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


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New Zealand Journal of Mathematics

Vol. 50, (2020), Pages 207-218

Borys Alvarez-Samaniego

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

Andres Merino

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

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.

Keywords Polish space; Cantor-Bendixson's derivative; cardinality.

Classification (MSC2000) 54E50, 54A25, 03E15.

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