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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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