A coarse characterization of the Baire macro-space

We prove that each coarsely homogenous separable metric space $X$ is coarsely equivalent to one of the spaces: the sigleton, the Cantor macro-cube or the Baire macro-space. This classification is derived from coarse characterizations of the Cantor macro-cube and of the Baire macro-space given in this paper. Namely, we prove that a separable metric space $X$ is coarsely equivalent to the Baire macro-space if any only if $X$ has asymptotic dimension zero and has unbounded geometry in the sense that for every $δ$ there is $ε$ such that no $ε$-ball in $X$ can be covered by finitely many sets of diameter $\le δ$.

No reviews yet.

No comments yet.