# A metric space with its transfinite asymptotic dimension omega + 1

## TL;DR

This paper constructs a specific metric space demonstrating that the transfinite asymptotic dimension can reach omega+1, providing a counterexample to the omega conjecture in geometric group theory.

## Contribution

It presents the first explicit example of a metric space with transfinite asymptotic dimension omega+1, disproving the omega conjecture.

## Key findings

- Transfinite asymptotic dimension can be omega+1.
- Counterexample to the omega conjecture.
- Both dimensions are omega+1 in the constructed space.

## Abstract

We construct a metric space whose transfinite asymptotic dimension and complementary-finite asymptotic dimension are both omega+1, where omega is the smallest infinite ordinal number. Therefore, we prove that the omega conjecture is not true.