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Universality of the Closure Space of Filters in the Algebra of All Subsets

Andrzej W. Jankowski
Studia Logica: An International Journal for Symbolic Logic
Vol. 44, No. 1 (1985), pp. 1-9
Published by: Springer
Stable URL: http://www.jstor.org/stable/20015194
Page Count: 9
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Universality of the Closure Space of Filters in the Algebra of All Subsets
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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 T₀-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 T₀-closure spaces. By this theorem we obtain the following characterization of the consequence operator of the classical logic: If L is a countable set and C: $\scr{P(L)}\rightarrow \scr{P(L)}$ is a closure operator on X, then C satisfies the compactness theorem iff the closure space $\langle \scr{L,O}\rangle $ 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 $2^{\omega}$ there exists a subset $X^{\prime}$ of irrationals and a subset $X^{\prime \prime}$ of the Cantor's set such that X is both a continuous image of $X^{\prime}$ and a continuous image of $X^{\prime \prime}$. We assume the reader is familiar with notions in [5].

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