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 [2] 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.The aim of this paper is to generalize the structure of a topological space, preserving its certain topological properties. The main idea is to consider the union and intersection of sets modulo "small" sets which are defined via ideals. Developing the concept of an i-topological space and studying structures with compatible ideals, we are concerned to clarify the necessary and sufficient conditions for a new space to be homeomorphic, in some certain sense, to a topological space.2000 AMS Classification: 54A05, 54E99.
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