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arXiv · 1907.09687

Inductive dimensions of coarse proximity spaces

Pawel Grzegrzolka·Jeremy Siegert
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Abstract

In this paper, we generalize Dranishnikov's asymptotic inductive dimension to the setting of coarse proximity spaces. We show that in this more general context, the asymptotic inductive dimension of a coarse proximity space is bigger or equal to the inductive dimension of its boundary, and consequently may be strictly bigger than the covering dimension of the boundary. We also give a condition, called complete traceability, on the boundary of the coarse proximity space under which the asymptotic inductive dimension of a coarse proximity space and the inductive dimension of its boundary coincide. Finally, we show that spaces whose boundaries are $Z$-sets and spaces admitting metrizable compactifications have completely traceable boundaries.

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Pawel Grzegrzolka, Jeremy Siegert. 2019-07-23. Inductive dimensions of coarse proximity spaces. https://doi.org/10.4064/ba250611-23-10

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