Some properties related to the Cantor-Bendixson derivative on a Polish space

Authors

  • Borys Alvarez-Samaniego Universidad Central del Ecuador
  • Andres Merino Pontificia Universidad Cat\'olica del Ecuador

Keywords:

Polish space, Cantor-Bendixson's derivative, cardinality

Abstract

We show a necessary and sufficient condition for any ordinal number to be a Polish space. We also prove that for each countable Polish space, there exists a countable ordinal number that is an upper bound for the first component of the Cantor-Bendixson characteristic of every compact countable subset of the aforementioned space. In addition, for any uncountable Polish space, for every countable ordinal number and for each nonzero natural number, we show the existence of a compact countable subset of this space such that its Cantor-Bendixson characteristic equals the previous pair of numbers. Finally, for every Polish space, we determine the cardinality of the partition, up to homeomorphisms, of the set of all compact countable subsets of the aforesaid space.

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Author Biographies

Borys Alvarez-Samaniego, Universidad Central del Ecuador

N\'ucleo de Investigadores Cient\'{\i}ficos
Facultad de Ciencias
Universidad Central del Ecuador (UCE)
Quito,
Ecuador

Andres Merino, Pontificia Universidad Cat\'olica del Ecuador

Escuela de Ciencias F\'isicas y Matem\'atica
Facultad de Ciencias Exactas y Naturales
Pontificia Universidad Cat\'olica del Ecuador (PUCE)
Quito,
Ecuador

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Published

2021-02-04

How to Cite

Alvarez-Samaniego, B., & Merino, A. (2021). Some properties related to the Cantor-Bendixson derivative on a Polish space. New Zealand Journal of Mathematics, 50, 207–218. Retrieved from https://nzjmath.org/index.php/NZJMATH/article/view/82

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Articles