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Transactions of the American Mathematical Society

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



Compact spaces and spaces of maximal complete subgraphs

Authors: Murray Bell and John Ginsburg
Journal: Trans. Amer. Math. Soc. 283 (1984), 329-338
MSC: Primary 54D30; Secondary 05C10, 06A10
MathSciNet review: 735426
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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$.

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Keywords: Compact space, graph, maximal complete subgraph, binary subbase, ordered set, maximal chain, one-point compactification
Article copyright: © Copyright 1984 American Mathematical Society