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

Taras O. Banakh; Ostap Chervak; Lubomyr Zdomskyy — 2013

Commentationes Mathematicae Universitatis Carolinae

We prove that for an unbounded metric space $X$ , the minimal character $𝗆 χ (\overset{ˇ}{X})$ of a point of the Higson corona $\overset{ˇ}{X}$ of $X$ is equal to $𝔲$ if $X$ has asymptotically isolated balls and to $max {𝔲, 𝔡}$ otherwise. This implies that under $𝔲 < 𝔡$ a metric space $X$ of bounded geometry is coarsely equivalent to the Cantor macro-cube $2^{< ℕ}$ if and only if $dim (\overset{ˇ}{X}) = 0$ and $𝗆 χ (\overset{ˇ}{X}) = 𝔡$ . 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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