Universality of the closure space of filters in the algebra of all subsets |
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Authors: | Andrzej W Jankowski |
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Institution: | (1) Institute of Mathematics, Warsaw University, Poland |
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Abstract: | In this paper we show that some standard topological constructions may be fruitfully used in the theory of closure spaces (see 5], 4]). These possibilities are exemplified by the classical theorem on the universality of the Alexandroff's cube for T
0-closure spaces. It turns out that the closure space of all filters in the lattice of all subsets forms a generalized Alexandroff's cube that is universal for T
0-closure spaces. By this theorem we obtain the following characterization of the consequence operator of the classical logic: If is a countable set and C: P( ) P( ) is a closure operator on X, then C satisfies the compactness theorem iff the closure space ![lang](/content/g2488553k5874083/xxlarge9001.gif) ,C is homeomorphically embeddable in the closure space of the consequence operator of the classical logic.We also prove that for every closure space X with a countable base such that the cardinality of X is not greater than 2 there exists a subset X of irrationals and a subset X of the Cantor's set such that X is both a continuous image of X and a continuous image of X .We assume the reader is familiar with notions in 5]. |
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