo de Investigadores Cientificos
Facultad de Ciencias,
Universidad Central del Ecuador (UCE)
Quito,
Ecuador.
Email: borys_yamil@yahoo.com,
balvarez@uce.edu.ec
Escuela de Ciencias Fisicas y Matematica
Facultad de Ciencias Exactas y Naturales
Pontificia Universidad Catolica del Ecuador
Apartado: 17012184, Quito,
Ecuador.
Email: aemerinot@puce.edu.ec
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 finitedimensional 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, the cardinality of the set of all the equivalence classes, up to homeomorphisms, of compact countable subsets of this metric space is less than or equal to alephone. We also show that for all cardinal number smaller than or equal to alephone, there exists a metric space with cardinality equals the aforementioned cardinal number.
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