Maximal independent collections of closed sets
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- by Harvy Lee Baker PDF
- Proc. Amer. Math. Soc. 32 (1972), 605-610 Request permission
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 $\cup \{ o|o$ 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.References
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H. L. Baker, Jr., Complete amonotonic collections of subcontinua of a compact continuum, Notices Amer. Math. Soc. 12 (1965), 91. Abstract #619-119.
—, Concerning complete amonotonic collections of subcontinua of a compact continuum, Notices Amer. Math. Soc. 12 (1965), 697. Abstract #626-24.
- Harvy Lee Baker Jr., Complete amonotonic decompositions of compact continua, Proc. Amer. Math. Soc. 19 (1968), 847–853. MR 232353, DOI 10.1090/S0002-9939-1968-0232353-3 —, Amonotonic decomposition of finite graphs, Pacific J. Math. (submitted).
- R. P. Dilworth, A decomposition theorem for partially ordered sets, Ann. of Math. (2) 51 (1950), 161–166. MR 32578, DOI 10.2307/1969503
Additional Information
- © Copyright 1972 American Mathematical Society
- 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