[Paper Review] Countable ordinal spaces and compact countable subsets of a metric space

Borys Álvarez-Samaniego, Andrés Merino|arXiv (Cornell University)|Feb 22, 2018

Advanced Topology and Set Theory3 citations

TL;DR

This paper establishes that every compact countable subset of a metric space is homeomorphic to a countable ordinal number, extending a classical result of Mazurkiewicz and Sierpinski. Using transfinite induction and Cantor-Bendixson derivatives, the authors prove that the number of homeomorphism classes of such subsets in any metric space is at most ℵ₁, and they construct examples showing this bound is tight for both countable and uncountable spaces.

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 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 $κ$ smaller than or equal to $\aleph_1$, there exists a metric space $(E_κ, d_κ)$ such that $|\mathscr{K}_{E_κ}|= κ$.

Motivation & Objective

• To extend the classical result of Mazurkiewicz and Sierpinski, which states that compact countable subsets of Euclidean spaces are homeomorphic to countable ordinals, to arbitrary metric spaces.
• To characterize the structure of compact countable subsets in metric spaces using transfinite induction and Cantor-Bendixson derivatives.
• To determine the maximum possible cardinality of the set of homeomorphism classes of compact countable subsets in any metric space.
• To construct metric spaces realizing every possible cardinality ≤ ℵ₁ for the set of homeomorphism classes of compact countable subsets.

Proposed method

• Employ transfinite induction on the Cantor-Bendixson derivative process to construct a homeomorphism between any compact countable subset of a metric space and a countable ordinal.
• Define the Cantor-Bendixson characteristic (α, p) for a countable closed set D, where α is the smallest ordinal such that D^(α) is finite and |D^(α)| = p.
• Use the Cantor-Bendixson characteristic to classify compact countable subsets up to homeomorphism, showing that two such sets are homeomorphic iff they have the same characteristic.
• Prove that the map assigning to each homeomorphism class its Cantor-Bendixson characteristic is injective, thereby bounding the number of classes by |ω₁ × ω| = ℵ₁.
• Construct specific metric spaces: finite discrete spaces for finite class counts, countable discrete spaces (like ℚ) for ℵ₀ classes, and uncountable discrete spaces for ℵ₀ classes.
• Use the density of ℚ in ℝ to show that ℚ supports ℵ₁ many non-homeomorphic compact countable subsets, each with distinct Cantor-Bendixson characteristics.

Experimental results

Research questions

• RQ1 Can every compact countable subset of a metric space be homeomorphically embedded into a countable ordinal with the order topology?
• RQ2 What is the maximum possible number of homeomorphism classes of compact countable subsets in a metric space?
• RQ3 For which cardinal numbers κ ≤ ℵ₁ does there exist a metric space whose set of homeomorphism classes of compact countable subsets has size exactly κ?
• RQ4 How does the Cantor-Bendixson derivative process fully classify compact countable subsets in metric spaces?

Key findings

• Every compact countable subset of a metric space is homeomorphic to a countable ordinal number, generalizing the result of Mazurkiewicz and Sierpinski from Euclidean spaces to arbitrary metric spaces.
• The set of homeomorphism classes of compact countable subsets in any metric space (E,d) has cardinality at most ℵ₁, i.e., |𝒦_E| ≤ ℵ₁.
• For every finite n ∈ ω, there exists a metric space (E_n, d_n) such that |𝒦_{E_n}| = n, achieved via finite discrete spaces.
• There exists a countable metric space (e.g., ℚ with the usual metric) such that |𝒦_ℚ| = ℵ₁, showing the bound ℵ₁ is sharp.
• There exists an uncountable metric space (e.g., ℝ with the discrete metric) such that |𝒦_ℝ| = ℵ₀, demonstrating that uncountability of the ambient space does not imply uncountably many classes.
• The Cantor-Bendixson characteristic (α, p) fully classifies compact countable subsets up to homeomorphism: two such sets are homeomorphic iff they have the same characteristic.