Dimension zero at all scales

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dc.creatorBrodskiy, N.
dc.creatorDydak, J.
dc.creatorHiges, J.
dc.creatorMitra, A.
dc.date2006-07-10
dc.date2007-06-09
dc.date.accessioned2026-07-07T09:23:24Z
dc.date.available2026-07-07T09:23:24Z
dc.descriptionWe consider the notion of dimension in four categories: the category of (unbounded) separable metric spaces and (metrically proper) Lipschitz maps, and the category of (unbounded) separable metric spaces and (metrically proper) uniform maps. A unified treatment is given to the large scale dimension and the small scale dimension. We show that in all categories a space has dimension zero if and only if it is equivalent to an ultrametric space. Also, 0-dimensional spaces are characterized by means of retractions to subspaces. There is a universal zero-dimensional space in all categories. In the Lipschitz Category spaces of dimension zero are characterized by means of extensions of maps to the unit 0-sphere. Any countable group of asymptotic dimension zero is coarsely equivalent to a direct sum of cyclic groups. We construct uncountably many examples of coarsely inequivalent ultrametric spaces.
dc.description17 pages, To appear in Topology and its Applications
dc.identifierhttps://arxiv.org/abs/math/0607241
dc.identifierhttp://arxiv.org/abs/math/0607241
dc.identifierTopology and its Applications, 154 (2007), 2729-2740
dc.identifier.urihttp://salesiana.dossiersoluciones.com/handle/123456789/155724
dc.subjectMetric Geometry
dc.subjectGeometric Topology
dc.titleDimension zero at all scales
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