[Paper Review] An embedding, an extension, and an interpolation of ultrametrics

Yoshito Ishiki|arXiv (Cornell University)|Aug 24, 2020

Advanced Topology and Set Theory46 references4 citations

TL;DR

This paper establishes $S$-valued ultrametric analogues of classical metric space theorems, including the Arens–Eells isometric embedding, Hausdorff extension, Niemytzki–Tychonoff compactness characterization, and interpolation theorems. It constructs isometric embeddings into $S$-valued ultra-normed modules over integral domains, proving that every $S$-valued ultrametric space embeds isometrically into a complete, $R$-independent, closed subset of such a module, generalizing prior results to arbitrary range sets $S$. The key contribution is a unified framework for ultrametric analogues of core metric theorems with strong algebraic and topological control.

ABSTRACT

The notion of the ultrametrics can be considered as a zero-dimensional analogue of ordinary metrics, and it is expected to prove ultrametric versions of theorems on metric spaces. In this paper, we provide ultrametric versions of the Arens--Eells isometric embedding theorem of metric spaces, the Hausdorff extension theorem of metrics, the Niemytzki--Tychonoff characterization theorem of the compactness, and the author's interpolation theorem of metrics and theorems on dense subsets of spaces of metrics.

Motivation & Objective

• To extend classical metric theorems—such as Arens–Eells embedding, Hausdorff extension, and Niemytzki–Tychonoff compactness characterization—to the setting of $S$-valued ultrametric spaces.
• To generalize the Arens–Eells isometric embedding theorem to $S$-valued ultrametrics by constructing embeddings into $S$-valued ultra-normed modules over integral domains.
• To establish an ultrametric version of the author’s interpolation theorem and theorems on dense $G_{\delta}$ subsets of spaces of metrics.
• To provide a framework for ultrametric analogues of metric theorems with control over algebraic structure (e.g., $R$-independence) and topological properties (e.g., completeness, closedness).

Proposed method

• Constructs an $S$-valued ultra-normed $R$-module $(V, \|\cdot\|)$ for any integral domain $R$ and range set $S$ with at least two elements.
• Defines an isometric embedding $I: X \to V$ such that $I(X)$ is closed in $V$ and $R$-independent, using module-theoretic constructions inspired by Lemin–Lemin universal ultrametric spaces.
• Utilizes the trivial valuation $t_R$ on $R$ to ensure compatibility between the ultra-norm $\|\cdot\|$ and the ring action, i.e., $\|r \cdot x\| = t_R(r) \|x\|$.
• Applies the ultrametric $d(x,y) = x \lor y$ on $S$ (with $d(x,x)=0$) as a base construction in key lemmas, drawing from Delhommé et al. for foundational $S$-valued ultrametric examples.
• Adapts the $\ell^\infty$-product metric and pullback metric constructions to define topologies on spaces of ultrametrics $\operatorname{UM}(X,S)$ and $\operatorname{M}(X)$.
• Introduces transmissible parameters $\mathfrak{G}$ and anti-$\mathfrak{G}$-transmissible properties to characterize dense $G_\delta$ subsets of $\operatorname{UM}(X,S)$, enabling interpolation and completeness results.

Experimental results

Research questions

• RQ1 Can the Arens–Eells isometric embedding theorem be generalized to $S$-valued ultrametric spaces with algebraic structure in the target space?
• RQ2 Does the Hausdorff extension theorem for metrics have a valid analogue in the ultrametric setting for arbitrary range sets $S$?
• RQ3 Can the Niemytzki–Tychonoff characterization of compactness be reformulated and proven for $S$-valued ultrametric spaces?
• RQ4 How can the author’s interpolation theorem for metrics be extended to the ultrametric context with control over $S$-valued distances and topological density?
• RQ5 What conditions ensure that the space $\operatorname{UM}(X,S)$ of $S$-valued ultrametrics is a Baire space or has dense $G_\delta$ subsets?

Key findings

• For any range set $S$ with at least two elements and any integral domain $R$, every $S$-valued ultrametric space $(X,d)$ admits an isometric embedding into an $S$-valued ultra-normed $R$-module $(V, \|\cdot\|)$ such that $I(X)$ is closed and $R$-independent.
• If $(X,d)$ is complete, the target module $(V, \|\cdot\|)$ can be chosen to be a complete metric space, extending the classical Arens–Eells result to ultrametric and algebraic settings.
• The constructed ultra-norm $\|\cdot\|$ is compatible with the trivial valuation $t_R$ on $R$, ensuring $\|r \cdot x\| = t_R(r) \|x\|$ for all $r \in R$, $x \in V$, which links the algebraic and metric structures.
• The space $\operatorname{UM}(X,S)$ of $S$-valued ultrametrics on $X$ admits a natural ultrametric $\mathcal{UD}_X^S$ under which it becomes a complete metric space, and the set of $d \in \operatorname{UM}(X,S)$ satisfying the anti-$\mathfrak{G}$-transmissible property is a dense $G_\delta$ subset.
• The set $\operatorname{CUM}(X,S)$ of complete $S$-valued ultrametrics is a $G_\delta$ subset of $\operatorname{UM}(X,S)$, and under suitable conditions, it is a Baire space.
• The paper proves that $\mathcal{UD}_X^S$-topology on $\operatorname{UM}(X,S)$ is metrizable and induces the same topology as the $\ell^\infty$-product metric, enabling convergence and completeness arguments in the space of ultrametrics.