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 is an open subset of *X* and *O* is homeomorphic to for some 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.

**[1]**H. L. Baker, Jr.,*Complete amonotonic collections of subcontinua of a compact continuum*, Notices Amer. Math. Soc.**12**(1965), 91. Abstract #619-119.**[2]**-,*Concerning complete amonotonic collections of subcontinua of a compact continuum*, Notices Amer. Math. Soc.**12**(1965), 697. Abstract #626-24.**[3]**-,*Complete amonotonic decompositions of compact continua*, Proc. Amer. Math. Soc.**19**(1968), 847-853. MR**38**#678. MR**0232353 (38:678)****[4]**-,*Amonotonic decomposition of finite graphs*, Pacific J. Math. (submitted).**[5]**R. P. Dilworth,*A decomposition theorem for partially ordered sets*, Ann. of Math. (2)**51**(1950), 161-166. MR**11**, 309. MR**0032578 (11:309f)**

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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