The Common Structure For Objects In Aperiodic Order And The Theory Of Local Matching Topology

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In aperiodic order, non-periodic but "ordered" objects such as tilings, Delone sets, functions and measures are investigated. In this article we depict the common structure of these objects by using the general framework of abstract pattern spaces. In particular, using the common structure we define local matching topology and uniform structure for objects such as tilings in quite a general space and a symmetry group. We prove Hausdorff property of the topology and the completeness of the uniform structure under a mild assumption. We also prove finite local complexity implies the compactness of the continuous hull and often the converse holds.