Taras Banakh, Ostap Chervak, Lubomyr Zdomskyy
On character of points in the Higson corona of a metric space

Comment.Math.Univ.Carolin. 54,2 (2013) 159-178.

Abstract:We prove that for an unbounded metric space $X$, the minimal character $\mathsf m\chi(\check X)$ 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 $2^{< \mathbb N}$ if and only if $\dim (\check X)=0$ and $\mathsf m\chi (\check X)= \mathfrak d$. 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.

Keywords: Higson corona, character of a point, ultrafilter number, dominating number
AMS Subject Classification: 03E17 03E35 03E50 54D35 54E35 54F45