Echeloned Spaces

TL;DR

Introduces echeloned spaces as an order-theoretic analog of metric spaces and develops a comprehensive framework connecting metrizability, morphisms, and automorphisms. The Fraïssé limit of finite echeloned spaces is constructed as $\mathbf F$, with $E(\mathbf F)\cong (\mathbb{\mathbb Q}_0^+,\le)$, and its edge-coloured graph $\mathcal T_{\mathbf F}$ is shown to be universal and homogeneous; the Ramsey property is established for finite ordered echeloned spaces, implying extreme amenability of the automorphism group. Universality of $\operatorname{Aut}(\mathbf F)$ is obtained via Katětov functors, yielding topological embeddings of $\operatorname{Sym}(\mathbb{N})$ into $\operatorname{Aut}(\mathbf F)$. An alternative categorical formulation via echeloned maps is also presented, illustrating the robustness and flexibility of the framework. These results bridge model theory, combinatorics, and topological dynamics in a unified theory of echeloned spaces and their symmetries.

Abstract

We introduce the notion of echeloned spaces - an order-theoretic abstraction of metric spaces. The first step is to characterize metrizable echeloned spaces. It turns out that morphisms between metrizable echeloned spaces are uniformly continuous or have a uniformly discrete image. In particular, every automorphism of a metrizable echeloned space is uniformly continuous, and for every metric space with midpoints the automorphisms of the induced echeloned space are precisely the dilations. Next we focus on finite echeloned spaces. They form a Fraisse class and we describe its Fraisse-limit both as the echeloned space induced by a certain homogeneous metric space and as the result of a random construction. Building on this we show that the class of finite ordered echeloned spaces is Ramsey. The proof of this result combines a combinatorial argument by Nesetril and Hubicka with a topological-dynamical point of view due to Kechris, Pestov, and Todorcevic. Finally, using the method of Katetov functors due to Kubis and Masulovic, we prove that the full symmetric group on a countable set topologically embeds into the automorphism group of the countable universal homogeneous echeloned space.

Figures (2)

• Figure 1: The construction of $\psi$
• Figure 2: The composition $K(\varphi_2)\circ K(\varphi_1)$

Theorems & Definitions (96)

• Definition 1.1
• Remark 1.2
• Lemma 1.3
• proof
• Corollary 1.4
• proof
• Lemma 1.5
• proof
• Definition 1.6
• Lemma 1.7