APD profiles and transfinite asymptotic dimension

TL;DR

This paper develops the theory of APD profiles for -pseudometric spaces, connecting them with transfinite asymptotic dimension, and provides characterizations and conditions for spaces with specific transfinite asymptotic dimensions.

Contribution

It introduces new connections between APD profiles and transfinite asymptotic dimension, offering characterizations and conditions for their bounds.

Findings

Characterization of spaces with transfinite asymptotic dimension +n

Sufficient condition for spaces with transfinite asymptotic dimension m n

Development of the theory of APD profiles for -pseudometric spaces

Abstract

We develop the theory of APD profiles introduced by J. Dydak for $\infty$-pseudometric spaces. We connect them with transfinite asymptotic dimension defined by T. Radul. We give a characterization of spaces with transfinite asymptotic dimension at most $\omega+n$ for $n\in\omega$ and a sufficient condition for a space to have transfinite asymptotic dimension at most $m\cdot \omega+n$ for $m,n\in\omega$, using the language of APD profiles.