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Keywords:
Higson corona; character of a point; ultrafilter number; dominating number
Higson corona; character of a point; ultrafilter number; dominating number
Summary:
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.
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