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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 

 

Maximal independent collections of closed sets


Author: Harvy Lee Baker
Journal: Proc. Amer. Math. Soc. 32 (1972), 605-610
MSC: Primary 54F05
DOI: https://doi.org/10.1090/S0002-9939-1972-0293578-5
MathSciNet review: 0293578
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Abstract: A theorem is proved which implies that if X is a separable metric space then there exists a countable maximal independent subset of the lattice of closed subsets of X. In the case where X has no isolated points this independent set is nontrivial in the sense that X does not belong to it and it contains no singletons. Furthermore, if X is a compact metric continuum such that $ \cup \{ o\vert o$ is an open subset of X and O is homeomorphic to $ {E^n}$ for some $ n > 1\} $ is dense in X then there exists a countable maximal such collection whose elements are connected. This complements previous work by the author which characterized continua for which there are such collections of a specialized nature.


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DOI: https://doi.org/10.1090/S0002-9939-1972-0293578-5
Keywords: Independent subset of a partially ordered set, amonotonic collection of sets, complete amonotonic decomposition of a continuum, pseudodevelopment of a space
Article copyright: © Copyright 1972 American Mathematical Society