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Tilings in topological spaces. (English) Zbl 0979.54014

Summary: A tiling of a topological space \(X\) is a covering of \(X\) by sets (called tiles) which are the closures of their pairwise-disjoint interiors. Tilings of \(\mathbb{R}^2\) have received considerabl attention (see [B. Grünbaum and G. C. Shepard, Tilings and patterns (1987; Zbl 0601.05001)] for a wealth of interesting examples and results as well as an extensive bibliography). On the other hand, the study of tilings of general topological spaces is just beginning [M. Breen, J. Geom. 21, 131-137 (1983; Zbl 0539.52014); M. J. Nielsen, Math. Ann. 284, No. 4, 601-616 (1989; Zbl 0662.52008); Geom. Dedicata 33, No. 1, 99-109 (1990; Zbl 0703.52009); Isr. J. Math. 81, No. 1-2, 129-143 (1993; Zbl 0780.52001)]. We give some generalizations for topological spaces of some results known for certain classes of tilings of topological vector spaces.

MSC:

54B99 Basic constructions in general topology
52C22 Tilings in \(n\) dimensions (aspects of discrete geometry)
46B20 Geometry and structure of normed linear spaces
54E52 Baire category, Baire spaces
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