Asymptotic dimension, decomposition complexity, and Haver’s property C. (English) Zbl 1297.54064
Decomposition complexity was introduced by E. Guentner et al. [Invent. Math. 189, No. 2, 315–357 (2012; Zbl 1257.57028)] using a game theoretic approach. It was shown that the finite decomposition complexity (FDC) implies property A. In this interesting paper the authors give the notions of straight Finite Decomposition Property (sFDC) (Definition 2.2), asymptotic property C (Definition 2.4), game theoretical asymptotic property C (Definition 4.1), and game theoretical property C (Definition 5.2). In Sections 2,3, and 4 they prove that:
(1) Asymptotic property C \(\Rightarrow\) sFDS
(2) Every discrete metric sFDC space can be isometrically embedded into a geodesic metric sFDC space.
(3) sFDS \(\Rightarrow\) property A (for metric spaces)
(4) Let \(Z\) be a metric space such that \(Z=X\cup Y\), where \(X\) and \(Y\) satisfy sFDC. Then \(Z\) also satisfies sFDC.
(5) A space \(X\) has the game theoretical asymptotic property C if and only if \({\text{ asdim}} X=0\).
In Section 5 of the paper the authors compare a game-theoretic approach with the standard in the classical dimension theory. They prove that:
(6) A compact metric space \(X\) has the game theoretical property C if and only if it is countable dimensional.
Editorial remark: In [Topology Appl. 227, 102–110 (2017; Zbl 1372.54026)], the authors fill a gap in the proof of Implication (3).
(1) Asymptotic property C \(\Rightarrow\) sFDS
(2) Every discrete metric sFDC space can be isometrically embedded into a geodesic metric sFDC space.
(3) sFDS \(\Rightarrow\) property A (for metric spaces)
(4) Let \(Z\) be a metric space such that \(Z=X\cup Y\), where \(X\) and \(Y\) satisfy sFDC. Then \(Z\) also satisfies sFDC.
(5) A space \(X\) has the game theoretical asymptotic property C if and only if \({\text{ asdim}} X=0\).
In Section 5 of the paper the authors compare a game-theoretic approach with the standard in the classical dimension theory. They prove that:
(6) A compact metric space \(X\) has the game theoretical property C if and only if it is countable dimensional.
Editorial remark: In [Topology Appl. 227, 102–110 (2017; Zbl 1372.54026)], the authors fill a gap in the proof of Implication (3).
Reviewer: Duanxu Dai (College Station)
MSC:
| 54F45 | Dimension theory in general topology |
| 53C23 | Global geometric and topological methods (à la Gromov); differential geometric analysis on metric spaces |
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