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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 depictthe common structure of these objects by using the general framework ofabstract pattern spaces. In particular, using the common structure we definelocal matching topology and uniform structure for objects such as tilings inquite a general space and a symmetry group. We prove Hausdorff property of thetopology and the completeness of the uniform structure under a mild assumption.We also prove finite local complexity implies the compactness of the continuoushull and often the converse holds.

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