On character of points in the Higson corona of a metric space

Abstract

We prove that for an unbounded metric space $X$, the minimal character $m\chi (\check{X})$ of a point of the Higson corona $\check{X}$ of $X$ is equal to $\mathfrak{u}$ if $X$ has asymptotically isolated balls and to $\max \{\mathfrak{u},\mathfrak{d}\}$ otherwise. This implies that under $\mathfrak{u}<\mathfrak{d}$ a metric space $X$ of bounded geometry is coarsely equivalent to the Cantor macro-cube $2^{<\IN}$ if and only if $\dim (\check{X})=0$ and $m\chi (\check{X})=\mathfrak{d}$. This contrasts with a result of Protasov saying that under CH the coronas of any two asymptotically zero-dimensional unbounded metric separable spaces are homeomorphic.

12 pages

References in corpus (1)

• A coarse characterization of the Baire macro-space