Metric spaces with complexity of the smallest infinite ordinal number

TL;DR

This paper investigates metric spaces characterized by the smallest infinite ordinal number, providing equivalent definitions and analyzing their complexity in product spaces, including specific calculations for wreath products.

Contribution

It introduces equivalent formulations for metric spaces with this complexity and determines the exact complexity of finite wreath product spaces.

Findings

The complexity of the finite product of wreath products equals the smallest infinite ordinal.

The complexity of (ZwrZ)wrZ is w+1.

Provides a new understanding of the structure of these metric spaces.

Abstract

In this paper, we are concerned with the study on metric spaces with complexity of the smallest infinite ordinal number. We give equivalent formulations of the definition of metric spaces with complexity of the smallest infinite ordinal number and prove that the exact complexity of the finite product of wreath product is the smallest infinite ordinal number. Consequently, we obtain the complexity of (ZwrZ)wrZ is w+1.