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

Borys Álvarez-Samaniego, Andrés Merino

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 all 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 each Polish space, we determine the cardinality of the partition, up to homeomorphisms, of the set of all compact countable subsets of the aforesaid space.

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

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 all 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 each Polish space, we determine the cardinality of the partition, up to homeomorphisms, of the set of all compact countable subsets of the aforesaid space.

Paper Structure

This paper contains 2 sections, 13 theorems, 19 equations.

Table of Contents

  1. Introduction
  2. On the cardinality of the equivalence classes, up to homeomorphisms, of compact countable subsets of a Polish space

Key Result

Proposition 1.1

Let $\alpha$ be an ordinal number. Then, $\alpha$ is a Polish space if and only if $\alpha$ is countable.

Theorems & Definitions (31)

  • Definition 1.1: Metrizable space
  • Definition 1.2: Completely metrizable space
  • Definition 1.3: Polish space
  • Proposition 1.1
  • proof
  • Definition 2.1: Cantor-Bendixson's derivative
  • Definition 2.2: Cantor-Bendixson's characteristic
  • Theorem 2.1
  • Remark 2.1
  • Proposition 2.2
  • ...and 21 more