Jerzy Dydak and his collaborators have published a very interesting series of papers recently, whose overall theme is that property A is the large scale analog of paracompactness. The papers that I have seen on the arXiv are these:
Cencelj, M., J. Dydak, and A. Vavpetič. “Asymptotic Dimension, Property A, and Lipschitz Maps.” Revista Matemática Complutense 26, no. 2 (July 1, 2013): 561–571. doi:10.1007/s13163-012-0102-2.
Cencelj, M., J. Dydak, and A. Vavpetič. Coarse Amenability Versus Paracompactness. ArXiv e-print, August 13, 2012. http://arxiv.org/abs/1208.2864.
Dydak, Jerzy. Coarse Amenability and Discreteness. ArXiv e-print, July 15, 2013. http://arxiv.org/abs/1307.3943.
Dydak, Jerzy, and Atish Mitra. “Large Scale Absolute Extensors.” arXiv:1304.5987 (April 22, 2013). http://arxiv.org/abs/1304.5987.
Recall that a Hausdorff topological space \(X\) is paracompact if every open cover of \(X\) has a refinement to a locally finite open cover. (It is metacompact, or weakly paracompact, if every open cover has a refinement to a pointwise finite open cover.) Most often though one applies paracompactness via the existence of partitions of unity: \(X\) is paracompact if and only if there exists a (locally finite) partition of unity subordinate to any open cover. The papers above elucidate what should be the “coarse version” of paracompactness both in the “covering” and the “partition of unity” interpretations, and in both cases they relate it to property A. Continue reading