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Large scale versus small scale. (English) Zbl 1306.54001

Hart, K. P. (ed.) et al., Recent progress in general topology III. Based on the presentations at the Prague symposium, Prague, Czech Republic, 2001. Amsterdam: Atlantis Press (ISBN 978-94-6239-023-2/hbk; 978-94-6239-024-9/ebook). 165-203 (2014).
The text is a nicely written introduction to coarse geometry from a topological point of view. It is accessible to non-experts. The main theme is the connection between the concepts of uniform category and those of coarse category. The former serve as a motivation for the definitions of the latter.
The coarse category is introduced from two equivalent points of view. By dualizing covers and by dualizing partitions of unity. It is convenient to have equivalent definitions as later dualizations usually require a specific point of view. The authors then focus on partitions of unity, developing the ideas of large scale paracompactness and connecting those to property \(A\) and related invariants. For example, they show that a large scale finitistic metric space has property \(A\) iff it is large scale paracompact.
In the last section the authors present the asymptotic dimension from three different points of view: by lifting properties, covers and from extensional point of view.
For the entire collection see [Zbl 1282.54001].
Reviewer: Ziga Virk (Litija)

MSC:

54-02 Research exposition (monographs, survey articles) pertaining to general topology
54E15 Uniform structures and generalizations
54F45 Dimension theory in general topology
54H99 Connections of general topology with other structures, applications
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