A rich hierarchy of functionals of finite types

TL;DR

This paper explores hierarchical structures of total, continuous functionals over metric spaces, highlighting the Urysohn space's role as a universal model for infinitary algorithm outputs, and establishes topological embeddings of these hierarchies.

Contribution

It introduces a framework for hierarchies of functionals over metric spaces and proves their embeddings into the Urysohn space, along with an effective density theorem.

Findings

Urysohn space can embed all separable metric spaces.

Hierarchies of functionals can be topologically embedded into the Urysohn hierarchy.

An effective density theorem for these hierarchies is established.

Abstract

We are considering typed hierarchies of total, continuous functionals using complete, separable metric spaces at the base types. We pay special attention to the so called Urysohn space constructed by P. Urysohn. One of the properties of the Urysohn space is that every other separable metric space can be isometrically embedded into it. We discuss why the Urysohn space may be considered as the universal model of possibly infinitary outputs of algorithms. The main result is that all our typed hierarchies may be topologically embedded, type by type, into the corresponding hierarchy over the Urysohn space. As a preparation for this, we prove an effective density theorem that is also of independent interest.