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Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.48.

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Compact spaces and spaces of maximal complete subgraphs
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by Murray Bell and John Ginsburg PDF
Trans. Amer. Math. Soc. 283 (1984), 329-338 Request permission

Abstract:

We consider the space $M(G)$ of all maximal complete subgraphs of a graph $G$ and, in particular, the space $M(P)$ of all maximal chains of an ordered set $P$. The main question considered is the following: Which compact spaces can be represented as $M(G)$ for some graph $G$ or as $M(P)$ for some ordered set $P$? The former are characterized as spaces which have a binary subbase for the closed sets which consists of clopen sets. We give an example to show that this does not include all zero-dimensional supercompact spaces. The following negative result is obtained concerning ordered sets: Let $D$ be an uncountable discrete space and let $\alpha D$ denote the one-point compactification of $D$. Then there is no ordered set $P$ such that $M(P) \simeq \alpha D$.
References
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  • P. C. Gilmore and A. J. Hoffman, A characterization of comparability graphs and of interval graphs, Canadian J. Math. 16 (1964), 539–548. MR 175811, DOI 10.4153/CJM-1964-055-5
  • J. Ginsburg, I. Rival and W. Sands, Antichains and finite sets that meet all maximal chains (to appear).
  • I. Juhász, Cardinal functions in topology, Mathematical Centre Tracts, No. 34, Mathematisch Centrum, Amsterdam, 1971. In collaboration with A. Verbeek and N. S. Kroonenberg. MR 0340021
  • J. van Mill, Supercompactness and Wallman spaces, Mathematical Centre Tracts, No. 85, Mathematisch Centrum, Amsterdam, 1977. MR 0464160
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Additional Information
  • © Copyright 1984 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 283 (1984), 329-338
  • MSC: Primary 54D30; Secondary 05C10, 06A10
  • DOI: https://doi.org/10.1090/S0002-9947-1984-0735426-5
  • MathSciNet review: 735426