Abstract. 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 R 2 have received considerable attention (see  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 (see [1,3,4,6]). We give some generalizations for topological spaces of some results known for certain classes of tilings of topological vector spaces.
In this paper we introduce the concept of directed fractal structure, which is a generalization of the concept of fractal structure (introduced by the authors). We study the relation with transitive quasiuniformities and inverse limits of posets. We define the concept of GF-compactification and apply it to prove that the Stone-Cech compactification can be obtained as the GF-compactification of the directed fractal structure associated to the Pervin quasi-uniformity.
Abstract. In this paper we study those regular fenestrations (as defined by Kronheimer in ) that are obtained from a tiling of a topological space. Under weak conditions we obtain that the canonical grid is also the minimal grid associated to each tiling and we prove that it is a T0-Alexandroff semirregular trace space. We also present some examples illustrating how the properties of the grid depend on the properties of the tiling and we pose some questions. Finally we study the topological properties of the grid depending on the properties of the space and the tiling.2000 AMS Classification: Primary 54B15; Secondary 05B45.
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